Quantum Mechanics: For Scientists and Engineers [1 ed.] 9781032117645, 9781003221432

Citation preview

QUANTUM MECHANICS FOR SCIENTISTS AND ENGINEERS

QUANTUM MECHANICS FOR SCIENTISTS AND ENGINEERS

Harish Parthasarathy Professor Electronics & Communication Engineering Netaji Subhas Institute of Technology (NSIT) New Delhi, Delhi-110078

First published 2022 by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 © 2022 Manakin Press Pvt. Ltd. CRC Press is an imprint of Informa UK Limited The right of Harish Parthasarathy to be identified as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Print edition not for sale in South Asia (India, Sri Lanka, Nepal, Bangladesh, Pakistan or Bhutan). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record has been requested ISBN: 978-1-032-11764-5 (hbk) ISBN: 978-1-003-22143-2 (ebk) DOI: 10.1201/9781003221432

Preface

This book covers the entire span of modern quantum mechanics including quantum field theory and more recent developments in quantum stochastics. The book starts with the Bohr-Sommerfeld correspondence principle which was the first model developed by Bohr to explain Rutherford's picture of the atom with electrons revolving around the nucleus without losing energy in the form of electromagnetic radiation (produced by accelerating particles according to the classical Ma.xwell theory) which would otherwise cause the energy of the electron to continuously decrease and collapse to the nucleus leading to the instability of all matter. Bohr explained this by postulating that in a stationary state the electron does not lose or gain energy and further that the angular momentum of the electron which is the line integral of the momentum around one closed orbit is quantized in integral multiples of h/21f where h is Planck's constant. Sommerfeld's generalization of this principle of Bohr to arbitrary action-angle varables is mentioned here so that the line integral of the action variable with respect to the angle variable over one complete circuit is an integral multiple of h/21r. The book then continues with the evolution of the Schrodinger equation from the Planck and De-Broglie hypothesis then proceeds to the matrix mechanics of Heisenberg and its equivalence with the Schrodinger wave mechanics along the lines outlined by Dirac in his celebrated book "The principles of quantum mechanics". In the process, we also discuss the interaction picture of Dirac which is something midway between the Schrodinger and the Heisenberg pictures. MIIJ{ Born's intepretation of the wave function in terms of probabilities is also discussed. The superposition principle which is at the heart of quantum mechanics is cliBcussed with illustrations from the Young double slit experiments for interference of photons and likewise, the Stern-Gerlach experiments that demonstrate the fact that electrons can sometimes behave like waves thus confirming De-Broglie's wave-particle duality. We then discuss the effect of an electromagnetic field on an atom using time dependent perturbation theory and also the Zeeman effect which predicts the splitting of the degenerate levels of an atom in a. constant magnetic field caused by the interaction of spin with the magnetic field. This leads us to the Pauli equation which is the Schrodinger equation for a two component wave function with spin-magnetic field interaction accounted for. A thorough discussion of a time independent quantum system perturbed by time varying potential using the Dyson series is presented which is later on used in the book to calculate scattering amplitudes for electrons, positrons and photons upto any order in perturbation theory. Time independent perturbation theory is also introduced as a tool for calculating approximately the shift in the energy levels and the energy eigenstates of a quantum system when perturbed by a potential. The problem of perturbation of degenerate states is also taken up by showing that if the unperturbed energy E is degenerate with a degeneracy of d, then on perturbation, it will split into d energy levels and whose eigensta.tes and energy eigenvalues are determined by the eigenvalues and eigenvectors of the "secular matrix" obtained by considering the matrix elements of the perturbing potential with respect to an orthonormal basis of the d dimensional vector space of all eigenfunctions of the unperturbed Hamiltonian having energy E. Using the Ca.uchySchwarz inequality, we give Weyl's proof of the Heisenberg uncertainty principle in its most general form. Three exactly solvable problems in quantum mechanics are presented in full detail, namely derivation the the energy eigenfunctions and energy eigenvalues of the Hamiltonian corresponding to (a.) a particle in a box, (b) the harmonic oscillator and (3) the non-relativistic Hydrdogen atom. The last two are based on solving ordinary differential equations using the power series method and for the Hydrogen atom case, the full derivation of the spherical harmonics as eigenfunctions of the angular momentum square operator is presented in full detail. Later on, we show how the spherical harmonics determined all the irreducible reprsenta.tions of the rotation group 80(3). This derivation gives an alternate method to obtain all the irreducible representations of the rotation group in comparison to the method of Wigner and Weyl who constructed the irreducible representations of SU(2) using polynomials in two complex variables. The book contains some major results in group representation theory like the Frobenius- Mackey's theory for constructing irreducible representations of the semidirect product of a normal Abelian group N with another subgroup H. This theory was applied by Wigner to construct the irreducible representations of the Poincare group which is generated by Lorentz transformations and spa.c&time translations. It is also known as the little group method where one starts with a. character of the Abelian group N, constructs its orbit under the dual action of H, then constructs the stability subgroup of H for which the chosen character is a fixed point, and then using an irreducible representation of this little group Ho and the chosen character xo, one constructs an irreducible representation of the sernidirect product Go = N ® 8 Ho of Nand H 0 and by the induced representation theory, the representation of G = N x •H is constructed using the Mackey method. Irreduciblity of such induced representations is proved and easier methods of describing such induced representations using functions on the character orbit with values in the vector space on which the irreducible representation of the little group acts are provided. The reader is referred to the excellent book by K.R.Parthasa.rathy on this subject (KRP:Mathematical Foundations of

Quantum Mechanics for Scientists and Engineers Preface

vi ii

Quantum Mechanics, Hindustan Book Agency). After the discussion of non-relativistic quantum mechanics using the Schrodinger method, we describe Feynman’s path integral approach to non-relativisti quantum mechanics. Feynman in his Ph.D thesis showed that solution to the Schrodinger equation for the evolving wave function under a Hamiltonian can equivalently be represented as a path integral of exp(2πiS[q]/h) over all paths q joining (t1 , q1 ) and (t2 , q2 ) where S[q] is t the action of the path t → q(t) with the prescribed end points. This action is obtained as the integral t12 L(t, q(t), q � (t))dt of the Lagrangian L of the particle or system of particles with the Lagrangian related to the Hamiltonian used in the Schrodinger equation via the Legendre transformation. Thus, Feynman’s approach is that along each path q there is a complex amplitude exp(2πiS[q]/h) for the particle/system of particles to go from (t1 , q1 ) to (t2 , q2 ) and the total probability amplitude to go from the initial point to the second in the given time is obtained by summing these amplitudes over all paths. This is an intuitively beautiful picture of presenting quantum mechanics because it displays the quantum interference along different paths and in addition, it clearly shows that in the limit when Planck’s constant converges to zero, the rapid oscillations of the phase 2πS[q]/h for small variations the path q cause cancellations of the probability amplitudes except for paths in the vicinity of the classical path q∗ where the variation of the action vanishes, ie where δS[q∗ ] = 0. A path q∗ satisfing this condition satisfied the classical Euler-Lagrange equation of classical mechanics which is equivalent to Newtonian classical mechanics. The mathematical problems with Feynman’s approach have till today defied mathematicians since the constant of proportionality in√the FPI is infinite and in addition, the path integral is 2 something like √ the integral of exp(−ix /2) over R where i = −1 and Feynman’s idea is equivalent to replacing this integral by 2πi. It gives the correct results predicted by Schrodinger’s equation but why it gives is not clear since the integral of exp(−ix2 /2) is oscillatory and hence does not converge. The Feynman path integral is the same as Kac t formula for u(t, x) = E[exp( 0 V (x + B(s))ds)f (x + B(t))] where B(.) is Brownian motion but with t replaced by it, ie ”complex time”. More general forms of Kac formula can be derived when B(.) is an arbitrary diffusion process or even any Markov process with a generator K. These ideas have been discussed in this book. After all these discussions, we come to relativistic quantum mechanics whose wave equation was first correctly proposed by Dirac. Before Dirac, we had the Klein-Gordon equation based on replacing the kinetic energy K by i∂/∂t + eV (t, r) and the mometum P by −i∇ + eA(t, r) in the Einstein energy-momentum relation K 2 − c2 P 2 − m2 c4 = 0 and treating this as an operator acting on a wave function ψ(t, r) set to zero. Earlier, Sommerfeld had solved the corresponding stationary state equation for the Hydrogen atom obtaining thereby relativistic corrections to the −1/n2 spectrum of Schrodinger. However, Dirac observed two major flaws with this equation. One, an evolution equation of quantum mechanics has to be first order in time, since if the evolution operator is T (t), then T (t+s) = T (t).T (s), t, s ≥ 0 and hence T (t) = exp(−itH) for some selfadjoint operator H. Which is the same as saying that T � (t) = −iHT (t). Of course, we also have T �� (t) = −H 2 T (t) but that would lead to a superpostion of two solutions exp(±itH) and unitarity of the evolution would be broken. One way to rectify this is to consider the wave operator as  K − c P 2 + m2 c2

where K = i∂/∂t + eV (t, r) and P replaced by −i∇ + eA(t, r) but the resulting equation could not be solved for any potential due to ∇2 occuring inside a square root. The other major problem with such an equation is that it is highly asymmetric in the space and time indices whereas special relativity requires that space and time should be treated on the same footing. Thus, Dirac proposed the factorization of the quadratic form E 2 −c2 (Px2 +Py2 +Pz2 )−m2 c4 into linear factors by the introducing of non-commuting 4 × 4 matrices known today as the famous Dirac γ matrices. The Dirac equation was solved by Dirac for the Coulomb potential thus obtaining negative as well as positive energy states, the positive energy states appear in a continuum and Dirac interpreted this as corresponding to a new kind of particle the positron having the same mass but opposite charge as that of an electron. Dirac proved that his relativistic wave equation for the electron is invariant under a representation of the Lorentz group, namely the ”Dirac representation” and he also proved that under conjugation followed by application of an appropriate unitary matrix, the resulting wave function also satisfies the Dirac wave equation but with the charge −e replaced by +e and no change in mass. This phenomenon is called charge conjugation and led Dirac to immediately conclude that this charge conjugated wave equation describes the motion of a positron. The discovery of the Positron by Dirac and its subsequent detection in an accelerator by Anderson came as a total surprise to the physics community since it led to the concept of antimatter, ie every elementary particle should have a corresponding antiparticle and that when a particle interacts with its antiparticle then the two may get annihilated to produce just a γ ray photon. The probabilities of such processes and other scattering processes where later on calculated systematically by Feynman in the process of developing quantum electrodynamics. This discovery of antimatter by Dirac must definitely rank as one of the greatest achievements in physics since the time of Newton. The discovery of the Dirac equation also led Dirac to conclude that the spin of the electron is a relativistic effect and he demonstrated that by replacing the momentum operator P by P + eA and the energy E by E + eV where A is the magnetic vector potential

Quantum Mechanics for Scientists and Engineers Preface

vii iii

and V is the electric potential, and premultiplying the resulting wave equation by its ”reversed version” , one obtains the original Klein-Gordon equation of the particle in an electromagnetic field plus terms that describe the interaction of the electric and magnetic field with the spin of the electron. Earlier, Pauli had artificially introduced spin terms into the non-relativistic Schrodinger equation to demonstrate the Zeeman splitting of the energy levels but Dirac’s approach led naturally to terms involving the interaction of the electronic spin with the external electromagnetic field. Moreover, for the development of relativistic quantum field theory, ie a quantum field theory that respects Lorentz invariance, it is the Dirac and Maxwell equations that must be used rather than the Schrodinger and Maxwell equations. In quantum field theory, wave functions become operator fields satisfying either commutation or anticommutation relations depending on whether the wave functions describe Bosons or Fermions. The process of replacing wave functions by operator valued fields is called second quantization and from a mathematical viewpoint, this can be achieved by considering multiple tensor products of Hilbert spaces and operators acting on such spaces. If the tensor product is the usual one, the operator fields describe distinguishable particles satisfying the Maxwell-Boltzmann statistics, if the tensor products are symmetrized, then they describe Bosons which are indistinguishable particles having integer spins and that any state can be filled by any number of such particles while if the tensor products are antisymmetrized, then they describe Fermions which are indistinguishable particles having half integral spins and any state can have only either no particle or one particle also known as the Pauli exlcusion principle. In the quantum field theory, for any kind of particle, we can have operators that create, annihilate or conserve the number of particles at any point in space-time or at any value of the four momentum. This leads us to a calculus of such operators and using the Pauli exclusion principle for Fermions, we can show that the Fermionic creation and annihilation operator fields satisfy canonical anticommutation relations while using the analogy of Bosons with the Harmonic oscillator algebra of creation and annihilation operators, we can show that the Bosonic creation and annihilation operator fields satisfy canonical commutation relations. The relation between spin and statistics is also one of the most striking features of quantum mechanics, namely that if we maximize the entropy of a system of particles subject to distinguishability of the particles and no restriction on the number of particles in a state, we get the Maxwell-Boltzmann distribution while if we maximize the entropy subject to indistinguishability and Pauli exclusion, we get the Fermi-Dirac statistics for Fermions and finally if we maximize the entropy subject to indistinguishability no Pauli exclusion, we get the Bose-Einstein statistics. The last of these, was first discovered by the great Indian physicist Satyendranath Bose who derived this statistics and showed that Planck’s law of black-body radiation for photons can be obtained from such statistics. Einstein communicated Bose’s paper to the most prestigious physics journal at that time ”Annalen-der-physik” after translating it into German and adding footnotes explaining the significance of Bose’s work at that time when only Planck’s law of black-body radiation was avaiable in the quantum theory and further that no rigorous derivation of this law from physical and mathematical principles was available. Our next theme in this book is the development of quantum electrodynamics. This involves first writing down the energy of the electromagnetic field in terms of the electric potential and magnetic vector potential in the frequency domain and then observing that for such a free electromagnetic field, the vector potential satisfies the wave equation which in the spatial frequency domain, is simply a sequence of second order in time ordinary differential equations for harmonic oscillators whose characteristic frequency is simply c|k| where k is the wave vector. The Lagrangian of the field is simply a quadratic functional of A(t, k) and ∂A(t, k)/∂t and and their complex conjugates and by applying the Euler-Lagrange equations to this Lagrangian, we get the same equation as mentioned above, namely a sequence of harmonic oscillator ode’s. We are thus led to conclude that the electromagnetic field is simply a continuous ensemble of harmonic oscillators. Application of the Legendre transformation to this Lagrangian gives us the Hamiltonian density of the field as a sum of position field square and momentum field square where the position field is the magnetic vector potential A(t, r). By postulating the canonical commutation relations between the position and momentum fields in the form [Aμ (t, r), πν (t, r� )] = iδνμ δ 3 (r − 0r� ) we are led into trouble since the form of the Lagrangian density of the electromagnetic field Fμν F μν implies that certain constraints on the position and momentum fields like for example since the Lagrangian does not depend on A0,0 ,the corresponding momentum π0 = 0 which is inconsistent with the canonical commutation relations. Further, by writing down the Euler-Lagrange equation of motion with a matter-field interaction Lagragian denstiy Jμ Aμ , we are lead to a relationship between the spatial momenta πr , r = 1, 2, 3 and and J μ involving only spatial derivatives of πr . It should be noted that the current field J μ is a matter field described in terms of the Dirac operator wave function: J μ = −eψ ∗ (t, r)γ 0 γ μ ψ(t, r) These constraints are therefore either primary ie, arising from the the structure of the Lagrangian or secondary, ie, arising from the Euler-Lagrange equations of motion. These constraints are incompatible with the canonical commutation relations. To rectify this situation, Dirac proposed a new type of bracket expressed in terms of the Poisson/Lie bracket and taking the constraints into account. He showed that this new bracket, known today as the Dirac bracket is conststent with the constraints and also yields the correct equations of motion for the unconstrained observables. Using

Quantum Mechanics for Scientists and Engineers Preface

viii iv

this he developed the first operator theoretic version of quantum electrodynamics. Feynman then applied his path integral formalism to fields like the electromagnetic four potential and the Dirac four component wave function and showed how one can derive from the path integral formalism, perturbative corrections to the electron and photon propagators arising from interaction between the matter field in the form of Dirac currents and the electromagnetic four potential. He gave a beautiful diagrammatic method for calculating the scattering matrix elements to any order in perturbation theory. The resulting formulae can directly be derived by considering an initial state |i > of photons, electrons and positrons having definite four momenta and spin/polarizations and likewise a final state |f >. These initial and final states can be obtained by acting on the vacuum appropriate creation operators, then writing the interaction terms in the total Hamiltonian between photon field and electron current as integrals expanding the time ∞ ordered exponential U (∞, −i∞) = T {exp(−i −∞ H(t� )dt� )} in the interaction picture as a power series in the interaction  energy − J μ (x)Aμ (x)d3 x where J μ (x) = −eψ ∗ (x)γ 0 γ μ ψ(x), then expressing ψ(x) and Aμ (x) in terms of creation and annihilation operators of the electron-positron and photon field and then using the standard commutation relations, evaluate the scattering matrix elements < f |U (∞, − inf ty)|i > in the interaction picture. This operator formalism was developed by Schwinger-Tomonaga and Dyson while Feynman developed the path integral approach to such calculations with finally Dyson proving the equivalence of the theories of Feynman and Schwigner-Tomonaga. The importance of the propagator computation in quantum field theory has been highlighted in this book. Namely, if φ(x) is a collection of fields and the action for such a collection of fields is expressed as S[φ] = SQ [φ] + �SN Q [φ] where SQ is a quadratic functional and SN Q is a cubic and higher degree functional, the latter coming from interactions, then the scattering matrix using FPI can be expressed as  S[φ∞ , φ−∞ ] = exp(iS[φ])Dφ =



ex(iSQ [φ])(1 + i�SN Q [φ] − �2 SN Q [φ]2 /2 + ...)Dφ

It is clear that each term in this infinite series is equivalent to calculating the moments of an infinite dimensional Gaussian distribution with complex variance and we know from basic probability theory, that even higher order moments of a Gaussian vector can be expressed as sums over products of the second order moments. The second order moment appearing here is the propagator of the field:  Dφ [x, y] = exp(iSQ [φ])φ(x)φ(y)Dφ These aspects of quantum electrodynamics have been covered in this book. We give explicit computations of the photon propagator in different gauges (Feynman, Landau and Coulomb gauges) and also for the electron propagator. These expressions are derived using both the methods, first the operator theoretic method and second, using the Feynman path integral for quadratic Lagrangians combined with the standard formulas for the second order moments of a multivariate Gaussian distribution. We also present modern developments in quantum field theory, starting with the standard YangMills group theoretic generalization of the Dirac or Klein-Gordon equations for matter fields interacting with the photon field. In this generalization, we assume that the wave function takes values in C4 ⊗ CN and a subgroup G of the unitary group U (N ) acts on this wave function. We introduce a Lie algebra theoretic covariant derivative which is an ordinary four gradient plus a connection field which takes values in the Lie algebra of the group G. This gauge covariant derivative acts on the matter field wave function and if the matter field wave function undergoes a local (ie space-time dependent) transformation g(x) ∈ G, then accordingly the gauge connection field in the covariant derivative has to undergo a certain transformation so that the covariant derivative of the matter field transforms simply by multiplication by g(x). This ensures that the Lagrangian density constructed out of functions of the matter field and its covariant derivative will be invariant under local G-transformations of the matter field and the gauge field provided that the Lagrangian is a G-invariant function of its agruments. Owing to the non-commutativity of the gauge fields, the gauge field tensor defined as the commutator of the covariant derivatives contains an extra quadratic nonlinear term in the gauge potentials apart from its four dimensional curl. This is a characteristic feature of non-Abelian gauge theories in contrast to the case of the electromagnetic field where the gauge group is U (1). In the electromagnetic field case, the matter field is the Dirac field since the gauge group is U (1), the local group transformation element g(x) is simply a modulus one complex number depending on the space-time variables. The corresponding gauge transformation of the gauge field potentials, namely, the electromagnetic four potential simplifies to the standard Lorentz gauge transformation Aμ → Aμ + ∂μ φ. Based on the Yang-Mills non-Abelian generalization of electromagnetism, Salam,Weinberg and Glashow were able to unify the electromagnetic forces and the weak forces, calling it the electro-weak theory. Their idea was to start with a

Quantum Mechanics for Scientists and Engineers Preface

ix v

non-massive vector gauge field Lagrangian corresponding to the weak forces in the nucleus, coupled to the matter field of Leptons (which include electrons) and then add symmetry breaking terms to this Lagrangian by coupling a scalar field to the Leptons. Symmetry breaking occurs when the scalar field is in its ground state which is a constant and this leads to mass terms involving quadratic non-derivative terms of the electronic field as well as to the gauge Bosonic fields which are components of the vector gauge field. In this context, we give a brief review of symmetry breaking both spontaneous and local versions. Symmetry breaking can occur when Hamiltonian/Lagrangian is invariant under a group G and the ground state is degenerate. If the Hamiltonian is invariant under G, then the subspace of ground states is invariant under G but any given ground state may not be invariant under G. If we take such a ground state and look upon the quantum state as this ground state plus a quantum perturbation, then the resulting Lagrangian will not be invariant under G since the ground state is not, but will be invariant under a subgroup H of G also called a broken subgroup. This sort of symmetry breaking produces massless particles called Goldstone Bosons. We can demonstrate this fact that symmetry breaking produces massless particles much better and in a more generalized framework using groupt theory. To do so, we first express the wave function of the system in terms of a unbroken H part and a broken G/H part. The unbroken part transforms according to H and the broken part can be viewed as a field with values in the coset space G/H. When the Lagrangian is expressed in terms of these components, it turns out that it does not contain any non-derivative quadratic components of the broken part while it contains non-derivative quadratic components of the unbroken part. This demonstrates that the broken part describes massless Goldstone Bosons while the unbroken part of the wave function describes massive particles. We also show that the G-symmetry of a Lagrangian can be broken by adding perturbative terms that are not G-invariant. Sometimes symmetry breaking can lead to massless particles being massive as in the electroweak theory of Salam, Weinberg and Glashow. This happens because of the coupling of the gauge fields to a scalar field. The gauge fields initially are massless but the coupling to the scalar field followed by evaluation of the Lagrangian in the ground state of the scalar field causes terms in the Lagrangian involving quadratic non-derivative terms of the gauge field to be present. These terms cause the gauge fields to become massive. The electroweak theory is an example of such a situation which gives masses to the gauge fields other than the electromagnetic field. All this is quantum field theory from the physical standpoint. We next explain how certain aspects of quantum field theory including the theory of quantum noise can be developed using rigorous mathematics, ie, functional analysis. In particular, we show how certain major stochastic processes in classical probability theory like the Brownian motion and Poisson processes are special cases of quantum stochastic processes, ie, a family of non-commuting observables in a special kind of Hilbert space, namely the Boson Fock space when viewed in specific states. The notion of a quantum probability space as a triplet (H, P (H), ρ) where H is a Hilbert space, P (H) is the lattice of orthogonal projection operators in H and ρ is a state in H, ie, a positive semidefinite unit trace operator in H is introduced as compared to a classical probability space (Ω, F, P ). After doing so, we construct the Boson Fock space which can describe an arbitrary number of Bosons using the symmetric tensor product of a particular Hilbert space. The Boson Fock space is the substratum for constructing basic quantum noise processes, ie, noncommuting family of operators which specialize to Brownian motion and Poisson processes when the state is appropriately chosen. The Boson Fock space or noise Bath space is coupled to the system Hilbert space via a tensor product. Then we follow the marvellous approach of R.L.Hudson and K.R.Parthasarathy of construting the creation, annihilation and conservation operator fields in the Boson Fock space. We show via physical arguments that the creation and annihilation operator fields can also be viewed fromt the standpoint of an infinite sequence of Harmonic oscillators, by constructing the creation and annihilation operator for each oscillator and defining the coherent vectors in terms of a superposion of the energy eigenstates of these oscillators and proving that the coherent states are eigenstates of the annihilation operator field now constructed as superpostions of the annihilation operators for the different oscillators. The creation operator is the adjoint of the annihilation operator and turns out to have the same action on coherent vectors as the complex derivative of the latter with respect to the complex numbers used to construct the coherent vector from the oscillator energy eigenstates. In the work of Hudson and Parthasarathy, coherent vectors in the Boson Fock space were constructed as a weighted direct sum of multiple tensor products of a fixed vector in the Hilbert space with itself and the creation, conservation and annihilation operators were defined using the generators of one parameter unitary groups derived from the Weyl operator by restricting to translation and unitary rotation. The Weyl operator in the Hudson-Parthasrathy (HP) theory is itself described by its action on coherent vectors. Coherent vectors without normalization in the HP theory were called exponential vectors. This approach to the construction of the basic noise fields is highly mathematical and we provide in this book some physical insight into this correspondence by making an isomorphism between the coherent vectors of the HP theory and coherent vectors constructed using eignstates of harmonic oscillators. Further, in our book, we construct the unitary rotation operators needed for constructing the conservation process by resorting to quadratic forms of the creation and annihilation operators of the harmonic oscillators. The theory of quantum stochastic calculus developed by Hudson and Parthasarathy introduces time dependent creation, annihilation and conservation processes which statisfy a quantum Ito formula for products of time differentials of these processes. The classical Ito formula for Brownian motion and the Poisson process is shown to arise as a special commutative version of the quantum Ito formula of HP. The HP theory says

x vi

Quantum Mechanics for Scientists and Engineers Preface

more, namely that in the general noncommutative case, the products of creation and annihilation operator differentials with conservation differentials need not be zero. This unlike the classical probability case where if B is Brownian motion and N is Poisson process, then dB.dN = 0. Our analogy of the HP theory with the quantum harmonic oscillator theory provides us with a direct route to quantum optics as described in the celebrated book by Mandel and Wolf on optical coherence and quantum optics. The idea is to first set up the famous Glauber-Sudarshan non-orthogonal resolution of the identity operator in the Boson Fock space as a complex integral of the coherent states |e(u) >< e(u)|, then to express the Hamiltonian of the system interacting with a photon bath as the sum of the system Hamiltonian which consists of spin matrices interacting with a constant classical magnetic field, the bath photon field Hamiltonian expressed as a quadratic form in the creation and annihilation operators and an interaction Hamiltonian expressed as a time varying linear combination of the products(tensor) between the atomic spin observables and the bath annihilation and creation variables, then assume that the density of ρ(t) of the system and bath can be expanded as a Glauber-Sudarshan integral:  ρ(t) = ρA (t, u) ⊗ |e(u) >< e(u)|du where ρA (t, u) is a finite dimensional matrix (of the same order as the spin observables of the atomic system). Finally, we substitute this expansion into the quantum equation of motion iρ� (t) = [HA + HF + HAF (t), ρ(t)] where HA is the atomic Hamiltonian, HF is the bath field Hamiltonian and HAF (t) is the atomic-bath field interaction Hamiltonian. Using properties of creation and annihilation operators acting on the exponential/coherent vectors and integration by parts, we then derive a partial differential equation for ρA (t, u) which may be called the fundamental equation of quantum optics. After these discussions, we proceed to one of the most modern techniques in time dependent quantum measurement theory. The time dependent HP theory, ie quantum stochastic calculus also has a nice physical interpretation. When the average values of the creation, and annihilation processes in a coherent state are calculated, we get time integrals of the product of the coherent state defining vector with the creation/annihilation process defining vector upto time t. This result can be interpreted physically by saying that the annihilation (creation) process acts on a coherent state and yields the total amplitude and phase of photons annihilated (created) upto time t taking into account the relative polarization of the photons in the coherent state with respect to that of the annihilation (creation) process defining vector. On the other hand, the conservation process average in a coherent state gives a time integral of a quadratic product of the coherent state defining vector upto time t which has the interpretation as being the number of photons in the state present upto time t. The conservation process in the HP calculus has the Poisson process interpretation in classical probability theory just as the creation and annihilation process are linked to Brownian motion. The whole subject of quantum probability and quantum stochastic processes can be viewed as an linear-algebrafunctional analytic approach to probability theory with generalizations to the non-commutative case. As an important application of the HP stochastic calculus, we present the celebrated work of V.P.Belavkin on quantum filtering. To do so, we first note that in quantum mechanics two or more observables may not be simultaneously measurable when they do not commute but when they commute, they can be simultaneously diagonalized and hence simultaneously measured. Measurement on a quantum system causes the state of the system to collapse to a state dictated by the outcome of the measurement or of the measurement is made by a set of projection valued operators (pvm) or more generallyn by a set of positive operators (povm), then the state of the system collapses to a state formed by superposing the collapsed states corresponding to each measurement outcome. Belavkin proposed a scheme of constructing a filtration on an Abelian Von-Neumann algebra of observables that satisfies the non-demolition property, ie, the algebra generated by the elements of the filtration upto time t is Abelian and also commutes with the states of the HP noisy Schrodinger equation at times s ≥ t. Such measurements, he called non-demolition measurements. The HP noisy Schrodinger equation determines a unitary evolution in the joint system and bath space h ⊗ Γs (L2 (R+ ) ⊗ Cd ). The dynamics of the unitary evolution is dictated by the system Hamiltonian, the fundamental creation, annihilation and conservation processes of the HP quantum stochastic calculus which are operators in the bath space and connect to the system dynamics via system operators. There is also a quantum Ito correction term to the system Hamiltonian in the form of an additive skew Hermitian operator that ensures unitarity of the evolution. It is known that if one computes the Heisenberg dynamics of a system observable using these unitary evolution operators, then the standard Heisenberg equations of motion are obtained along with noise correction terms which and the system observables after finite time t evolves to an observable in the tensor product of the system Hilbert space and the noise bath space. Specifically, if U (t) denotes the evolution operator of the HP noisy Schrodinger equation and X is a system observable, then after time t it evolves to jt (X) = U (t)∗ XU (t) which satisfies the noisy Heisenberg equations of motio. Belavkin proposed that if we take an input noise process Yin (t) which is a commuting family of operators in the bath space and define the output noise process Yout (t) = U (t)∗ Yin (t)U (t), then by virtue of the fact that the unitarity of U (t) depends only on system operators which commute with bath operators, it follows that Yout (t) = U (T )∗ Yin (t)U (T ) for all T ≥ t from

Quantum Mechanics for Scientists and Engineers Preface

xi vii

which it is easy to see that Yout (t) commutes with Yout (s) for all t, s ≥ 0 and further that Yout (t) commutes with js (X) for all s ≥ t. This commutativity enables us to jointly measure jt (X), Yout (s), s ≤ t and hence define the conditional expectation πt (X) = E(jt (X)|Yout (s), s ≤ t). The family πt (X), t ≥ 0 of operators forms an Abelian family since πt (X) is a function of Yout (s), s ≤ t and the latter form a commuting family of operators. Belavkin derived a stochstic differential equation for πt (X) driven by the process Yout (.) and its derivation was greatly simplified using quantum versions of the Kallianpur-Striebel formula and other methods based on orthogonality properties of the conditional expectation estimate by John Gough and Kostler. The resulting equation for πt (X) is the fundamental quantum filtering equation and is the non-commutative generalization of Kushner’s equation for classical non-linear filtering. These aspects have been discussed in this book. It should be noted that although this version of the filtering equations takes place in the observable domain, it could be directly transformed to the density domain. This is analogous to the situation of classical filtering where we can describe the evolution of the conditional moments of the state or more generally of the conditional expectation of any function of the state at time t given measurements upto time t or equivalently of the conditional probability density of the state at time t given measurements upto time t. To obtain the Belavkin stochastic differential equation for the conditoinal density of the state at time t given output measurements upto time t, we must simply note that we can write πt (X) = T r(ρt X) where ρt can be viewed as a density matrix in the system Hilbert space that is a function of the output measurement process upto time t, or equivalently, simply as a classical random process with values in the space of system space density operators. Classical random because the measurement operators commute. We substitute πt (X) = T r(ρt X) into the Belavkin observable version of the filtering equation and using the arbitrariness of X, we derive a classical stochastic differential equation for the system state valued classical random process ρt driven by Yout (t). Such equations are called ”Stochastic Schrodinger equations” (Luc Bouten, Ph.D thesis on quantum optics filtering and control). We present a generalization of Belavkin’s work by first constructing a family of p commuting inputmeasurement processes which are expressible as linear combinations of the creation, annihilation and conservation processes. Such processes have independent increments in coherent states and the quantum Ito’s formula leads in general to the fact that any integer power of the differentials of such processes is non zero just as in the classical case (dN )k = dN, k = 1, 2, ... where N (.) is a Poisson process. The Belavkin filter is constructed by assuming it to have the form  Gmkt (X)(dYmout (t))k dπt (X) = Ft (X)dt + k≥1,m=1,2,...,p

with Ft (X)andGmkt (X) being measurable with respect to the algebra generated by the output measurments upto time t. The coefficients Ft (X), Gmkt (X) are determined by applying the quantum Ito formula to the orthogonality equation E((jt (X) − πt (X))Ct ) = 0

where

dCt =

 m,k

fmk (t)Ct (dYmout (t))k , t ≥ 0, C0 = 1

and using arbitrariness of the complex valued functions fmk (t). It should be noted that in the absence of any measurement, ie, when the system evolves according to the HP noisy Schrodinger equation, we can take a system observable X, obtain its noisy evolution jt (X) and then choose a pure states of the form |fk ⊗ φ(u) >, k = 1, 2 of the system and the bath where |fk > is a system state and |φ(u) > is a bath coherent state, then compute the matrix elements < f1 ⊗ φ(u)|jt (X)|f2 ⊗ φ(u) > and write down the evolution of this quantity. More generally, we can compute the system state at time t defined via the duality equation T r(ρs (t)X) = T r((ρs (0) ⊗ |φ(u) < φ(u)|)jt (X)) or equivalently, as or equivalently, as

ρs (t) = T rB (U (t)(ρs (0) ⊗ |φ(u) >< φ(u)|)U (t)∗ ) < f1 |ρs (t)|f2 >= T r(ρs (t)|f2 >< f1 |) =

T r((ρs (0) ⊗ |φ(u) >< φ(u)|)jt (|f2 >< f1 |))

where |fk >, k = 1, 2 are system states. The resulting differential equation for the system state ρs (t) is the generalized GKSL equation (Gorini, Kossakowski, Sudarshan, Lindblad). If u = 0, ie, the bath is in vaccuum, then the ordinary GKSL equation is obtained. Using the dual of the GKSL equation, we get a description of the evolution of system observables when corrupted by bath noise in such a way that the system observable always remains a system observable, ie, averaging out over the bath noise variables is being performed at each time. The dual GKSL can be used to describe non-Hamiltonian quantum dissipative systems, for example, damped quantum harmonic oscillators or lossy

Quantum Mechanics for Scientists andPreface Engineers

xii viii

quantum transmission lines. We then introduce the notion of quantum control of the Belavkin stochastic Schrodinger equation in the sense of Luc Bouten. This involves taking non-demolition Belavkin measurements on a quantum system evolving according to the HP noisy Schrodinger equation, then considering at each time point t, the Belavkin filtered and controlled state ρc (t), applying the Belavkin filter evolution by making a non-demolition measurement dY (t) in time [t, t + dt] to obtain the Belavkin filtered state at time t + dt as ρ(t + dt) = ρc (t) + Ft (ρc (t))dt + Gt (ρc (t))dY (t) c = exp(iZdY (t)) in the time interval [t, t + dt] to the Belavkin filter output and then applying a control unitary Ut,t+dt ρ(t + dt) to get the controlled state at time t + dt as c c∗ ρ(t + dt)Ut,t+dt ρc (t + dt) = Ut,t+dt

Here, Z is a system observable that commutes with dY (t). More precisely, Z should be replaced by U (t + dt)∗ ZU (t + dt) to make it commute with dY (t). We can then show by application of the quantum Ito formula that the system operator Z can be selected so that the evolution from ρc (t) to ρc (t + dt) is such that a major portion of the noise in the Belavkin filter is removed and the evolution of ρc (t) nearly follows that of the noiseless Belavkin equation, ie, the GKSL equation. The next topic discussed in this book is that of designing quantum gates using scattering theory experiments. The basic idea is to realize a unitary quantum gate using the quantum scattering matrix. The system consists of a free projectile having Hamiltonian H0 = P 2 /2m and the interaction potential energy between the projectile and the scattering centre is V (Q) so that the total Hamiltonian of the projectile interacting with the scattering centre is H = H0 + V . The projectile arrives from the infinite past (t → −∞) from the input free particle state |φin >. This free state evolves according to H0 , ie, at time t this free particle state is exp(−itH0 )|φin >. After interacting with the scatterer, it goes to the input scattered state |ψin > which evolves according to H, ie, at time t, this input scattered state is exp(−itH)|ψin >. It follows that these two states coincide in the remote past, ie, limt→−∞ (exp(−itH)|ψin > −exp(−itH0 )|φin >) = 0 from which, we easily deduce that where

|ψin >= Ω− |φin > Ω− = slimexp(itH).exp(−itH0 )

After interacting with the scatterer for a sufficient long time, we ask the question, what is the probability amplitude of the projectile being in the free particle state |φout >?. To evaluate this, we must first determine the scattered state, ie, the output scattered state |ψout > that develops as t → ∞ to |φout >. It is clear that the condition required for this is that limt→∞ (exp(−itH)|ψout > −exp(−itH0 )|φout >) = 0 or equivalently,

where

|ψout >= Ω+ |φout > Ω+ = slimt→∞ exp(itH).exp(−itH0 )

The domains of Ω− and Ω+ will not in general be the entire Hilbert space L2 (R3 ). In fact, determining the domains of definition of Ω± is a hard problem in operator theory and nice treatments of this delicate problem have been given in the masterful monographs of T.Kato (Perturbation theory for linear operators) and W.O.Amrein (Hilbert space methods in quantum mechanics). The scattering matrix S is an operator that maps free input particle states to free output particle states so that the scattering amplitude for the process |φin >→ |φout > is given by < φout |S|φin >=< ψout |ψin >= or equivalently,

< φout |Ω∗+ Ω− |φin > S = Ω∗+ Ω−

We define and derive a formula for the operator R in the from

R=S−I

< λ, ω � |R|λ, ω >=< λ, ω � |V − V (H − λ)−1 V |λ, ω >

Quantum Mechanics for Scientists and Engineers Preface

xiii ix

where λ is the energy of the total system or equivalently of the free projectile coming from t = −∞. The energy is conserved during the scattering process. The state |λ, ω > represents the incident free projectile having energy P 2 /2m = λ and with incoming momentum P being directed along the direction ω ∈ S 2 and likewise |λ, ω � > represents the free projectile state having energy λ and outgoing momentum P being directed along ω � ∈ S 2 . It should be noted that the scattering operator S conserves the energy since it commutes with H0 . This follows from the relations Ω− exp(−itH0 ) = exp(itH)Ω− , Ω+ exp(−itH0 ) = exp(itH)Ω+ for all t ∈ R, from which follows

exp(itH0 )Ω∗+ Ω− exp(−itH0 ) = Ω∗+ Ω−

which gives formally, on differentiating w.r.t t at t = 0,

[H0 , S] = 0 The design of the quantum gate is based on choosing a potential V so that for a given energy λ, the matrix ((< λ, ω � |R|λ, ω >))ω,ω is as close as possilble to Ug − I where Ug is a given N × N unitary matrix and N is the number of discretized directions ω chosen on the unit sphere. Another topic of importance discussed in this book is a rough idea about how a unified quantum field theory can be developed encompassing gravitation, electromagnetism, the electronpositron field of matter and if possible other Yang-Mills fields. The theory developed relies heavily on identifying the correct connection for the covariant derivative of spinor fields in the presence of a curved space-time metric. The construction of this gravitational connection for Dirac spinors can be achieved in terms of the tetrad components of the gravitational metric and the Lorentz generators in the Dirac spinor representation. This construction can be found in Weinberg’s book (Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity). This connection can also be used for Yang Mills field. Once this has been done, it is an easy matter to construct the Lagrangian density of the Dirac or Yang-Mills field in curved space-time. This Lagrangian density is added to the Einstein-Hilbert Lagrangian density of the gravitational field and to the Lagrangian density of the electromagentic and Yang-Mills gauge fields, and to the Lagrangian density of scalar Higgs field taking once again into account the gravitational connection using the Tetrad. This total Lagrangian density is a function of the Dirac wave function, the Yang-Mills wave function, the tetrad components of the gravitational metric and the gauge fields, namely the electromagnetic four potential, ie, the U (1) gauge fields for the Dirac equation and the non-Abelian SU (N ) gauge fields for the Yang-Mills equations. The corresponding four dimensional action integral is then constructed and Feynman diagrammatic rules are formulated using the path integral approach for determining the probability amplitude for scattering/absorption-emission processes involving gravitons, photons, electrons, positrons, and particles associated with the Yang-Mills fields. The basic idea for calculating these amplitudes is to use Feynman’s trick: Identify the parts Sq of the action S that are quadratic in the fields retain them in the exponent exp(iS). The cubic and higher degree terms of the fields appearing in S denoted by Sc are considered as perturbations thus writing down exp(iS) = exp(iSq )(1 + iSc − Sc2 /2 − iSc3 /6 + ...) and the path integral is evaluated using the basic Wick theorem which roughly states that higher moments of a Gaussian distribution can be decomposed into a sum over products of second order moments, ie, propagators. Other schemes of quantization of all the fields can be found in the textbook by Thiemann (Modern canonical quantum general relativity). Finally, we discuss the modern theory of Supersymmetry which is one of the mathematically successful attempts to unify the various field theories like electromagnetism, Dirac’s relativistic quantum mechanics, the scalar field theories of Klein-Gordon and Higgs, the Yang-Mills gauge theories and even general relativity. The idea here is to introduce four anticommuting Majorana Fermionic variables θ and to define a superfield S[x, θ] as a function of both the bosonic space-time coordinates x = (xμ ) and the four Fermionic variables θ = (θa ). When the superfield is expanded in powers of θ, all terms involving five or more θ� s vanish owing to the anticommutativity of the four θ� s. Hence, the superfield S is a fourth degree polynomial in θ and the coefficients of θa , θa θb , θa θb θc , θ0 θ1 θ2 θ3 , 0 ≤ a < b < c ≤ 4 are functions ¯ a a = 0, 1, 2, 3 that are of x only. Then following Salam and Stratadhee, we introduce supersymmetric generators La , L vector field vector field in super-space (x, θ). These generators satisfy the standard anticommutation relations required for supersymmetry generators, ie, their anticommutators are bosonic generators: ¯ b } = γ μ ∂/∂xμ {La , L ab The general form of these generators can be obtained using standard group theoretic arguments: Define the composition of superspace variables as    (xμ , θa ).(xμ , θa ) = (xμ + xμ� + θT Γμ θ, θa + θa )

Quantum Mechanics for Scientists and Engineers Preface

xiv x

Then, the supersymmetry generators (which are supervector fields) are calculated by expressing ∂ f (x, θ).(x� , θ� )) ∂xμ and

∂ f ((x, θ).(x� , θ� )) ∂θa evaluated at x� = 0, θ� = 0 in terms of ∂x∂ μ and ∂θ∂a . The supersymmetry generators when acting on a superfield induce transformations on the component superfield and it is noted that only the coefficient of the fourth degree term in θ suffers a change that is a total differential. In fact, if the coefficient of the fourth degree in θ is split into a sum of two components, one we call the D component and the other we write as a constant times the D’Alembertian acting on the scalar superfield component, then the D component suffers a change that is a perfect divergence. Hence, the four dimensional space-time integral of the D component remains invariant under a supersymmetry transformation and hence this integral can be used as a candidate action for a supersymmetric field theory. It is also noted that if we impose conditions that one part of the θ3 component, namely the λ component (called the gaugino) and the D component is zero, and further if the gauge component Vμ (x) which appear as coefficients of the θ2 portions is a perfect gauge, ie, a perfect divergence, then after a supersymmetry transformation again λ and D vanish. Hence we obtain a subclass of the class of all superfields, namely the Chiral superfields which can be expressed as the sum of a left Chiral and a right Chiral superfield. Chiral fields are characterized by the property that their λ and D components vanish and the Vμ component is a perfect gauge. We further note following the exposition of Wienberg that a left Chiral superfield can be expressed as a function of xμ+ and θL only and likewise, a right Chiral superfield can be expressed T as a function of xμ− and θR where xμ± = xμ ± θL �γ μ θR and θL = (1 + γ5 )θ/2, θR = (1 − γ5 )θ/2 are respectively the left chiral and right chiral projections of the Fermionic parameter θ. There are just two linearly independent left chiral Fermionic parameters and likewise two linearly independent right chiral Fermionic parameters. This means that cubic and higher degree terms in θL vanish and likewise cubic and higher terms in θR vanish. Further, left chiral superfields are characterized by the property that certain ”right” superdifferential operators DR acting on these superfields give zero and right chiral superfields are characterized by the property that certain ”left” superdifferential operators DL acting on these superfields give zero. Here DR and DL are defined as respectively the right chiral and left chiral projections (mutliplication by (1 + γ5 )/2 and (1 − γ5 )/2) of a super vector field D obtained by changing a sign in the expression of the supersymmetry generators. The proofs of these facts follow by showing first that DR xμ+ = 0, DL xμ− = 0 and then noting that DR (anylef tsuperf ield) = 0, DL (anyrightsuperf ield) = 0 since left superfields are functions of x+ , θL 2 only and right superfields are functions of x− , θR only. Further it is readily shown that the coefficient of θL (called the F -term) in a left chiral superfield ΦL (x, θ) suffers change by a perfect divergence under a supersymmetry transformation and hence its space-time integral is a candiadate supersymmetric action. After this, we come to the crucial point: A kinematic Lagrangian density should canonically by a quadratic function of component superfields just as kinetic energies are quadratic functions of momenta/velocities and potential energies of harmonic oscillators are quadratic functions of positions, or equivalently, the Klein-Gordon Lagrangian density is a quadratic function of the space-time derivatives of the field minus a mass term which is a quadratic function of the field itself. Now, if we start with a supefield S[x, θ] and consider the D-component of the superfield S[x, θ]∗ S[x, θ], then we get terms that are bilienar in the D and C terms but no term that is quadratic or higher in the D term. Likewise, we get terms that are bilinear in the λ and ω terms (λ is a cubic component and ω is a first degree component) but no terms that are quadratic or higher in the λ term. Now if we build our action from these terms, then standard Gaussian path integral considerations show that we must evaluate our path integrals by setting the variational derivatives of the action with respect to the various component fields to zero. But the variational derivative of the above mentioned action w.r.t. the D term of S ∗ S is the C term and likewise, the variational derivative of S ∗ S w.r.t the λ term is the ω terms. So we are led to the disastrous consequence that C = 0 and ω = 0, ie, we cannot have scalar fields like the Klein-Gordon field or Fermionic fields like the Dirac field. To rectify this problem, we assume that D = 0, λ = 0 in S while constructing the action [S ∗ S]D . In other, words, to construct matter field Lagrangians, we take our basic matter superfield S as a Chiral field. For a given superfield S[x, θ], the component superfields C(x) (the scalar field) (zeroth power of θ coefficient), the Ferminoic field ω(x) (first power of θ coefficient) and the other fields M (x), N (x) (which are coefficients of the second power of θ) constitute the matter fields and the other components Vμ (x) (one set of components of the second power of θ) which is also called the gauge field, λ(x) (one part of the Fermion field appearing as a coefficient of the third power of θ) also called the gaugino field (and interpreted as the superpartner of the gauge field) and the auxiliary field D(x) appearing as a coefficient of the fourth power of θ constitute the gauge part of the superfield. To get a gauge invariant theory of supefields, first we must form out of a left Chiral superfield Φ[x, θ] (built out of the matter fields C, ω, M, N ) and a matrix Γ[x, θ] built out of the gauge superfields Vμ , λ, D the D component [Φ∗ ΓΦ]D of Φ∗ ΓΦ (which will of course be supersymmetry invariant since it is the D component of a superfield) and the transformation law of the matter part Φ and the gauge part Γ under a gauge transformation is defined in such a way so that [Φ∗ ΓΦ]D remains invariant. This forms the part of the

Quantum Mc:dianiai for Scientists and Engineers

XV

total Lagrangian density that describes matter like the scalar field, the Dirac Fermionic field of electrons and positrons etc., and the interaction of the matter fields with the gauge fields. Finally, another superfi.eld W[x, 9] iB constructed out of the gauge and auxiliary components v,., >.., D such that W[x, 9] is a left Chiral field (and hence its F-component is supersymmetry invariant) and its F component has the fonn

c1F,..,F"" + c1XT"f" D,.>.. + c3D2 which guarantees gauge invariance of the F-pa.rt of W. Here, iF,.., [=a,.+ iV,., a.,+ iV.,] where ir1 an Abelian gauge theory, Yl' is simply a function of x so that F,.., = Vv,l" - \'s.,v iB the electromagne!ic field of photons and :FD,.>.. is the sum of>..T-y~' >..,,. which iB like a kinematic part of the Dirac Lagrangian density and >..Ti-y~'>..VI' which is like the interaction Lagrangian between the Dirac field of Fermions and the photon electromagnetic four potential field VI'. In a non-Abelian gauge theory, V,. is a Lie algebra valued function of x and then [VI', V.,] =F 0. It follows then that the structure constants of this Lie algebra will enter into the picture while defining and F~'v'D~'>... In this case, the above gauge Lagrangian density will describe particles arising in the Yang-Mills theories like Electrowea.k theories etc. When the matter and matter-gauge interaction Lagrangian [4?*f4?]D iB added to the above gauge Lagrangian, we obtain a Lagrangian density that is both supersymmetry and gauge invariant. In such a theory, if path integrals are evaluated with respect to certain auxiliary fields, we obtain equations like the Dirac relativistic wave equation with mass dependent on the scalar field, the Maxwell photon equations, the Yang-Mills equations and even gravitational fields can be included by more additions to the superfield. In short, supersymmetry provides the ideal ground for unifying all the known theories into a single framework, studying irlteractions between the particles associated with each theory and finally quantizing such a unified field theory using the Feynman path irltegral. We do not give all the details here for they can be found in the masterpiece ofWienberg (Supersymmetry, Cambridge University Press).

Table of Contents 1. Wave and Matrix Mechanics of Schrodinger, Heisenberg and Dirac

1-6

2. Exactly Solvable Schrodinger Problems, Perturbation THeory, Quantum Gate Design 7-16 3. Open Quantum Systems, Tensor Products, Fock Space Calculus, Feynman’s Path Integral 17-22 4. Quantum Field THeory, Many Particle Systems 23-48 5. Quantum Gates, Scattering THeory, Noise in Schrodinger and Dirac’s Equations 49-58 6. Quantum Stochastics, Filtering, Control and Coding, Hypothesis Testing and Non-Abelian Gauge Fields 59-74 7. Master Equations, Non-Abelian Gauge Fields 75-88 Appendix 89-208

Chapter

1

Wave and Matrix Mechanics of Schrodinger, Heisenberg and Dirac [1] The De-Broglie Duality of particle and wave properties of matter. A plane wave in one dimension is expressed by the following complex amplitude: ψ(t, x) = A.exp(i(kx − ωt))

where k = 2π/λ with λ as the wavelength and ν = ω/2π is the frequency. According to De-Broglie, we can associate a particle having momentum p = h/λ = hk/2π with such a wave where h is Planck,s constant. According to Planck, we can associate a quantum of energy E = hν = hω/2π with such a wave. Thus, we have (ih/2π)

∂ψ = (hω/2π)ψ, = Eψ ∂t

(−ih∇/2π)ψ = (hk/2π)ψ = pψ In the presence of an external potential V (x), E should be taken as p2 /2m + V (x) and hence, Eψ = (p2 /2m + V )ψ = (−h2 ∇2 /8π 2 m + V (x))ψ Although the above plane wave ψ cannot satisfy this equation for general V (x), we assume that the actual De-Broglie matter wave ψ associated with the particle is a solution to the above equation.. This is called the one dimensional Schrodinger equation. In short, by putting together De-Broglie’s matter-wave duality principle and Planck’s quantum hypothesis, we have heuristically derived the Schrodinger wave equation. [2] Bohr’s correspondence principle. Bohr’s correspondence principle is that associated with each observable in classical mechanics is an observable in quantum mechanics that follows certain quantization axioms which can be used to explain the energy spectra of atoms and quantum oscillators. For example, consider an electron of charge −e and mass m moving in a circular orbit of radius r around the nucleus. Its mometum is p = mv and the correspondence principle implies that Γ pdq = nh, n ∈ Z where q is the position coordinate and the lhs is the line integral around a complete orbit of the electron. Thus, we derive 2πpr = nh, p = nh/2πr, n ∈ Z Thi is the same as

mvr = nh/2π

According to the centripetal force law that keeps a particle in an orbit, mv 2 /r = KZe2 /r2 , K = 1/4π0 Thus, we get or and

n2 h2 /4π 2 mr3 = KZe2 /r2 r = n2 h2 /4π 2 mKZe2 v 2 = KZe2 /mr = 4π 2 K 2 Z 2 e4 /n2 h2

and finally, we get the energy spectrum of the particle: E == En = p2 /2m − KZe2 /r = mv 2 /2 − Ze2 /r = KZe2 /2r − KZe2 /r = −KZe2 /2r = −2π 2 mK 2 Z 2 e4 /n2 h2 , n = 1, 2, ...

This spectrum first derived by Bohr, agreed with experiments.

[3] Bohr-Sommerfeld’s quantization rules. If q is a canonical position coordinate vector and p the canonical momentum vector, then for cyclic motion, the Bohr-Sommerfeld’s quantization rules state that  p.dq = nh, n ∈ Z Γ

Quantum Mechanics for Quantum Scientists and Engineers Mechanics

2 2

where Γ is a closed loop. A special case of this rule was applied earlier by Bohr to derive the spectrum of the Hydrogen atom. For example, if the system is described by action-angle variables I, θ, then we get  2π I(θ)dθ = nh, n ∈ Z 0

Sommerfeld applied this to relativistic quantum mechanics according to which the Hamiltonian of the particle is given by  H(q, p) = c p2 + m2 c2 − eV (q) Writing H = E and solving for p, we get

p = ±[(E + eV (q))2 + c2 p2 + m2 c4 ]1/2 = ±p(q, E) If p given by this expression vanishes at q = q1 , q2 , then the Bohr-Sommerfeld quantization rule states that the energy spectrum of this relativistic particle is given by  q2 2 p(q, E)dq = nh, n ∈ Z q1

Sommerfeld used this to derive the relativistic spectrum of the Hydrogen atom which improves upon the non-relativistic spectrum given by Bohr. [4] The principle of superposition of wave functions and its application to the Young double slit diffraction experiment. In quantum mechanics, a pure state in the position representation is described by a complex valued wave function ψ(x), x ∈ R3 . Given two such wave functions ψ1 (x), ψ2 (x), we can construct another wave function ψ(x) = c1 ψ1 (x) + c2 ψ2 (x), where c1 , c2 are two complex numbers. |ψk (x)|2 is proportional to the intensity of the wave ψk at x, k = 1, 2 and |ψ(x)|2 = |c1 |2 |ψ1 (x)|2 + |c2 |2 |ψ2 (x)|2 + 2Re(¯ c1 c2 ψ¯1 (x)ψ2 (x)) is proportional to the intensity of the wave ψ at x. In particular, we can take c1 = c2 = 1, then the intensity of ψ is the intensity of ψ1 plus the intensity of ψ2 plus an interference term 2Re(ψ¯1 (x)ψ2 (x)). This last term is a purely quantum mechanical effect. This fact has been marvellously illustrated by Feynman using the Young double slit experiment: We may two slits in a cardboard sheet and place an electron gun behind this sheet. On the other side of the sheet is a screen than can record the impact of electrons. If the first slit is open and the second closed, then the electron intensity pattern on the screen shows a maximum value at the portion of the screen directly in front of the first slit and decays down as the distance from the first slit increases on both the sides. Likewise, if slit one is closed and slit two is open, then the electron intensity on the screen in front of slit two shows a maximum. We denote the former intensity pattern by I1 (x) and the latter intensity pattern by I2 (x). Now, when both the slits are open, if the electron were to behave as classical particles, we should expect the intensity pattern on the screen to be I1 (x) + I2 (x), ie, maxima at both the points on the screen in front of the first and second slit respectively. But this is not what we observe. Instead we observe a maximum on the screen at a point in front of the middle between the two slits, ie at a point on the screen that is equidistant from the two slits. Further, as we move away from this intensity maximum point, the intensity shows a sinusoidal variation. This can be explained only by the presence of the above interference term. For example, if ψ1 (x) = A1 .exp(ik1 x), ψ2 (x) = A2 .exp(ik2 x) where x is the distance on the screen from the central point P that is equidistant from both the slits, then the intensity pattern on the screen when both the slits are open is given by I(x) = |ψ1 (x) + ψ2 (x)|2 = |A1 |2 + |A2 |2 + 2|A1 ||A2 |cos((k1 − k2 )x + φ) where

φ = arg(A1 ) − arg(A2 )

It follows that if φ = 0 (which will be the case when the two slits are equidistant from the electron gun, then I(x) will show a maximum at x = 0 and its spatial variation will be sinusoidal with the distance between two successive maxima or between two successive minima being given by 2π/|k1 − k2 |. Thus we are forced to conclude that electrons exhibit wave behaviour at some time which is completely in accord with the De-Broglie matter-wave duality principle. The Heisenberg uncertainty principle can also be explained using this setup. The difference between the two electron momenta parallel to the screen is Δp = h|k1 − k2 |/2π according to De-Broglie and the position measurement uncertainty is the distance between an intensity maximum and an intensity miniumum on the screen, ie, Δx = π/|k1 − k2 |. Thus, Δp.Δx ≈ h/2 This means that if we attempt to measure the momentum difference accurately then the position measurement will become less accurate and vice versa.

Quantum Quantum Mechanics for Scientists and Engineers

3 3

[5] Schrodinger’s wave mechanics and Heisenberg’s matrix mechanics. In wave mechanics, the state of a quantum system at time t is defined by a normalized vector |ψ(t) > in a Hilbert space H and this vector satisfies a ”Schrodinger wave equation”: i

d|ψ(t) > = H(t)|ψ(t) > dt

where H(t) is a possibly time varying self-adjoint operator in H. If H(t) = H is time independent, then the solution can be expressed as |ψ(t) >= exp(−itH)|ψ(0) > where exp(−itH) is a unitary operator in H and may be defined via resolvents: exp(−itH) = limn→∞ (1 + itH/n)−n We note that even if H is an unbounded operator, this definition of the exponential may make sense using the theory of resolvents and spectra. (A complex number z is said to be in the resolvent set ρ(H) of H if (z − H)−1 exists and is a bounded operator. The complement of ρ(H) denoted by σ(H) is called the spectrum of the operator H. Since here H is Hermitian, we have the spectral representation  H = d xE(dx) R

so that exp(−itH) = and



(1 + itH/n)−n = so for any |f >∈ H, we have � (1 + itH/n)−n f − exp(−itH)f �2 =



R

R



exp(−itx)E(dx)

(1 + itx/n)−n E(dx)

|(1 + itx/n)−n − exp(−itx)|2 < f |E(dx)|f >

which converges by the dominated convergence theorem to zero as n → ∞. We note that if T is any densely defined operator in H, bounded or unbounded, and if for z ∈ ρ(T ), we have an inequality of the form � (z − T )−1 �≤ f (z) then

� (1 + itT /n)−n �≤� (1 + itT /n)−1 �n ≤ nf (in/t)/|t|

and this may remain bounded as n → ∞. On the other hand, (1 + itT /n)n is unbounded if T is unbounded and hence we cannot defined exp(itT ) as the limit of this as n → ∞ (Reference: T.Kato, Perturbation theory for linear operators, Springer). In wave mechanics, the observables like position, momentum, angular momentum, energy etc. are represented by selfadjoint operators in the Hilbert space H and these do not vary with time while the state |ψ(t) > varies with time. Hence, if X is an observable, then its average at time t is given in the Schrodinger wave mechanics picture by < ψ(t)|X|ψ(t) >. On the other hand, in the Heisenberg matrix mechanics picture, observables change with time while the state remains the same and hence, the average of X at time t in the Heisenberg picture is < ψ(0)|X(t)|ψ(0) >. Since the physics must be the same no matter which model we use, we must have < ψ(t)|X|ψ(t) >=< ψ(0)|X(t)|ψ(0) > − − −(1) and differentiating this equation w.r.t. time gives i < ψ(t)|[H(t), X]|ψ(t) >=< ψ(0)|X � (t)|ψ(0) > Writing then gives us

|ψ(t) >= U (t)|ψ(0) > X � (t) = iU (t)∗ [H(t), X]U (t) = [iU (t)∗ H(t)U (t), X(t)]

Quantum Mechanics for Scientists Engineers Quantumand Mechanics

4 4

We could also directly write (1) as < ψ(0)|U (t)∗ XU (t)|ψ(0) >=< ψ(0)|X(t)|ψ(0) > and hence

X(t) = U (t)∗ XU (t)

from which we obtain on differentiation, X � (t) = iU (t)∗ [H(t), X]U (t) = i[U (t)∗ H(t)U (t), X(t)] This is Heisenberg’s equation of matrix mechanics. In particular, if H(t) = H does not vary with time, we get U (t)∗ HU (t)∗ = H and Heisenberg’s equation of matrix mechanics becomes X � (t) = i[H, X(t)] with solution,

X(t) = exp(itH).X.exp(−itH)

[6] Dirac’s replacement of the Poisson bracket by the quantum Lie bracket. The Poisson bracket between two observables u(q, p) and v(q, p) satisfies {u, vw} = {u, v}w + v{u, w} and likewise for the first argument. If we agree that an analogous bracket [.] exists for non-commutative quantum observables with the same ordering preserved, then we must have [uv, wz] = [uv, w]z + w[uv, z] = u[v, w]z + [u, w]vz + wu[v, z] + w[u, z]v on the one hand and on the other hand, [uv, wz] = [u, wz]v + u[v, wz] = [u, w]zv + w[u, z]v + u[v, w]z + uw[v, z] Equating these two expressions gives [u, w]vz + wu[v, z] = [u, w]zv + uw[v, z] Hence

[u, w](vz − zv) = (uw − wu)[v, z]

It follows from the arbitrariness of the four observables u, v, w, z that the quantum bracket [.] should have the form [u, w] = c(uw − wu) where c is a complex constant. In other words, the quantum bracket that replaces the classical Poisson bracket must be proportional to the Lie bracket. [7] Duality between the Schrodinger and Heisenberg mechanics based on Dirac’s idea. In Schrodinger’s wave mechanics, the state of the system at any time t is described by a density operator ρ(t) in a Hilbert space H. In other words, ρ(t) ≥ 0, T r(ρ(t)) = 1. Further, it satisfies the Schrodinger-Liouville- Von-Neumann equation of motion: iρ� (t) = [H(t), ρ(t)] This follows by writing the spectral decomposition of ρ(t) as  |ψk (t) > pk < ψk (t)| ρ(t) = k

in diagonal form with

p�k s

constant and



k

pk = 1, pk ≥ 0. The ψk (t)� s satisfy the Schrodinger wave equation i|ψk� (t) >= H(t)|ψk (t) >

Quantum Quantum Mechanics for Scientists and Engineers

and hence

5 5

−i < ψk� (t)| =< ψk (t)|H(t)

so that

iρ� (t) = =

 k

We can write its solution as



(i|ψk� (t) > pk < ψk (t)| + i|ψk (t > pk < ψk� (t)|)

k

(H(t)|ψk (t) > pk < ψk (t)| − |ψk (t)pk < ψk (t)|H(t)) = [H(t), ρ(t)] ρ(t) = U (t)ρ(0)U (t)∗

where U (t) is a unitary operator satisfying the Schrodinger equation U � (t) = −iH(t)U (t) In this picture, observables do not vary with time. Thus, the average value of an observable X at time t is given by  pk < ψk (t)|X|ψk (t) > < X > (t) = T r(ρ(t)X) = k

In the Heisenberg picture, the observable X changes at time t to X(t) while the state ρ(0) does not change. Hence, in this picture the average of the observable at time t is < X > (t) = T r(ρ(0)X(t)) and the two pictures must give the same average. Thus, T r(ρ(0)X(t)) = T r(ρ(t)X) = T r(U (t)ρ(0)U (t)∗ X) = T r(ρ(0)U (t)∗ XU (t)) and hence from the arbitrariness of ρ(0), it follows that dynamics of observables in the Heisenberg picture must be given by X(t) = U (t)∗ XU (t) and hence where

˜ X(t)] X � (t) = iU (t)∗ (H(t)X − XH(t))U (t) = i[H(t), ˜ H(t) = U (t)∗ H(t)U (t)

In particular, if H(t) = H is time independent, then U (t) = exp(−itH) and we get X � (t) = i[H, X(t)] for the Heisenberg dynamics and

iρ� (t) = [H, ρ(t)]

for the Schrodinger dynamics. [8] Quantum dynamics in Dirac’s interaction picture. Suppose H(t) = H0 + V (t) is the system Hamiltonian. The Schrodinger evolution operator U (t) satisfies iU � (t) = H(t)U (t) We write

U (t) = U0 (t)W (t), U0 (t) = exp(−itH0 )

Then, substituting this into the above Schrodinger evolution equation gives iW � (t) = V˜ (t)W (t), V˜ (t) = U0 (t)∗ V (t)U0 (t)

Quantum Mechanics for Scientists Engineers Quantum and Mechanics

6 6

This leads us to the Dirac interaction picture where a state |ψ > evolves according to the Hamiltonian V˜ (t) while an observable X evolves according to the Hamiltonian H0 . This keeps the physics invariant since if X is an observable and |ψ > is the state at time 0, then in the Dirac interaction picture, the average of the observable in the state at time t is given by (The subscript i stands for evolution of states or observables in the interaction picture. Thus |ψi (t) >= W (t)|ψ =< ψ|W (t)∗ (U0 (t)∗ XU0 (t))W (t)|ψ > where W (t) = T {exp(−i Now, as a check



t

V˜ (τ )dτ )}

0

d (U0 (t)W (t)) = U0� (t)W (t) + U0 (t)W � (t) = dt −iH0 U0 (t)W (t) + U0 (t)(−iV˜ (t))W (t)

= −iH0 U0 (t)W (t) − iU0 (t)U0 (t)∗ V (t)U0 (t)W (t)

= −iH0 U0 (t)W (t) − iV (t)U0 (t)W (t) = −iH(t)U0 (t)W (t)

˜ (t) satisfies the same equation as U (t) and hence must equal U (t). Thus, U0 (t)W (t) = U (t) as In other words, U0 (t)U expected and this shows that < ψi (t)|Xi (t)|ψi (t) >=< ψ|U (t)∗ XU (t)|ψ > which is in agreement with the Schrodinger and Heisenberg pictures. [9] The Pauli equation: Incorporating spin in the Schrodinger wave equation in the presence of a magnetic field. We describe how the Pauli equation can be used to calculate the average electric dipole moment and average magnetic dipole moment of an atom in an external electromagnetic field. This calculation enables us to calculate the electric polarization field and the magnetization field of a material. It will be a consequence of this calculation that the average electric dipole moment and hence polarization depends on both the external electric and magnetic fields and likewise for the average magnetic dipole moment and magnetization. The electric dipole moment observable is −er, r = (x, y, z) being the electron position relative to the nucleus. The magnetic dipole moment observable is μ = (e/2m)(L+gσ/2) where L = (Lx , Ly , Lz ) is the angular momentum observable vector and σ = (σx , σy , σz ) is the vector of Pauli spin matrices. A heuristic justification of this fact [10] The Zeeman effect: Let H0 be the Hamiltonian of an atom that commutes with the orbital and spin angular momentum operators (Lx , Ly , Lz ) = L (ie, H0 is rotation invariant as happens when H0 = p2 /2m0 + V (r) where V depends only on the radial coordinate) and (σx , σy , σz ) = σ. Then the eigenstates of the Hamiltonian operator H = H0 + e(L + gσ, B)/2m0 where B is a constant magnetic can be calculated as follows. We may assume without loss of generality that B is along the z axis. Then, H = H0 + (eB/2m)(Lz + gσz ) and since L2 , Lz , σz , H0 mutually commute, the energy eigenstates of H are labelled by four quantum numbers |n, l, m, s > where l(l+1) is an eigenvalue of L2 , m is an eigenvalue of Lz and s is an eigenvalue of σz . If E(n, l) is an energy eigenvalue of H0 ie of H without the magnetic field, then this eigenvalue has a degeneracy of 2(2l + 1) corresponding to the 2l + 1 eigenvalues of Lz for a given eigenvalue l(l + 1) (ie,+ m = −l, −l + 1, ..., l − 1, l)of L2 and the two eigenvalues ±1/2 of σz . When the magnetic field is turned on, these 2(2l + 1) degenerate eigenstates split into the the same number of nondegenerate eigenstates with eigenvalues E(n, l, m, s) = E(n, l) + (eB/2m0 )(m + gs), m = −l, −l + 1, ..., l − 1, l, s = ±1/2. The eigenstate of H corresponding to the eigenvalue E(n, l, m, s) is denoted by |n, l, m, s >. [11a] The spectrum of the Hydrogen atom. This is described by the stationary Schrodinger equation ∇2 ψ(r, θ, φ) + 2m(E + e2 /r)ψ(r, θ, φ) = 0 Writing

ψ(r, θ, φ) = R(r)Ylm (θ, φ)

2

Chapter |n, l, m, s >

Exactly Solvable Schrodinger Problems, Perturbation THeory, Quantum Gate Design l+1

=− −

2

l E(n, l, m, s) = E(n, l) + (eB/2m0 )(m + gs), m = −l, −l + 1, ..., l − 1, l, s = ±1/2. .

[11a] The spectrum of the Hydrogen atom. This is described by the stationary Schrodinger equation ∇2 ψ(r, θ, φ) + 2m(E + e2 /r)ψ(r, θ, φ) = 0 Writing

ψ(r, θ, φ) = R(r)Ylm (θ, φ)

where Ylm are the spherical harmonics, we get using ∂ 2 ∂ r − L2 /r2 ∂r partialr where L2 is the squared angular momentum operator: ∂ ∂ 1 ∂2 1 L2 = − sin(θ) − sin(θ) ∂θ ∂θ sin2 (θ) ∂φ2 ∇2 = r−2

Ylm is an eigenfunction of both the commuting operators L2 and Lz : L = r × p = −ir, L2 = −(r × ∇)2 Lz Ylm = mYlm , L2 Ylm = l(l + 1)Ylm Exercise: Verify that L given by the above differential expression coincides with 2

−(r × ∇)2 = (ypz − zpy )2 + (zpx − xpz )2 + (xpy − ypx )2 , The radial equation for R(r) thus becomes

px = −i∂x , py = −i∂y , pz = −i∂z

R�� (r) + 2R� (r)/r − l(l + 1)R(r)/r2 + 2m(E + e2 /r)R(r) = 0

or equivalently,

r2 R�� (r) + 2rR� (r) + 2m(Er2 + e2 r − l(l + 1)/2m)R(r) = 0

As r → ∞, this equation approximates to

r2 R�� (r) + 2mEr2 R(r) ≈ 0

or equivalently,

R�� (r) = −2mER(r)

Since for bound states, E must be negative, it follows that as r → ∞, we have R(r) ≈ C.exp(−αr), α = writing the exact solution as √ R(r) = f (r)exp(−αr), α = −2mE we get

R� = (f � − αf )exp(−αr), R�� = (f �� − 2αf � + α2 f )exp(−αr)

The Schrodinger equation now becomes

r2 (f �� − 2αf � + α2 f ) + 2r(f � − αf ) + (−α2 r2 + 2me2 r − l(l + 1))f = 0

or

r2 f �� + 2r(1 − αr)f � + (2me2 r − l(l + 1))f = 0

We solve this equation by the power series method. Substitute  f (r) = c(n)rn+s n≥0

Then



n≥0

(n + s)(n + s − 1)c(n)rn+s + 2 +2me

2

or equivalently,



n≥0



n≥0

c(n)r



n≥0

(n + s)c(n)rn+s − 2α

n+s+1

− l(l + 1)



c(n)r



(n + s)c(n)rn+s+1

n≥0

n+s

n≥0

((n + s)(n + s + 1) − l(l + 1))c(n)rn+s

=0

√ −2mE. So,

Quantum Mechanics for Scientists and Engineers

8 8

+



(2me2 − 2(n + s)α)c(n)rn+s+1 = 0

Equating coefficients of equal powers of r gives (s(s + 1) − l(l + 1))c(0) = 0, and for all n ≥ 1,

((n + s)(n + s + 1) − l(l + 1))c(n) + (2me2 − 2(n + s − 1)α)c(n − 1) = 0

We must assume c(0) �= 0 for otherwise, this recursion would imply that c(n) = 0∀n ≥ 0 which would mean that the wave function vanishes. Then, we get s(s + 1) = l(l + 1) so s = l since s = −l − 1 would give a singularity at r = 0 and the wave function would fail to be square integrable when l > 0. Further, for n large, ie, n → ∞, the above difference equation asymptotically is equivalent to c(n) ≈ 2αc(n − 1)/n or equivalently,

c(n) ≈ (2α)n /n!

and hence

f (r) =

 n

so

c(n)rn+l ≈ rl exp(2αr)

f (r)exp(−αr) ≈ rl exp(αr)

which is not square integrable, in fact, it diverges exponentially as r → ∞. The only way out is that c(n) = 0 for all n ≥ n0 for some n0 ≥ 1. Let n0 be the smallest such integer. Then we get from the above recursion by putting n = n0 , and c(n0 − 1) �= 0, me2 = (n0 + s − 1)α = (n0 + l − 1)α Thus,

E = −me4 /2(n0 + l − 1)2

Thus, we get the result that the possible energy levels of the hydrogen atom are En = −me4 /2n2 , n = max(1, l), l + 1, l + 2, ..., This result was first derived rigorously by Schrodinger in this way although it was earlier obtained by Niels Bohr using adhoc arguments like the correspondence principle. [11b] The spectrum of particle in a 3 − D box. This is described by the stationary state Schrodinger equation −∇2 ψ(r)/2m = Eψ(r), r = (x, y, z) ∈ [0, a] × [0, b] × [0, c] with boundary conditions that ψ(r) vanishes on the boundary, ie when either x = 0, a or y = 0, b or z = 0, c. The solution to this boundary valued problem is obtained by separation of variables and the result is the set of orthonormal wave functions ψnmk (r) = ((2/a)(2/b)(2/c))1/2 sin(nπx/a)sin(mπy/b)sin(lπz/c), n, m, k = 1, 2, 3, ... with the corresponding energy eigenvalues E = π(n2 /a2 + m2 /b2 + k 2 /c2 )1/2

[11c] The spectrum of a quantum harmonic oscillator. A quantum Harmonic oscillator has the Hamiltonian H = p2 /2m + mω 2 q 2 /2

Quantum Mechanics for Scientists and Engineers

9

where

9

[q, p] = i

We define the annihilation and creation operators by √ √ a = (p − imωq)/ 2m, a∗ = (p + imωq)/ 2m Then,

a∗ a = p2 /2m + mω 2 q 2 /2 − ω/2 = H − ω/2, aa∗ = H + ω/2

Thus,

[a, a∗ ] = ω

Now let |E > be any normalized eigenstate of H with eigenvalue E. Then we have 0 ≤< E|a∗ a|E >=< E|H − ω/2|E >= E − ω/2 Hence E ≥ ω/2

ie, the minimum energy level of the oscillator is ω/2. This is attained when and only when a|E >= 0. In other words, we have a|ω/2 >= 0 and hence

(d/dq + mωq) < q|ω/2 >= 0

or equivalently, < q|ω/2 >= C.exp(−mω 2 q 2 /2) with C being the normalizing constant chosen so that   | < q|ω/2 > |2 = |C|2 exp(−mω 2 q 2 )dq 1 =< ω/2|ω/2 >= R

R

= |C| = (π/m) 2

so we may take

1/2

ω

−1

C = (π/mω 2 )1/4

[12] Time independent perturbation theory: Calculation of the energy levels and eigenfunctions of non-degenerate and degenerate systems using perturbation theory. The perturbed Hamiltonian is ∞  �k Vk H = H0 + k=1

The stationary state wave functions for H are expanded as

|ψ >= |ψ0 > +



k≥1

�k |ψk >

and the corresponding perturbed energy level is also expanded as a power series:  E = E0 + �k Ek k≥1

Substituting these expressions into the eigenvalue problem H|ψ >= E|ψ > and equating coefficients of � , k = 0, 1, 2, ... gives us the series k

H0 |ψ0 >= E0 |ψ0 >,

Quantum Mechanics for Scientists Engineers Quantum and Mechanics

10 10

(H0 − E0 )|ψk > +

k 

Vr |ψk−r > +

r=1

k  r=1

Er |ψk−r >, k ≥ 1

For each eigenvalue E0 = E0 (m) of H0 , let |m, r >, r = 1, 2, ..., d(m) denote an orthonormal basis of eigenstates of N (H0 − E0 (m)). Thus, the eigenvalue E0 (m) has a degeneracy of d(m). So we can write d(m)

|ψ0 >=



c(m, r)|m, r >

r=1

for E0 = E0 (m). The O(�) equation (H0 − E0 (m))|ψ1 > +V1 |ψ0 >= E1 |ψ0 > − − −(1) then gives on premultiplying by < m, r|,  c(m, r) < m, k|V1 |m, r >= E1 c(m, r)δ[k − r] r

and hence, it follows that the for the unperturbed energy level E0 (m), the possible first order perturbations E1 = E1 (m, s), s = 1, 2, ..., d(m) to the energy levels are given by the eigenvalues of the d(m) × d(m) secular matrix ((< d(m) m, k|V1 |m, r >))1≤k,r≤d(m) . Further, if cs (m) = ((cs (m, r)))r=1 is the eigenvector of this secular matrix corresponding d(m) to the eigenvalue E1 (m, s), then then the corresponding unperturbed state is given by r=1 cs (m, r)|m, r >. We note d(m) that the constants cs (m, r) can be chosen so that the states r=1 cs (m, r)|m, r >, s = 1, 2, ..., d(m) form an orthonormal basis for the d(m) dimensional vector space N (H0 − E0 (m)). Further, we get from (1) by premultiplying by < l, r| for l �= m, (E0 (l) − E0 (m)) < l, r|ψ1 > + < l, r|V1 |ψ0 >= E1 < l, r|ψ0 >= 0 so that

< l, r|ψ1 >= which implies that the unperturbed state |ψ0 >=



< l, r|V1 |ψ0 > E0 (m) − E0 (l)

r cs (m, r)|m, r

> gets perturbed to

|ψ0 > +�|ψ1 > +O(�2 ) where |ψ1 >= and



l�=m,r=1,2,...,d(m)

|l, r >< l, r|V1 |ψ0 > /(E0 (m) − E0 (l)) d(m)

< l, r|V1 |ψ0 >=



k=1

cs (m, k) < l, r|V1 |m, k >

We now calculate the second order pertrubation to the energy levels and the the corresponding eigenfunctions. [13] Time dependent perturbation theory: Calculation of the transition probabilities of a quantum system from one stationary state of the unperturbed system to another stationary state in the presence of a time varying interaction potential. The perturbed Hamiltonian has the form H(t) = H0 +

∞ 

�m Vm (t)

m=1

The Schrodinger evolution operator U (t) of this system is expanded as a perturbation series:  �m Um (t) U (t) = U0 (t) + m≥1

Substituting this into the evolution equation

iU � (t) = H(t)U (t)

Quantum Mechanics for Scientists and Engineers

11

and equating equal powers of the perturbation parameter � gives us the sequence of differential equations: iU0� (t) = H0 U0 (t), � (t) − H0 Um (t) = iUm

Thus

m 

k=1

Vk (t)Um−k (t), m ≥ 1

U0 (t) = exp(−itH0 ), Um (t) = −i

m  

k=1

0

t

U0 (t − s)Vk (s)Um−k (s)ds, m ≥ 1 − − − (1)

Let now |n > be an eigenstate of the unperturbed system with energy E(n): H0 |n >= E(n)|n > Then when the perturbation is switched on, the transition probability amplitude from state |n > to state |m > in time [0, T ] with m �= n is given by < m|U (T )|n >= −i

r 

�k < m|Uk (T )|n > +O(�r+1 )

k=1

where U1 (T ), ...., Ur (T ) are successively determined from (1). For example,  t U0 (t − s)V1 (s)U0 (s)ds, U1 (t) = −i 0

U2 (t) = −i =−



0 to |m > for m �= n in time T is given by  T exp(−i(E(m)(t − s) − E(n)s)) < m|V1 (s)|n > ds|2 + O(�3 ) PT (n → m) = �2 | < m|U1 (T )|n > |2 + O(�3 ) = �2 | 0

= �2 |



0

T

exp(i(E(m) − E(n))s) < m|V1 (s)|n > ds|2 + O(�3 )

In the limit as T → ∞, this probability converges to where Vˆ1 (ω) =

∞ 0

�2 |Vˆ1 (E(m) − E(n))|2 + O(�3 )

V1 (t)exp(iωt)dt is the one sided Fourier transform of V1 .

[14] The full Dyson series for the evolution operator of a quantum system in the presence of a time varying potential. [15] The transition probabilities in the presence of a stochastically time varying potential. If the potentials Vk (t), k = 1, 2, ... appearing in section (13) are operator valued stochastic processes, then the various perturbation terms Uk (t), k = 1, 2, ... in the expansion of the evolution operator will also be operator valued stochastic of terms, each term of which is a multilinear functional of Vr (.), r = 1, 2, ... processes and in fact, Uk (t) will be a sum  with Vrj (.) appearing mj times such that j mj rj = k. The transition probability from state |n > to state |m > in time T will then be given by PT (n → m) = E[< m|U (T )|n > |2 ] = E[(|sumk≥1 �k < m|Uk (T )|n > |2 ] where the expectation is taken w.r.t the probability distribution of the random potentials Vk (.), k = 1, 2, .... [16] Basics of quantum gates and their realization using perturbed quantum systems.

Quantum Mechanics for Scientists and Engineers

12

If H0 is the unperturbed Hamiltonian and we apply a perturbation time varying potential p 

V (t) =

fk (t)Vk

k=1

where the fk (t)� s are control functions, then the solution to the Schrodinger evolution unitary operator U (t) defined by U � (t) = −iH(t)U (t), H(t) = H0 + δV (t) upto O(δ m ) is given by W (T ) = W (T, f ) = I +



U (t) = U0 (t)W (t), U0 (t) = exp(−itH0 ),  (−i)r δ r fk1 (t1 )...fkr (tr )V˜k1 (t1 )...V˜kr (tr )dt1 ...dtr + O(δ m ) 0 converges weakly to zero. However since � en − em �= √ 2, n �= m, the sequence |en > does not converge strongly and in fact, no subsequence of this sequence can converge strongly. Now define the operators Tn = |en >< en |, n = 1, 2, .... Then < f |Tn |g >=< f |en >< en |g >→ 0, hence wlimTn = 0. Further � Tn f �2 = | < en |f > |2 → 0

hence slimTn = 0. Further, � Tn �= 1 and hence Tn does not converge in the operator norm. Assuming the spectral theorem stated above, we can define for any Borel set B in R the orthogonal projection  E(B) = B dE(x). This means that for all f, g ∈ H, < f |E(B)|g >= B d < f |E(x)|g > where d < f |E(x)|g > defines a signed measure for each f, g. Note that the function x →< f |E(x)|g > has bounded variation since |d < f |E(x)|g > | = | < f |dE(x)|g > | ≤� f �< g|dE(x)|g >1/2 =� f �� dE(x)g � =� f � .(d(� E(x)g �)2 )1/2

where we use the fact that (dE(x))2 = dE(x) = dE(x)∗ and that for a < b, � (E(b) − E(a))g �2 =� E(b)g �2 − � E(a)g �2 We also have for < f |f >= 1, |d < f |E(x)|g > |2 = | < f |dE(x)|g > |2 ≤< dE(x)g|dE(x)g >=� dE(x)g �2 so that |d < f |E(x)|g > | ≤� dE(x)g �

Quantum Mechanics for Scientists and Engineers

14

Note that x →� E(x)g � is a non-decreasing function bounded above by � g � and hence is a function of bounded variation or more precisely, R d � E(x)g �=� g �< ∞. We have further, for < f |f >= 1 that | < f |E(x)|g > | ≤� E(x)g �

and also

| < f |E(y)|g > − < f |E(x)|g > | = | < f |(E(y) − E(x))|g > | ≤� (E(y) − E(x))g �

Now x →� E(x)g � is a function of bounded variation and x →< g|E(x)|g >=� E(x)g �2 is also a function of bounded variation since it is a non-decreasing function bounded above by � g �2 . Now, 4Re(< f |E(x)|g >) =< f + g|E(x)|f + g > − < f − g|E(x)|f − g > implying that x → Re(< f |E(x)|g >) is of bounded varation an likewise x → Im(< f |E(x)|g >) is also of bounded variation obtained by replacing f with i.f . Hence x →< f |E(x)|g > has bounded variation which enables us to define spectral integrals weakly using Riemann-Stieltjes integrals. Further, if  xdE(x) H= R

then for c ∈ R, the projection Pc onto R((H − c)+ ) is given using the formula  (H − c)+ = (x − c)dE(x) x≥c

as

Pc = E(c)

This formula enables us to constructively describe the spectral measure associated with a Hermitian operator. In fact, in the proof of the spectral theorem, we define Pc = f (H) = (H −c)+ by taking f (x) = max(x − c, 0) = (|x − c| + (x − c))/2 and then proving that c → Pc is a spectral measure such that cdPc = H. See T.Kato, ”Perturbation theory for linear operators”, Springer, for details. Remark: If z is any complex number with Im(z) �= 0, then for Hermitian H,  (H − z)−1 = (x − z)−1 dE(x) so that the spectral norm of (H − z)−1 is given by � (H − z)−1 �= max(|x − z)−1 : x ∈ R} = |Im(z)|−1 < ∞ In paricular, (H − z)−1 is a bounded operator, ie z ∈ ρ(H) where ρ(H) is the resolvent set of H. More generally, it can be shown that if H is a closed symmetric operator, then ±i belong to ρ(H) iff H is Hermitian. Remark: If T is an operator in a Hilbert space, its graph is defined by Gr(T ) = {(x, T x) : x ∈ D(T )} T is said to be closed if Gr(T ) is closed, ie if xn ∈ D(T ), xn → x, T xn → y imply x ∈ D(T ) and y = T x. An operator T is said to be closable if the closure Cl(Gr(T )) of Gr(T ) is also the graph of some operator T˜. It is easy to see that this happens iff xn ∈ D(T ), xn → x, T xn → 0 imply x = 0. Then we say that T is closable with T˜ being the closure of T . A operator T in H is said to be essentially selfadjoint if T is closable with its closure T˜ being selfadjoint. If E is a subset of D(T ) such that {(x, T x) : x ∈ D(T )} is dense in Gr(T ), then E is called a core for T . It is easy to see that this happens iff for any x ∈ D(T ) there exists a sequence xn ∈ E such that � xn − x �2 + � T xn − T x �2 → 0, or equivalently, iff for any x ∈ D(T ), < (x, T (x)), (z, T (z)) >= 0 for all z ∈ E implies x = 0. [19] The general theory of Events, states and observables in the quantum theory. The basic axiomatic set up that defines a quantum probability space is the triplet (H, P (H), ρ) where H is a Hilbert space, P (H) is the lattice of orthogonal projections in H and ρ is a state in H, ie, a positive definite matrix of unit trace. This is to be compared with a classical probability space defined by the triplet (Ω, F, P ) where Ω is the sample space, F is a σ − algebra of subsets of Ω and P is a probability measure on F. The elements of F are called events in classical probability and likewise the elements of P (H), namely the orthogonal projections are the events in quantum probability. The elements of Ω in classical probability are called the elementary outcomes and likewise the elements of

Quantum Mechanics for Scientists and Engineers

15

H after normalizing are the elementary outcomes in quantum probability. Just as an event E in classical probability is a set of elementary outcomes, likewise in quantum probability, an event P which is an orthogonal projection can be viewed as an aggregated of elements of H using the spectral diagonalization method:  P = |ψa >< ψa ||, |ψa >∈ H, < ψa |ψb >= δab a∈I

P is viewed as the aggregate of |ψa >, a ∈ I. Finally if μ is any measure on P (H), ie, μ : P (H)  → [0, 1] is a map such that if Pn , n = 1, 2, ... are pairwise orthogonal orthogonal projections in H, then μ( n Pn ) = n μ(Pn ), μ(0) = 0, μ(I) = 1, then Gleason’s theorem asserts that there exists a positive definite operator ρ of unit trace such that μ(P ) = T r(ρP ), P ∈ P (H) This proves the generality of the setup of a quantum probability space. An observable in the quantum probability space (Ω, F, P ) is a Hermitian operator X in H. It has spectral representation  X = xEX (dx) and the probability that when X is measured, the outcome will fall in the range (a, b] is T r(ρEX ((a, b])). In other words, FX (x) = T r(ρE((−∞, x])) is the probability distribution of X in the state ρ. This is to be compared with the classical setup where an observable X is a random variable in the the probability space (Ω, F, P ), ie a measurable map X : (Ω, F) → (R, B(R)). The probability distribution of X in the ”state” P , is FX (x) = P X −1 ((−∞, x])) = P ({ω ∈ Ω : X(ω) ≤ x}) To show that classical random variables are special cases of quantum observables, we define the spectral measure EX in L2 (R) by EX (B) = χB , B ∈ B(R)

ie EX (B) is multiplication by the indicator of the set B. Then we obviously have  xEX (dx) X= R

Then, for any functionf : R → R, we have

f (X) =

and hence (f (X)g)(y) =





f (x)EX (dx)

f (x)χ(x,x+dx] (y)g(y) = f (y)g(y)

ie f (X) is the operator of multilication by f (x). The crucial difference between classical and quantum probability is that in the former, all observables commute while in the latter, they need not commute. Thus, if X, Y are two classical random variables defined over the same probability space (Ω, F, P ), then both have same spectral measure:

X=



EX (B) = EY (B) = χB ,  xEX (dx), Y = yEY (dy), EX = EY

Hence, we can define the joint probability of X taking values in a Borel set B1 and Y taking values in a Borel set B2 as P (X ∈ B1 , Y ∈ B2 ) = P (X −1 (B1 ) ∩ Y −1 (B2 )) and in particular the joint probability distribution of the pair (X, Y ) is FXY (x, y) = P (X −1 ((−∞, x]) ∩ Y −1 ((−∞, y])) On the other hand, in quantum probability two observables X, Y need not commute, ie XY − Y X �= 0. Then the spectal measures of X, Y will differ:   X=

xEX (dx), Y =

yEY (dy), EX �= EY

Quantum Mechanics for Scientists and Engineers

16

and hence, we cannot speak of the event that (X, Y ) takes values in B1 × B2 in a given state. In fact, we have the basic Heisenberg uncertainty inequality T r(ρX 2 )T r(ρY 2 ) ≥ T r(ρXY )2 = (Re(T r(ρXY ))2 + (Im(T r(ρXY ))2 = (T r(ρ(XY + Y X)))2 /4 + (T r(ρ(XY − Y X)/i))2 /4 ≥ (T r(ρ[X, Y ]/i))2 /4

In particular, if [X, Y ] = ic, ie,

T r(ρX 2 ).T r(ρY 2 ) ≥ c2 /4

Thus, if X is replaced by X − T r(ρX) and Y by Y − T r(ρY ), we get that the product of the variances of X and Y in any state ρ is always ≥ c2 /4 and hence we cannot measure them simultaneously. We can measure any single Hermitian function of X, Y by not jointly two or more functions unless they commute. If X, Y commute, then they have the same spectral measure (jointly diagonable) and hence we can represent   X = xEX (dx), Y = f (x)EX (dx) for some real valued function f and hence T r(ρEX (B1 Y in B2 . If X, Y do not commute, then the quantity



f −1 (B2 ))) is the joint probability of X taking values in B1 and

ψX (t1 , t2 ) = T r(ρ.exp(i(t1 X + t2 Y ))) need not be positive definite and hence it is not a characteristic function of any probability measure in R2 . However, for any a, b ∈ R, ψ(t) = T r(ρ.exp(it(aX + bY )) is positive definite in t and hence the characteristic function of the observable aX + bY . Thus, any single real linear combination or more generally, any single Hermitian matrix valued function of (X, Y ) is measurable with a definite characteristic function but two different linear combinations may not have a joint characteristic function that is positive definite and hence the 2-D inverse Fourier transform of such a function need not be a probability measure on R2 . Exercise: Choose a general 2 × 2 state   p1 z ρ= z¯ p2 so that

p1 , p2 ≥ 0, p1 + p2 = 1, p1 p2 − |z|2 ≥ 0

These conditions ensure that ρ is a state in C2 , ie, positive definite of unit trace. Take the three Pauli matrices     0 1 0 −i , σ = , σ1 = 2 10 i 0   1 0 σ3 = 0 −1 Calculate the joint characteristic function of these three observables in the state ρ, ie, ψ(t1 , t2 , t3 ) = T r(ρ.exp(i(t1 σ1 + t2 σ2 + t3 σ3 ))) and show that there exists a ρ such that ψ is not positive definite. [20] [21] [22] tics. [23] [24]

The evolution of the density operator in the absence of noise. The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation for noisy quantum systems. Distinguishable and indistinguishable particles: The Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statisThe relationship between spin and statistics. Tensor products of Hilbert spaces: Let H1 , H2 be two Hilbert spaces. For (x1 , x2 ), (y1 , y2 ) ∈ H1 × H2 , define K((x1 , x2 ), (y1 , y2 )) =< x1 , y1 >1 < x2 , y2 >2

By Schur’s theorem, K is a positive definite kernel on H1 × H2 . Hence, by Kolmogorov’s consistency theorem, there exists a Gaussian process {λ(x1 , x2 ), (x1 , x2 ) ∈ H1 × H2 } on H1 × H2 such that E(λ(x1 , x2 )λ(y1 , y2 )) = K((x1 , x2 ), (y1 , y2 ))

Chapter

3

Open Quantum Systems, Tensor Products, Fock Space Calculus, Feynman’s Path Integral [20] [21] [22] tics. [23] [24]

The evolution of the density operator in the absence of noise. The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation for noisy quantum systems. Distinguishable and indistinguishable particles: The Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statisThe relationship between spin and statistics. Tensor products of Hilbert spaces: Let H1 , H2 be two Hilbert spaces. For (x1 , x2 ), (y1 , y2 ) ∈ H1 × H2 , define K((x1 , x2 ), (y1 , y2 )) =< x1 , y1 >1 < x2 , y2 >2

By Schur’s theorem, K is a positive definite kernel on H1 × H2 . Hence, by Kolmogorov’s consistency theorem, there exists a Gaussian process {λ(x1 , x2 ), (x1 , x2 ) ∈ H1 × H2 } on H1 × H2 such that Quantum Mechanics E(λ(x1 , x2 )λ(y1 , y2 )) = K((x1 , x2 ), (y1 , y2 )) or equivalently, if < ., . > denotes the standard inner product in the underlying probability space, we can write < λ(x1 , x2 ), λ(y1 , y2 ) >= K((x1 , x2 ), (y1 , y2 )) =< x1 , y1 >1 < x2 , y2 >2 λ is called the tensor product of H1 with H2 . It is easy to prove that the maps xk → λ(x1 , x2 ), k = 1, 2 are linear. This is proved by expanding � λ(c1 x + c2 y, z) − c1 λ(x, z) − c2 λ(y, z) �2

and likewise for the second argument. [25] Symmetric and antisymmetric tensor products of Hilbert spaces, the Fock spaces. Let H be a finite dimensional Hilbert space of dimension N . Choose an orthonormal basis {|ei >: i = 1, 2, ..., N } for this space. Let ⊗ be a tensor product of this Hilbert space with itself as many times as needed. Using the definition of the inner product < u1 ⊗ ... ⊗ un , v1 ⊗ ... ⊗ vn >= Πni=1 < ui , vi > we easily see that

{ei1 ⊗ ... ⊗ ein : 1 ≤ i1 , ..., in ≤ N }

is an orthonormal basis of the Hilbert space H⊗n and hence dimH⊗n = N n . The state |ei1 ⊗ ... ⊗ ein > corresponds to a state of n distinguishable particles in which the k th particle is in the state |eik >. Since there may be repetitions of an index in i1 , ..., ik , it follows that more than one particle can occupy a given state. Now suppose that the n particles are indistinguishable but are bosons, ie, more than one particle can occupy a given state. Then a given state of the n-particle system is specified by saying how many particles occupy each distinct state |ei >. Such a state  should be invariant under a permutation of the particles, ie, it must be superpositions of vectors in H⊗n of the form σ∈Sn |uσ1 ⊗ ... ⊗ uσn > where uk ∈ H. The set of all such superpositions forms a Hilbert space H⊗n , called the n-fold symmetric tensor product of H with itself. An orthonormal basis for this space is |er11 ...erNN >, r1 , ..., rN ≥ 1 N ⊗ ... ⊗ e⊗r > multiplied by the normalizing 0, r1 + ... + rN = n. where |er11 ...erNN > equals the symmetrization of |e⊗r 1 N factor (r1 !...rN !)−1 (n!/(r1 !...rN !))−1/2 = (n!r1 !...rN !)−1/2 . It is easy to see from this that dimH⊗sn is the number of ways in which n indistinguishable balls can be placed in N boxes and this number is the coefficient of xn in the expansion (1 + x + x2 + ...)N = (1 − x)−N . We thus get     −N N +n−1 (−1)n = dimH⊗s n = n n Finally, we consider the situation of Fermions which are also indistinguishable particles butno state can have more than one particle. Such a state consisting of n particles is a superposition of states of the form σ∈Sn sgn(σ)uσ1 ⊗ ... ⊗ uσn , ie, it is antisymmetric with respect to interchange of particles. The set of all such superpositions forms a subspace of H⊗n and is denoted by H⊗an and an orthonormal basis for this space is |ei1 ∧ ei2 ∧ ... ∧ ein >, 1 ≤ i1 < ... < in ≤ N , where  u1 ∧ ... ∧ un = (n!)−1/2 sgn(σ)uσ1 ⊗ ... ⊗ uσn σ∈Sn

uantum Mechanics for Scientists and Engineers

18 where

It is easily seen that if any two of the u�i s are identical, then this n-fold wedge product is zero and this corresponds to the fact that no state have more than one particle, also called the Pauli exclusion principle for Fermions. The above results are sometimes expressed by saying that bosonic wave functions are symmetric under particle interchange while Fermionic wave functions are antisymmetric under particle interchange. We are now in a position to derive the MaxwellBoltzmann, Bose-Einstein and Fermi-Dirac statistics respectively for distinguishable particles, Bosons and Fermions. Suppose our quantum system has energy levels E1 , E2 , ... with respective degeneracies of g1 , g2 , .... The number of ways of placing nk distinguishable articles in the k th energy level Ek is gknk . Hence, the total number of ways of placing n distinguishable particles in these states such that nk particles occupy the energy level Ek is Ω(n1 , n2 , ...) = (

n! (g n1 ...gknk ...) n1 !n2 !... 1

The most {nk } is obtained by maximizing Ω(n1 , n2 , ...) subject to the number and energy conprobable distribution  straints k nk = n, k gk nk = E. Using Stirling’s formula for the factorial and incorporating the above constraints using Lagrange multipliers yields the fact that the most probable distribution is obtained by maximizing    18 Quantum Mechanics F [n1 , n2 , ..., a, b] = (nk log(gk ) − nk log(nk )) − a( nk − n) − b( Ek nk − E) k

Setting

∂F ∂nk

k

k

= 0 gives log(gk ) − log(nk ) − 1 = a + bEk

or

nk = Cgk .exp(−bEk ), C = exp(−a − 1)

  This is called the Maxwell-Boltzmann distribution C, b are determined by the conditions k nk = n, k Ek nk = E. Suppose now that we are concerned with bosons, ie, indistinguishable particles with no constraint on the number of particles in each state. The number of ways in which nk bosons can be distributed in the energy level Ek having gk k −1 boxes is nk +g and hence if no constraint is placed on the number of particles, the most probable distribution {nk } nk with a constraint on the total energy E is obtained by maximizing  nk + gk − 1  F = log Ek nk − E) − b( nk k

k

or using Stirling’ formula and neglecting constants,   ((nk + gk )log(nk + gk ) − nk log(nk )) − b( Ek nk − E) F = k

Setting

∂F ∂nk

k

= 0 gives log(nk + gk ) − log(nk ) − bEk = 0

or

1 + gk /nk = bEk , nk = gk /(exp(bEk ) − 1)

If we put a constraint on the total number of particles n, then the function to be maximized is    ((nk + gk )log(nk + gk ) − nk log(nk )) − a( nk − n) − b( Ek nk − E) F = k

Setting

∂F ∂nk

or

k

k

= 0 then gives log(nk + gk ) − log(nk ) − a − bEk = 0 nk =

gk exp(a + bEk ) − 1

This is the Bose-Einstein distribution. The constants a, b are determined from the constraints   nk = n, E k nk = E k

k

[26] Coherent/exponential vectors in the Fock spaces. Let H0 = L2 (R) and consider the standard position and momentum operators, q, p = −id/dq acting in this space. Construct the annihilation and creation operators a, a∗ in this space as √ √ a = (q + ip)/ 2, a∗ = (q − ip)/ 2 Show that

[a, a∗ ] = 1

a Quantum Mechanics for Scientists and Engineers

19

Now consider copies of H0 and in the k th copy, let the annihilation and creation operators be denoted by ak , a∗k . Thus, in the tensor product of p such copies H0⊗p , we have defined operators a1 , ..., ap , a∗1 , ..., a∗p satisfying [ai , a∗j ] = δij , [ai , aj ] = 0, [a∗i , a∗j ] = 0. Let |n1 , ..., np > be the tensor product of the state |n1 >, ..., |np > where |nk > is the state in the k th √ √ copy of H0 such that ak |nk >= nk |nk − 1 >, a∗k |nk >= nk + 1|nk + 1 > and a∗k ak |nk >= nk |nk >, ak a∗k |nk >= p (nk + 1)|nk >. Define for z ∈ C , the state   |e(z) >= z1n1 ...zpnp |n1 , ..., np > / n1 !...np ! n1 ,...,np ≥0

=



n1 ,...,np ≥0

∗np 1 z1n1 ...zpnp a∗n 1 ...ap |0 > /n1 !...np !

= exp(z.a∗ )|0 >, z.a∗ = z1 a∗1 + ... + zp a∗p Show using

< n1 , ..., np |m1 , ..., mp >= Πpk=1 δ[nk − mk ] = δ[n − m]

that for z, u ∈ Cp ,

< e(z)|e(u) >= exp(< z, u >)

Now, define for z ∈ Cp , n ≥ 0, 

ψ(z ⊗n ) =

√

n! z1n1 ...zpnp |n1 , ..., np > n1 !...np !



n1 +...+np =n

Then show using the multinomial theorem that for z, u ∈ Cp , < ψ(z ⊗n ), ψ(u⊗n ) >=< z, u >n Now extend the map ψ linearly to the Boson Fock space Γs (Cp ) =



(Cp )⊗s n

n≥0

by defining for |f (z) >=

 z ⊗n √ ∈ Γs (Cp ) n! n≥0

ψ(|f (z) >) = |e(z) >

or equivalently,

√ ψ(z ⊗n / n!) = n



z n1 ...zp p 1 |n1 , ..., np >, n ≥ 0 n1 !...np ! n1 +...+np =n

Show that ψ defines a Hilbert space isomorphism between Γs (Cp ) and L2 (Rp ). This gives a physical interpretation of the coherent states in Boson Fock space in terms of the p-dimensional Harmonic oscillator algebra. [25] Creation, conservation and annihilation operators in the Boson Fock space. [26] The general theory of quantum stochastic processes in the sense of Hudson and Parthasarathy. [27] The quantum Ito formula of Hudson and Parthasarathy. [28] The general theory of quantum stochastic differential equations. [29] The Hudson-Parthasarathy noisy Schrodinger equation and the derivation of the GKSL equation from its partial trace. [30] The Feynman path integral for solving the Schrodinger equation. Let X(t), t ≥ 0 be a Markov process with values in R having generator K, ie E[φ(X(t + dt))|X(t) = x] = φ(x) + Kφ(x)dt + o(dt) Note that by including generalized functions such as the Dirac Delta function and its derivatives, we may represent K as an integral kernel:  Kφ(x) = K(x, y)phi(y)dy We define

u(s, t, x) = E[f (X(t))exp(



s

t

V (X(s))ds)|X(s) = x], 0 ≤ s ≤ t

Quantum Mechanics for Scientists and Engineers

20

, s

Then an easy application of the Markov property shows that

and hence,

u(s, t, x) = (1 + V (x)ds)E[(u(s + ds, t, X(s + ds))|X(s) = x] + o(ds)  = (1 + V (x)ds)(u(s, t, x)ds + u,s (s, t, x) + ds K(x, y)u(s, t, y)dy) + o(ds) ∂s u(s, t, x) + V (x)u(s, t, x) +

Further,



K(x, y)u(s, t, y)dy = 0

u(t, t, x) = f (x)

In particular, if X(t) is Brownian motion, we have K(x, y) = (1/2)δ �� (x − y) and the above becomes

∂s u(s, t, x) + V (x)u(s, t, x) + (1/2)∂x2 u(s, t, x) = 0

which is the Feynman-Kac formula. Since we are assuming that the Markov process is time homogeneous, ie its generator K is time independent, it follows that u(s, t, x) = u(0, t − s, x). We denote this by v(t − s, x) and then we get  ∂t v(t, x) = V (x)v(t, x) + K(x, y)v(t, y)dy, t ≥ 0, v(0, x) = f (x) √ −1, ie define w(t, x) = v(it, x) in the above formula to get  i∂t w(t, x) = −V (x)w(t, x) − K(x, y)w(t, y)dy

Feynman formally replaced t by it, i =

Now replace V by −V to get the generalized Schrodinger equation  i∂t w(t, x) = V (x)w(t, x) − K(x, y)w(t, y)dy, w(0, x) = f (x) where w(t, x) = E[exp(− = E[exp(−i





t

it

V (X(s))ds)f (X(it))|X(0) = x]

0

V (X(is)ds)f (X(it))|X(0) = x]

0

Formally, this formula can be interpreted as follows: v(t, x) is approximated as  t v(t, x) = E[exp(− V (X(s))ds)f (X(t))|X(0) = x] ≈ 0



where

exp(

n 

(−V (xk )) + (log(K))(xk , xk+1 ))δsk )f (sn )Π0≤k≤n dxk

k=0

0 = s0 < s1 < ... < sn = t

and log(K) is the operator logarithm of K (not log(K(x, y))). [31] Comparison between the Hamiltonian (Schrodinger-Heisenberg) and Lagrangian (path integral) approaches to quantum mechanics. The Schrodinger-Heisenberg approaches to quantum mechanics are Hamiltonian approaches. Feynman proposed an alternative approach to non-relativistic quantum mechanics that is based on the Lagrangian. To see how this proceeds, assume that the Hamiltonian of the system is H(t, q, p) . The Lagrangian is then L(t, q, q � ) = (p, q � ) − H(t, q, p) with p=

∂L ∂q �

Quantum Quantum Mechanics Mechanics for Scientists and Engineers

Thus,

2121

< t + dt, q �� |t, q � >=< t, q �� |exp(−iH(t, q, p)dt)|t, q � >

We can applying the commutation relations between q and p, assume that in the Hamiltonian H(t, q, p), all the p� s appear to the left of all the q � s. Then we get  < t + dt, q �� |t, q � >= < t, q �� |t, p� > dp� < t, p� |exp(−iH(t, q, p)dt)|t, q � > =C =C





exp(i(q”, p� ))exp(−iH(t, q � , p� )dt)|t, q � > dp�

exp(i((q �� , p� ) − H(t, q � , p� )dt)).exp(−i(q � , p� ))dp�

=C



exp(i[(q �� − q � , p� )/dt − H(t, q � , p� )]dt))dp�

By composing these infinitesimal amplitudes, we get the transition amplitude for finite time as

C

K(t2 , q2� |t1 , q1� ) =< t2 , q2� |t1 , q1� >=



exp(i

q(t1 )=q1 ,q(t2 )=q2



t2

t1

(p(t), q � (t)) − H(t, q(t), p(t)))dt)Πt1 ≤t≤t2 dq(t)dp(t)

This is indeed the formula for the time evolution operator kernel in the position representation, ie  t2 H(t)dt)}|t1 , q1� > K(t2 , q2� |t1 , q1� ) =< t2 , q2� |T {exp(−i t1

where T {} is the time ordering operator. In the special case, when the Hamiltonian has the form H(t, q, p) = p2 /2m + V (q) the integral over p becomes a Gaussian integral and therefore it can be replaced by evaluating the action integral at the stationary point, ie at p(t) given by d ((p(t), q � (t)) + p2 (t)/2m) = 0 dp(t) ie

q � (t) = p(t)/m, p(t) = mq � (t)

Thus, in this special case, K(t2 , q2� |t1 , q1� ) = C



q(t1 )=q1 ,q(t2 )=q2

exp(i



t2

t1

(mq � (t)2 /2 − V (q(t)))dt)πt1 at time t2 in which the field is exactly another given function φf (r), r ∈ R3 of the spatial variables. The corresponding Hamiltonian will then have the form   H(t) = H0 (φ(x), ∇φ(x), π(x))d3 x − f (x)V (φ(x))d3 x (Note: x = (t, r), d3 x = d3 r, d4 x = d3 xdt = d3 rdt) where H0 is the Hamiltonian density corresponding to the Lagrangian density L0 (ie obtained by applying the Legendre transform to L0 ):  H0 = (1/2) (π 2 + (∇φ)2 + m2 φ2 )d3 x The transition probability amplitude from |φi >→ |φf > in the time duration [t1 , t2 ] can be calculated using the Feynman path integral formula: < φf |S[t2 , t1 ]|φi >=   exp(−( Ldtd3 x))Πr∈R3 ,t∈(t1 ,t2 ) dφ(x) C =C where



φ(t1 ,.)=φi ,φ(t2 ,.)=φf

exp(iS0 )(1 + i



[t1 ,t2 ]×R3

f (x)V (φ(x))dtd3 x + (i2 /2!) S0 =



L0 dtd3 x = (1/2)





f (x)f (x)� φ(x)φ(x� )dtdt� d3 xd3 x� + ..)Πdφ(x)

(∂μ φ∂ μ φ − m2 φ2 )dt d3x

By expanding V (φ(x) as a power series in φ(x), the computation of the above path integral reduces to computing the moments of a complex infinite dimensional zero mean Gaussian distribution sinc S0 is a quadratic functional of φ. In particular, we note that the odd moments of a symmetric Gaussian distribution are zero and the even moments can be computed by summing the products of the second moments taken over all partitions of the product fields into pairs. Thus, computation of the second moments of such a Gaussian distribution beomes significant, ie,  D(x, y) = C exp(iS0 )φ(x)φ(y)Πz∈R4 dφ(z) if we are interested in transitions from t = −∞ to t = +∞. From standard methods in quantum mechanics, it is easily seen that D(x, y) =< 0|T {φ(x)φ(y)}|0 >

provided that we use the interaction representation which removes the effect of the unperturbed Hamiltonian H0 . If we use the Schrodinger representation, then we would have to compute D as D(x, y) =< 0|T {U (∞, −∞)φ(x)φ(y)}|0 >= < 0|U (∞, tx )φ(x)U (tx , ty )φ(y)U (ty , −∞)|0 >

Quantum )}|0 >= Mechanics for Scientists and Engineers

26

assuming tx ≥ ty and where U is the unperturbed Schrodinger evolution operator. Here |0 > is the vacuum state of the field. The function D(x, y) is called the propagator. The complete propagator taking into account interactions is defined as  Dc (x, y) = C exp(iS)φ(x)φ(y)Πz dφ(z) where

S = S0 +



f (x)V (φ(x))d4 x =



Ld4 x

We can write a perturbative expansion for Dc as  Dc (x, y) = exp(iS0 )(1 + iS1 + i2 S12 /2! + ..)φ(x)φ(y)Πz dφ(z) where S1 =



f (x)V (φ(x))d4 x

is the perturbation to the action caused by external field coupling. Even if there is no external field, but there is a small perturbation to the Lagrangian density/Hamiltonian density, the above series expansion can be used to determine the complete propagator It was Feynman’s genius to recognize that the various perturbation terms in Dc can be calculated easily using a digrammatic method which could be applied to more complex situations like quantum electrodynamics wherein the quantum fields are the electromgnetic four potential Aμ (x) and the Dirac four component spinor wave function ψ(x). Let us now formally compute the propagator of the unperturbed KG field:  S0 = φ(x)[(1/2)∂μ ∂ μ − m2 /2)δ 4 (x − y)]φ(y)d4 xd4 y =



φ(x)K(x, y)φ(y)d4 xd4 y

and hence a simple Gaussian second moment evaluation gives  D(x, y) = exp(iS0 [φ])φ(x)φ(y)Πz dφ(z) = C1 (det(iK))−1/2 .K −1 (x, y) In other words D(x, y) is proportional to K −1 (x, y) where K −1 is the inverse Kernel of K:  K −1 (x, y)K(y, z)d4 y = δ 4 (x − z) We can write K(x, y) = K(x − y) and then defining its four dimensional Fourier transform:  ˆ K(p) = K(x)exp(−ip.x)d4 x, p.x = pμ xμ = p0 x0 − p1 x1 − p2 x2 − p3 x3 we get Clearly, and hence Thus,

K −1 (p) = 1/K(p) K(x) = (1/2)∂μ ∂ μ − m2 /2)δ 4 (x) ˆ K(p) = (pμ pμ − m2 )/2 ˆ D(p) =

where Finally,

C0 p2 − m2

p2 = pμ pμ = p02 − p12 − p22 − p32 D(x, y) = D(x − y) = C0 /(2π)4



exp(ip.x) 4 d p p2 − m2

The corrected (complete) propagator:   Dc (x, y) = exp(iS0 [φ])φ(x)φ(y)(1 + i f (x)V (φ(z))d4 z + ...)Πu dφ(u)

Quantum Mechanics for Scientists and Engineers

27

Clearly, we can write this in operator kernel notation as Dc = D + DΣD + DΣDΣD + ... using the property of moments of a Gaussian distribution. For example, if V (φ) = φ4 and f = c0 , then in the Gaussian average of the  product φ(x)φ(y) f (z)φ(z)4 d4 z, we get the coupling terms 4 < φ(x)φ(z) >< φ(z)2 >< φ(z)φ(y) > so if we define Σ(z) = 4f (z)0 < φ(z)2 >, we can write   < φ(x)φ(y)( f (z)φ(z)4 d4 z) >= D(x − z)Σ(z)D(z − y)d4 z Likewise, for the next perturbation term



f (z)φ(z)4 d4 z)2 >=

< φ(x)φ(y)( 

f (z1 )f (z2 ) < φ(x)φ(y)φ4 (z1 )φ4 (z2 ) > d4 z1 d4 z2

Again, this can be expressed using the Gaussian moments formula as a sum of terms of the form  f (z1 )f (z2 ) < φ(x)φ(z1 ) >< φ(z1 )3 φ(z2 )3 >< φ(z2 )φ(y) > d4 z1 d4 z2 and



f (z1 )2 < φ(x)φ(z1 ) >< φ(z1 )2 φ(z2 )4 >< φ(z1 )φ(y) > d4 z1 d4 z2

etc. Now, each term < φ(z1 )m φ(z2 )m > is a product of propagators D(z1 − z2 ) and D(0) so the above general form is valid. [b] Quantization of the electromagnetic field. The Lagrangian density is 1 LF = − Fμν F μν , Fμν = Aν,μ − Aμ,ν 4 Thus, We get

F0r = Er , F12 = −B3 , F23 = −B1 , F31 = −B2 LF =

1 2 (E − B 2 ) 2

as required. We compute the canonical momenta: πr =

∂LF = −F0r ∂Ar,0

and π0 =

∂LF =0 ∂A0,0

This is inconsistent with the canonical commutation relations [Aμ (t, r), πν (t, r )] = iδνμ δ 3 (r − r ) Hence we must use the Dirac brackets which are modifications of the Poisson/Lie bracket when constraints are taken into account. Thie first constraint is π0 = 0 and the second constraint is obtained from the equations of motion: ∂r

∂LF = −J 0 ∂A0,r 

where J μ the four current density is a matter field unlike the Aμ s which are the Maxwell fields. These equations can be expressed as χ1 = ∂r πr + J 0 = 0 and since time derivatives do not appear here, this equation should be regarded as a constraint, ie, a relationship between the matter field and the electromagnetic field. The above equation is obtained by adding to the field Lagrangian density, the matter-field interaction Lagrangian Lint = −J μ Aμ

Now, we work in the Coulomb gauge (we are free to impose a gauge condition on the potentials that leaves the actual electric and mangetic field invariant). In this gauge, divA = 0, ie, χ2 = Ar,r = 0

27 Quantum Mechanics for Scientists and Engineers

28

The Maxwell equations in this gauge imply that which has solution

∇2 A0 = −J 0

A0 (t, r) =



J 0 (t, r� ) 3 � d r |r − r� |

and since both sides of the above equation are taken at the same time, we can regard A0 as a matter field. Hence, the quantized electromagnetic field is described by only three position fields Ar , r = 1, 2, 3 and once we impose the Coulomb gauge condition, there are only two degrees of freedom for the position fields. We now calculate the field Hamiltonian when it interacts with matter. The Hamiltonian density is with L = LF + Lint , H = πr Ar,0 − L = −F0r Ar,0 + (1/4)Fμν F μν + J μ Aμ 1 F0r (Ar,0 + A0,r ) + (1/4)Frs Frs + J μ Aμ 2 Thus, making use of the matter equation A0,rr = −J 0 , the constraint χ1 and neglecting a 3-dimensional divergence (which will not affect the total Hamiltonian), we get H=

1 F0r F0r + F0r A0,r + (1/4)Frs Frs + J μ Aμ 2

= π 2 /2 + (∇ × A)2 /2 − F0r,r A0 + J μ Aμ

= (1/2)(π 2 + (∇ × A)2 ) + πr,r A0 + J μ Aμ = (1/2)(π 2 + (∇ × A)2 ) − J 0 A0 + J μ Aμ

The term J 0 A0 is a pure matter field while the two terms within the bracket are pure field terms. This simplifies to H = (1/2)(π 2 + (∇ × A)2 ) − J.A where

J.A = J r Ar

There is no pure matter term in this Hamiltonian. We define π⊥ = π − ∇A0 ie

π⊥r = πr − A0,r

Then,

divπ⊥ = πr,r − A0,rr = −J 0 + J 0 = 0

Thus π⊥ is a solenoidal field. Then, we can express H=

1 2 (π + (∇ × A)2 ) − J.A + (∇A0 )2 /2 − − − (1) 2 ⊥

since the term (∇A0 , π⊥ ) on perfoming a 3 − D integration is zero because it is a perfect 3-D divegence: (∇A0 , π⊥ ) = div(A0 π⊥ ) (1) is our final form of the Hamiltonian of the electromagnetic field interacting with an external current source. We note that the last term (∇A0 )2 /2 is a pure matter field. Hence, if we are bothered only about the electromgnetic field and its interaction with matter, the Hamiltonian density is HF = (1/2)(π 2 + (∇ × A)2 ) where we have renamed π⊥ as π for convenience of notation. Our constraints are divπ = 0, divA = 0.

Quantum Mechanics for Scientists and Engineers 28

29

Dirac brackets for constraints: Suppose Q1 , ..., Qn , P1 , ..., Pn are the unconstrained positions and momenta of a system. The constraints are Qj = Pj = 0, j = n+1, ..., n+p. Without loss of generality, we are choosing our constrained variables as new positions and momenta. The Poisson bracket relations are {f, g} =

n+p  i=1

f,Qi g,Pi − f,Pi g,Qi )

In particular, we get the contradiction f,Qi = −f,Pi , {f, Pi } = f,Qi , i > n since Qi = Pi = 0, i > n. In order to rectify this problem, Dirac introduced a new kind of bracket defined as follows: Let χij = {ηi , ηj } = Jij J is the standard symplectic matrix of size 2p × 2p. where

η = [Qn+1 , ..., Qn+p , Pn+1 , ..., Pn+p ]T Qn+i , Pn+i , i = 1, 2, ..., p, ie ηi are functions of Qi , Pi , i = 1, 2, ..., n and the bracket {., .}ef f is calculated using Qi , Pi , i = 1, 2, ..., n and regarding Qn+i , Pn+i as functions of Qj , Pj , j ≤ n. The bracket {f, g}P is computed using Qi , Pi , i ≤ n and taking Qn+i = 0, Pn+i = 0: n  (f,Qi g,Pi − f,Pi g,Qi ) {f, g}P = i=1

We have

C = χ−1 = −J

as 2p × 2p matrices. Then, the Dirac bracket is defined as

{f, g}D = {f, g} + We see that for k ≤ n,

 {f, ηi }Jij {ηj , g} i,j

{f, Qk }D = {f, Qk } = −f,Pk

since

{ηj , Qk } = 0, k ≤ n

Note that {., .} is the unconstrained Poisson bracket. Again, we note that {f, Pk }D = {f, Pk } = f,Qk since {Pk , ηj } = 0, k ≤ n. Further, for i, j ≥ 1, we have {f, ηi }D = {f, ηi } + = {f, ηi } − since



 k,l

 k,l

{f, ηk }Jkl {ηl , ηi }

{f, ηk }Ckl χli = 0

Ckl χli = δki

l

We note that

{f, g}D = {f, g} −

 {f, ηi }Jij {ηj , g} i,j

   (f,Qi + (f,ηj ηj,Qi ))(g,Pi + g,ηj ηj,Pi ) = i≤n

j

−interchangeof f andg

j

Quantum Mechanics for Scientists and Engineers

30

+ =







f,ηi Jij g,ηj +

i

(f,Qi +



{f, ηi }Jij {ηj , g}

(f,ηj ηj,Qi ))(g,Pi +

j

i≤n



g,ηj ηj,Pi )

j

This formula tells us that the Dirac bracket between two observables is calculated using the Poisson bracket w.r.t. the unconstrained variables and by regarding the constrained variables as functions of the unconstrained variables. More generally, suppose Q = (Q1 , ..., Qn ), P = (P1 , ..., Pn ) are arbitrary canonical coordinates and the constraints are of the form χi (Q, P ) = 0, i = 1, 2, ..., r Define the Poisson bracket {., .} as usual w.r.t Q, P , and then define the Dirac bracket {f, g}D = {f, g} − where

r 

i,j=1

{f, χi }Cij {χj , g}

((Cij )) = (({χi , χj }))−1

Then,

{f, χk }D = {f, χk } − = {f, χk } − as required. Further,

 i,j

 i

{f, χi }Cij {χj , χk }

{f, χi }δik = 0

{f, Qk }D = −f,Pk −



f,Pi Cij χj,Pk

i,j

It is then not hard to show that the rhs is the same as −dPk f , ie, the partial derivative of f w.r.t. Pk where we regard f as an independent function of Q, P and the χ�i s and then defined  dPk f = f,Pk + f,χj χj,Pk j

Now, we evaluate the Dirac bracket between πi and A taking into account the constraints: j

χ1 = πi,i = 0, χ2 = Ai,i = 0 We get

{χ1 (t, r), χ2 (t, r� )} = i∇2 δ 3 (r − r� )

The inverse kernel of the rhs is K(r − r� ) = i/4π|r − r� |. Further,

3 3 {Am (t, r), χ1 (t, r� )} = −iδkm δ,k (r − r� ) = iδ,m (r − r� )

{Am (t, r), χ2 (t, r� )} = 0, {χ1 (t, r), πm (t, r� )} = 0,

Hence,

k 3 3 δ,k (r − r� ) = iδ,m (r − r� ) {χ2 (t, r), πm (t, r� )} = iδm

{Am (t, r), πk (t, r� )}D = iδkm δ 3 (r − r� ) − = iδkm δ 3 (r − r� ) + = iδkm δ 3 (r − r� ) −



(





d3 r�� d3 r��� {Am (t, r), χ1 (t, r�� )}K(r�� − r��� ){χ2 (t, r��� ), πk (t, r� )}

3 3 d3 r�� d3 r��� δ,m (r − r�� )K(r�� − r��� )δ,k (r��� − r� )

∂2 K(r�� − r��� ))δ 3 (r − r�� )δ 3 (r��� − r� )d3 r�� d3 r��� ∂xm ∂xk

= iδkm δ 3 (r − r� ) −

∂2 K(r − r� ) ∂xm ∂xk

Quantum Mechanics for Scientists and Engineers 30

31

= iδkm δ 3 (r − r� ) + K,mk (r − r� )

where

K(r) = i/4π|r|

Quantum electrodynamics using creation and annihilation operators for photons, electrons and positrons: We work in the Coulomb gauge so that divA = 0 and this implies ∇2 A0 = −J 0 , ie, A0 is a matter field. The Maxwell wave equation for A in the absence of matter, ie charge and current densities is given by ∇2 A − A,0 = 0 and the general solution to this is  Ak (t, r) = er (K, σ)[a(K, σ)exp(−i(|K|t − K.r)) + e¯r (K, σ)a(K, sigma)∗ exp(i(|K|t − K.r))]d3 K Here, the summation 3 is over σ = 1, 2 corresponding to only two linearly independent polarizations of the photon, ie, divA = 0 implies r=1 K r er (K, σ) = 0. The energy of the electromagnetic field in the Coulomb gauge is   HF = (1/2) (E 2 + B 2 )d3 x = (1/2) [(A2,t + (∇ × A)2 ]d3 x =



2|K|2 |e(K, σ)|2 a(K, σ)∗ a(K, σ)d3 K

once we make use of the fact that |K × e(K, σ)| = |K||e(K, σ)|. For this to be interpretable as the sum of energies of harmonic oscillators, each oscillator in the spatial frequency domain having energy |K|, ie, the frequency of the wave. This means that we must have |e(K, σ)| = (2|K|)−1/2 in order to ensure that

HF =



|K|a(K, σ)∗ a(K, σ)d3 K

We can cross check this result as follows. Assuming that the a(K, σ)� s satisfy the canonical commutation relations: [a(K, σ), a(K � , σ � )∗ ] = δ 3 (K − K � )δσ,σ it follows from the Heisenberg equations of motion that a(t, K, σ),t = i[HF , a(t, K, σ)] = −i|K|a(t, K, σ), a∗ (t, K, σ),t = i[HF , a(t, K, σ)∗ ] = i|K|a∗ (t, K, σ) These equations imply

a(t, K, σ),t = −|K|2 a(t, K, σ)

a∗ (t, K, σ),tt = −|K|2 a∗ (t, K, σ)

which are the correct equations for the spatial Fourier transform of the vector potential arrived from the wave equation. Another way to check these commutation relations which we leave as an exercise, is to start with the Lagrangian density LF = (1/2)(A,t )2 − (1/2)(∇ × A)2 so that the momentum density is πk (t, r) =

∂LF = Ak,t ∂Ak,t

then apply the canonical commutation relations k 3 [Ak (t, r), πm (t, r� )] = iδm δ (r − r� )

and verifiy that these relations are satisfied by the above Fourier integral representation of A assuming the canonical commutation relations between a(K, σ) and a(K � , σ � ). We leave this verification as an exercise to the reader.

Quantum Mechanics for Scientists and Engineers 31

32

Now consider the second quantized Dirac field described by the four component ield operators ψ(x), ψ(x)∗ where x = (t, r), t ∈ R, r ∈ R3 . In the absence of any classical or quantum electromagnetic field ,ψ satisfies the Dirac equation [iγ μ ∂μ − m]ψ(x) = 0 or equivalently,

[γ μ pμ − m]ψ = 0, pμ = i∂μ

The solutions to ψ are plane waves:  ψ(x) = (u(P, σ)a(P, σ)exp(−ip.x) + v(P, σ)b(P, σ)∗ exp(ip.x))d3 P where

p.x = pμ xμ = E(P )t − P.r, E(P ) =

 m2 + P 2 , p = (π μ ) = (E, P ), u(P, σ), v(P, σ) ∈ C4

Here, the summation is over σ = ±1/2 corresponding to the fact that Dirac’s equation can be expressed as [i∂0 − (α, P ) − βm]ψ(x) = 0, P = −i∇ and hence if P denotes an ordinary 3-vector (not an operator), then u(P )exp(−ip.x) satisfies the Dirac equation iff [p0 − (α, P ) − βm]u(P ) = 0 and likewise, v(P )exp(ip.x) satisfies the Dirac equation iff (−p0 + (α, P ) − βm)v(P ) = 0 Thus, u(P ) is an eigenvector of the matrix HD (P ) = (α, P ) + βm with eigenvalue p0 and v(−P ) is an eigenvector of HD (P ) with eigenvalue p0 . matrix, it has four real eigenvalues taking all multiplicities into account. These Now since HD (P ) is a 4 × 4 Hermitian √ eigenvalues are ±E(P ), E(P ) = m2 + P 2 with each one have a multiplicity of two. We denote the corresponding mutually orthogonal eigenvectors by u(P, σ), v(−P, σ), σ = ±1/2. On applying second quantization, the free Dirac Hamiltonian becomes  HDQ = ψ(x)∗ ((α, −i∇) + βm)ψ(x)d3 x and it is easy to verify that the normalizations of u(P, σ) and v(P, σ) are chosen so that  HDQ = E(P )(a(P, σ)∗ a(P, σ) + b(P, σ)b(P, σ)∗ )d3 P

and if we postulate the anticommutation relations {a(P, σ), a(P � , σ � )∗ } = {b(P, σ), b(P � , σ � )∗ } = δσ,σ δ 3 (P − P � ) then and only then we can ensure the canonical anticommutation relations (CAR) {ψl (t, r), πm (t, r� )} = iδlm δ 3 (r − r� ) where πm is the canonical momentum associated with the canonical position field ψm . From the free Dirac Lagrangian density LD = ψ(x)∗ (i∂0 − (α, −i∇) − βm)ψ(x) we infer that

= ψ(x)∗ γ 0 (iγ μ ∂μ − m)ψ(x) πm (x) =

so that the CAR gives

∂LD = iψl (x)∗ ∂ψl,0

{ψl (t, r), ψm (t, r� )∗ } = δlm δ 3 (r − r� )

Thus in particular, we can subtract an infinite constant from the second quantized Dirac Hamiltonian to get  HDQ = E(P )(a(P, σ)∗ a(P, σ) − b(P, σ)∗ b(P, σ)) − − − (1) This equation has the following nice interpretation: a(P, σ)∗ creates an electron with momentum P and spin σ, a(P, σ) annihilates an electron with momentum P and spin σ. b(P, σ)∗ creates positron with momentum P and spin σ while

Quantum Mechanics for Scientists and Engineers 32

33

b(P, σ) annihilates a positron with momentum P and spin σ. a(P, σ)∗ a(P, σ) is the number operator density for electrons and b(P, σ)∗ b(P, σ) is the number operator for positrons. Since the presence of an additional electron increases the energy of the Dirac sea of electrons by E(P ) while the presence of an additional positron decreases the energy of the Dirac sea by E(P ), equn (1) has the correct physical interpretation for the energy of the second quantized Dirac field. Now suppose we have a collection of photons, electrons and positrons. The total Lagrangian density is then L = LEM + LD + Lint

so that

= (−1/4)Fμν F μν + ψ ∗ γ 0 (γ μ (i∂μ + eAμ ) − m)ψ LEM = (−1/4)Fμν F μν , LD = ψ ∗ γ 0 (iγ μ ∂μ − m)ψ, Lint = −J μ Aμ ,

J μ = −eψ ∗ γ 0 γ μ ψ

J μ is the Dirac four current density. It is easily verified to be conserved even when an electromagnetic field is present. In other words, we can verify using the Dirac equation [γ μ (i∂μ + eAμ ) − m]ψ = 0 that

∂μ J μ = 0

ie, the current is conserved. We can further show that the matrices K μν = (−1/4)[γ μ , γ ν ] satisfy the same commutation relations as do the standard skew-symmetric generators of the Lorentz group do. Hence these matrices furnish a representation of the Lie algebra of the Lorentz group. Let D denote the corresponding representation of the Lorentz group. D is called the Dirac spinor representation of the Lorentz group and if Λ is any Lorentz transformation, we write D(Λ) = exp(ωμν K μν ) where

Λ = exp(ωμν Lμν )

with ω a skew symmetric matrix and Lμν the standard generators of the Lorentz group: (Lμν )αβ = η μα η νβ − η μβ η να Further, we note the following:

D(Λ)γ μ D(Λ)−1 = Λμν γ ν

and hence, the Dirac equation is invariant under Lorentz transformations ie if xμ → Λμν xν and ψ(x) → D(Λ)ψ(x), Aμ → Λμν Aν , then the Dirac equation remains invariant. Further, the existence of the positron follows from the fact that if we start with the Dirac equation, conjugate it and multiply by the unitary matrix iγ 2 , then we get γ μ )(iγ 2−1 )(−i∂μ + eAμ ) − m]iγ 2 ψ¯ = 0 [(iγ 2 )(¯ It is easily verified that this equation is the same as [γ μ (i∂μ − eAμ ) − m]ψ˜ = 0 where

ψ˜ = iγ 2 ]ψ¯

In other words ψ˜ satisfies the Dirac equation in an electromagnetic field but with the charge e replaced by −e or equivalently, −e replaced by e. This observation led Dirac to conclude the existence of the positron, namely the antiparticle of the electron, having the same mass but opposite charge as that of the electron. The positron was discovered in an accelerator later by Anderson. Another property of the Dirac equation in an external electromagnetic field is obtained by considering (γ μ (i∂μ + eAμ ) + m)(γ ν (i∂ν + eAν ) − m)ψ = 0

Quantum Mechanics for Scientists and Engineers 33

34

If Aμ = 0, this reduces to the free KG equation [∂μ ∂ μ + m2 ]ψ = 0 since In case however Aμ �= 0, we get or equivalently,

{γ μ , γ |nu } = 2η μν [γ μ γ ν (i∂μ + eAμ )(i∂ν + eAν ) − m2 ]ψ = 0

((1/2){γ μ , γ ν } + (1/2)[γ μ , γ ν ])[(i∂μ + eAμ )(i∂ν + eAν ) − m2 ]ψ = 0

or since {γ μ , γ ν } = 2η μν , we get

[(∂μ − eAμ )(∂ μ − ieAμ ) + m2 + (1/4)[γ μ , γ ν ][∂μ − ieAμ , ∂ν − ieAν ]]ψ = 0 or since

we can write

[∂μ − ieAμ , ∂ν − ieAν ] =

−i(Aν,μ − Aμ,ν ) = −iFμν [(∂μ − eAμ )(∂ μ − ieAμ ) + m2 − (i/4)[γ μ , γ ν ]Fμν ]ψ = 0

This equation is the same as the KG equation in an external electromagnetic field obatined by replacing ∂μ by ∂mu −ieAμ except for the last term which displays explicitly the interaction of the electromagnetic field Fμν (whose non-zero components are the electric and magnetic fields) with the spin of the electron described by the antisymmetric ”spin tensor” (−i/4)[γ μ , γ ν ]. Exercise: Write down explicitly in terms of the components of the electric and magnetic fields, the spin-field interaction component and display in particular, the spin magnetic dipole moment and the spin electric dipole moment. The photon and electron propagator: We make this calculation using first the Feynman path integral for fields and then leave as an exercise to demonstrate the same result using the operator expansion of the fields. First, note that the photon propagator is defined in space-time as Dμν (x, y) =< 0|T {Aμ (x)Aν (y)}|0 > and the electron propagator as

Slm (x, y) =< 0|T {ψl (x)ψm (y)∗ }|0 >

Here we are using the Lorentz gauge in which case even A0 is a component of the electromagnetic field potential, not a matter field. In the absence of matter, Aμ (x) satisfies the wave equation ∂μ ∂ μ Aα = 0 and this equation has solutions    Aμ (x) = [a(K, σ)eμ (K, σ)exp(−ik.x)/ 2|K| + a(K, σ)∗ e¯μ (K, σ)exp(ik.x)/ 2|K|]d3 K

where

The Lorentz gauge condition implies

k.x = kμ xμ = |K|t − K.r ∂μ Aμ = 0 kμ eμ (K, σ) = 0

which means that eμ has three degrees of freedom. Formally, we do not take the canonical momentum π0 as zero, even though the Lagrangian density LEM of the electromagnetic field does not depend on A0,0 and hence implies π0 = 0. The way out is to introduce a small perturbing Lagrangian density to the Lagrangian density of the electromagnetic field involving A0,0 replace LEM by this perturbed Lagrangian density and define the canonical momenta as πμ =

∂LEM ∂Aμ,0

Quantum Mechanics for Scientists and Engineers 34

35

Then, we introduce the commutation relations [Aμ (t, r), Aν (t, r� )] = δνμ δ 3 (r − r� ) This is satisfied provided

[a(K, σ), a(K � , σ � )∗ ] = δσ,σ δ 3 (K − K � )

Then, we find that since a(K, σ)|0 >= 0, < 0|a(K, σ)∗ = 0, we have  eν (K � , σ � )(exp(−i(k.x − k � .x� ))/2|K|)d3 Kd3 K � Dμν (x, x� ) = θ(t − t� ) < 0|a(K, σ)a(K � , σ � )∗ |0 > eμ (K, σ)¯ +

=





θ(t� − t) < 0|a(K � , σ � )a(K, σ)∗ |0 > eν (K � , σ � )¯ eμ (K, σ)(exp(i(k.x − k � .x� ))/2|K|)d3 Kd3 K �  = θ(t − t� ) eμ (K, σ)¯ eν (K � , σ � )δσ,σ δ 3 (K − K � )/2|K|)(exp(−i(k.x − k � .x� ))/2|K|)d3 Kd3 K �  +θ(t� − t) eν (K � , σ � )¯ eμ (K, σ)δσ,σ δ 3 (K − K � )(exp(i(k.x − k � .x� ))/2|K|)d3 Kd3 K �

[eμ (K, σ)¯ eν (K, σ)θ(t−t� )exp(−i|K|(t−t� )+iK.(r−r� ))+eν (K, σ)¯ eμ (K, σ)θ(t� −t)exp(i|K|(t−t� )−iK.(r−r� ))]d3 K/2|K|

The normalization condition

es (K, σ) = δrs , r, s = 1, 2, 3 er (K, σ)¯

(summation over σ = 1, 2, 3) had to be imposed (look at the treatment in the Coulomb gauge) in order to guarantee the correct commutation relations for the creation and annihilation operators following from the canonical commutation relations (CCR) for the position and momentum fields. From this we get  Drs (x, x� ) = δrs (2|K|)−1 [θ(t − t� )exp(−ik.(x − x� )) + θ(t� − t)exp(ik.(x − x� ))]d3 K where k 0 = |K|. This is a function of only x − x� and so, we can denote it by Drs (x − x� ). Now, consider k 0 to be a variable and consider the identity  dk 0 /(k 02 − |K|2 ) = πi/|K| Γ

where Γ is a contour along the real axis from −∞ to +∞ making a small encirclement of the pole at |K| above the real axis but excluding the pole at −|K| and then completed into a big infinite semicircle below the real axis (so that on this semicircle, k 0 has a negative imaginary part). Then it is clear that for t > t� , the contour integral  exp(−ik.(x − x� ))dk 0 /(k 02 − |K|2 ) = iπexp(−i|K|(t − t� ) + iK.(r − r� ))/|K| Γ

and likewise for the other term t� > t. Thus, it follows from the above formula that  −δrs Drs (x).exp(−ik.x)d4 x = 02 k − |K|2 + i� and to preserve Lorentz invariance, we may assume  ˆ μν (k) = Dμν (x).exp(−ik.x)d4 x = D where

ημν ημν = 2 k 02 − |K|2 + i� k + i�

k 2 = k 02 − |K|2 = kμ k μ

We can repeat this calculation for the electron propagator and show that  Sˆlm (p) = Slm (x)exp(−ip.x)d4 x = (γ 0 γ μ pμ − m)−1

These results can be derived directly from the FPI:   Dαβ (y, z) = exp((−i/4) Fμν (x)F μν (x)d4 x)Aα (y)Aβ (z)DAμ

Quantum Mechanics for Scientists and Engineers 35

36

Fμν (x)F μν (x) = (Aν,μ − Aμ,ν )(Aν,μ − Aμ,ν ) = 2Aν,μ Aν,μ − 2Aν mu Aμ,ν

= −2Aν ∂μ ∂ μ Aν + 2Aν ∂ ν ∂μ Aμ

on ignoring perfect divergences which do not contribute to the action integral. Thus,  S[A] = (−1/4) Fμν (x)F μν (x)d4 x =

or equivalently in the Fourier domain



K μν (x − y)Aμ (x)Aν (y)d4 xd4 y

S[A] = where



ˆ μν (k)Aˆμ (k)Aˆν (k)∗ d4 k K

ˆ μν (k) = k 2 η μν − k μ k ν K

ˆ The matrix K(k) is singular and hence its inverse cannot be evaluated. However, we evaluate its pseudo-inverse and use ˆ μν (k)kν = 0, we use as the photon propagator, the solution to the equation it as the propagator, or since K ˆ D(k) ˆ K(k) = P (k) where P (k) is the orthogonal projecton onto the spatial variable subspace, ie, Pμν (k) = I − kμ kν /k 2 or equivalently,

P μν (k) = I − k μ k ν /k 2

We are using the fact that the second moment matrix of a Gaussian distribution is its covariance matrix, ie, in this case, ˆ the inverse/pseudo-inverse of the matrix K(k). For the electron propagator, we find via the operator formalism that taking  ψl (x) = (ul (p, σ)a(p, σ)exp(−ip.x) + vl (p, σ)b(p, σ)∗ exp(ip.x))d3 P

with

p0 = E(P ) =

 P 2 + m2

and u(P, .), v(−P, .) eigenfunctions of the free Dirac Hamiltonian (α, P ) + βm, αr = γ 0 γ r , β = γ 0 , normalized in such a way that the CAR for ψl , πl = iψl imply {a(P, σ), a(P � , σ � )∗ } = δ 3 (P − P � )δσ,σ , {b(P, σ), b(P � , σ � )∗ } = δ 3 (P − P � )δσ,σ , {a(P, σ), b(P � , σ � )} = 0,

{a(P, σ), b(P � , σ � )∗ } = 0

the electron propagator is given by

Slm (x, y) =< 0|T (ψl (x)ψm (y)∗ )|0 >=   [ (ul (P, σ)um (P, σ)∗ exp(−ip.(x − y)) + vl (P, σ)vm (P, σ)∗ )exp(ip.(x − y))]d3 P l

= where



Sˆlm (p)exp(ip.(x − y))d4 p

ˆ S(p) = (γ 0 γ μ pμ − m + i�)−1

We leave this derivation as an exercise to the reader. Alternatively using the FPI,   Slm (x, y) = exp(i ψ(x)∗ (iγ 0 γ μ ∂μ − m)ψ(x)d4 x)ψl (y)ψm (z)∗ DψDψ ∗

Quantum Mechanics for Scientists and Engineers 36

37

gives directly the answer using the standard formula for the second moment of a Gaussian distribution. Now we discuss interactions. Suppose the initial state of the field is |i >= |pim , σim , p�il , σil� , kin , sin , m = 1, 2, ..., Ni1 , l = 1, 2, ..., Ni2 , n = 1, 2, ..., Ni3 > where pim , σim are the four momenta and spins of the mth electron, p�il , σil� are the four momenta and spins of the lth positron and kin , sin are the four momenta and helicities of the nth photon. Then, we can write |i >= Πm,l,n a(pim , σim )∗ b(p�il , σil� )∗ c(kin , sin )∗ |0 > where we are using the notation c(k, s) for the photon annihilation operators. Likewise for the final state. Since transitions in the state occur only because of interactions, we work in the interaction representation in which the Hamiltonian of the electron-positron-photon field is given by  HI (t) = −e Aμ (x)ψ(x)∗ (iγ 0 γ μ ∂μ ψ(x)d3 x

Substituting the operator expressions for the quantum fields Aμ , ψ, ψ ∗ , it follows that HI can be expressed as a trilinear functional of (c(k, s), c(k, s)∗ ), (a(P, σ), b(P, σ)∗ ), (a(P, σ)∗ , b(P, σ)). The transition probability amplitude from |i > at time t → −∞ to |f > at time t → +∞ is given by  ∞ < f |T {exp(−i HI (t)dt)}|i > = δ(f − i) +

∞ 

(−i)n

n=1

−∞



−∞ as Fourier integrals of the position space  field operators. To proceed further, we observe that (LEM + LD )d4 x is a quadratic functional of the position fields and Lint is small. So perturbation theory gives for the above FPI,     exp(i (LEM + LD )d4 x)(1 + i( Lint d4 x) + (i2 /2!)( Lint d4 x)2 + ...)F (ψ, ψ ∗ , Aμ )DψDψ ∗ DAμ where

Lint = e



ψ ∗ (iγ 0 γ μ ∂μ − m)ψAμ d4 x =−



J μ Aμ d4 x

Quantum Mechanics for Scientists and Engineers 37

38

Then the various terms in the intergral are evaluated using the standard formulae for the moments of a Gaussian  distribution on noting that exp(i (LEM + LD )d4 x) is a Gaussian density functional, it being a quadratic functional of ψ, ψ ∗ , Aμ . The propagators then enter naturally into the picture when we express the higher moments of even order as products of second moments, ie propagators. Renormalization, an example: Consider the Hamiltonian H = H0 − g|e >< e| Let |k >, k = 1, 2, ... denote the energy eigenstates of H0 with H0 |k >= Ek |k > Let |ψ > denote an energy eigenstate of H with energy eigenvalue E. Then, H|ψ >= E|ψ > gives H0 |ψ > −g|e >< e|ψ >= E|ψ >

or

|ψ >= g(H0 − E)−1 |e >< e|ψ >  |k > (Ek − E)−1 < k|e >< e|ψ > =g k

The normalization condition < ψ|ψ >= 1 then implies  (Ek − E)−2 | < k|e > |2 | < e|ψ > |2 = 1 g2 k

This implies that | < e|ψ > |2 = (g 2

 k

(Ek − E)2 | < k|e > |2 )−1

If the above sum is divergent, then we would get < e|ψ >= 0 which may nor be the case. To avoid such divergences, we make an ultraviolet cutoff meaning thereby that the sum over k is truncated to k such that Ek ≤ Λ where Λ is a finite positive constant. For a given ultraviolet cutoff Λ, we may thus define a renormalized coupling constant g = g(Λ) so that  (Ek − E)2 < k|e > |2 )−1 g 2 (Λ) = ( and get with this cutoff imposed that

k:Ek ≤Λ

| < e|ψ > |2 = 1

Thus, divergence problems are avoided by redefining the coupling constant. In quantum field theory, when we calculate the transition probability amplitudes like vacuum polarization, self energy of the electron or anomalous magnetic moment of the electron, the integrals obtained using the Feynman diagrams for the various Dyson series terms diverge. So to ge meaningful answers, we renormalize the fields and coupling constants like charge and mass, by scaling these with constant factors and then split the resulting Lagrangian density into a term not involving the coupling constants Z and an interaction term involving the Lagrangian terms scaled by Z − 1. The latter terms are regarded as perturbations and we expand the resulting exponential in powers of Z − 1 and calculate the modified matrix elements or propagators. Finally, Z may be made to tend to infinity in such a way so as to cancel out the infinities arising in the matrix elements or propagators computed without the Z. This method was first demonstrated by Dyson to lead to the experimentally correct values for the above phenomena. [33] Dirac’s wave equation in a gravitational field. [34] Canonical quantization of the gravitational field. Let Λ(x) be a local Lorentz transformation and let Λ → D(Λ) be the Dirac spinor representation of the Lorentz group. Let gμν (x) be the metric of curved space-time and ηab the Minkowski metric of flat space-time. Let Vμa (x) be the associated tetrad, ie, ηab Vμa (x)Vνb (x) = gμν (x)

Quantum Mechanics for Scientists and Engineers 38

39

((Vμa (x))) can be regarded as a locally inertial frame, ie, ξ μ → Vμa ξ μ = ξ a transforms a vector field ξ μ (x) in curved space-times to a Minkowski vector, ie each component ξ a is a scalar. Let ((γ a )) be the Dirac Gamma matrices. They determine the Dirac equation in flat space-time: (iγ μ ∂μ − m)ψ = 0 orin the presence of an electromagnetic field, (γ μ (i∂μ + eAμ (x)) − m)ψ = 0 This equation is invariant under global Lorentz transformations ,ie, if Λ is a constant 4 × 4 Lorentz transformation matrix, then D(Λ)(γ μ (i∂μ + eAμ ) − m)ψ = 0 implies

which implies or equivalently, where

[D(Λ)γ μ D(Λ)−1 (i∂μ + eAμ ) − m]D(Λ)ψ = 0 [Λμν γ ν (i∂μ + eAμ ) − m]D(Λ)ψ = 0 [γ ν (i∂ν� + eA�ν (x� )) − m]ψ � (x� ) = 0 

x μ = Λμν xν ,

so that

i∂ν� = Λμν ∂μ

and

A�ν (x� ) = Λμν Aμ (x), ψ � (x� ) = D(Λ)ψ(x) 

ie, ψ (x ) = D(Λ)ψ(x) satisfies Dirac’s equation in the Lorentz transformed system (x μ ) with the electromagnetic field also transformed in accordance with the Lorentz transformation Λ. Note that the Lorentz generators of the Dirac representation are given by J μν = (i/4)[γ μ , γ ν ]. We wish to define a Dirac equation in curved space-time that is invariant under local Lorentz transformations in accordance with the equivalence principle of general relativity. To do this, we must use the tetrad which will transform the non-inertial metric to the inertial metric. So we assume our curved space-time Dirac equation to be given by �

�

[γ a Vaμ (i∂μ + iΓμ (x) + eAμ (x)) − m]ψ(x) = 0 where Γμ (x) is a 4×4 matrix interpreted as the gravitational connection of space-time in the Dirac spinor representation. Applying a local Lorentz transformation D(Λ(x)) gives [D(Λ(x))γ a D(Λ(x))−1 Vaμ (x)(iD(Λ(x))(∂μ + Γμ (x))D(Λ(x))−1 or

If we define

+eAμ (x) − m]D(Λ(x))ψ(x) = 0 [Λab (x)γ b Vaμ (x)(i(∂μ + D(Λ(x))Γμ (x)D(Λ(x))−1 ) + iD(Λ(x))(∂μ D(Λ(x))−1 ) +eAμ (x)) − m]D(Λ(x))ψ(x) = 0 

Vaμ (x) → Λba (x)Vbμ (x) = Vbμ (x)

as the transformation of the tetrad under the local Lorentz transformation Λ(x), then we can express the above equation as  [γ b Vbμ (x)(i(∂μ + Γμ (x)) + eAμ (x)) − m]D(Λ(x))ψ(x) = 0

where Γμ (x) is the transformed gravitational connection under the local Lorentz transformation Λ(x): Γ�μ (x) = D(Λ(x))Γμ (x)D(Λ(x))−1 + D(Λ(x))(∂μ D(Λ(x))−1 ) Equivalently, if ω(x) = ((ωab (x))) is an infinitesimal local Lorentz transformation so that Λ(x) = I + dD(ω(x)) = I + ωab (x)J ab

Quantum Mechanics for Scientists and Engineers 39

40

and with neglect of O(� ω(x) �2 ) terms,

D(Λ(x))−1 = I − ωab (x)J ab

then under the infinitesimal local Lorentz transformation Λ(x), the gravitational connection Γμ (x) transforms to Γ�μ (x) = (I + ωab (x)J ab )Γμ (x)(I − ωcd (x)J cd ) −(I + ωab (x)J ab )ωcd,μ (x)J cd

= Γμ (x) + ωab (x)[J ab , Γμ (x)] − ωab,μ (x)J ab

We need to look for a Dirac gravitational connection Γμ (x) that satisfies such a transormation law under an infinitesimal local Lorentz transformation I + ω(x). It is easily seen that Γμ (x) = J ab Vaν (x)Vνb,μ (x) does the job. Indeed, under I + ω(x), this transforms to J ab (Vaν + ωac V νc )(Vνb + ωbd Vνd ),μ d = J ab Vaν Vνb,μ + J ab (Vaν Vν,μ ωbd + V νc Vνb,μ ωac ) d ωbd + V νc Vνb,μ ωac ) = Γμ + J db ωbd,μ + J ab (Vaν Vν,μ

This is seen to coincide with

Γμ − ωab,μ J ab + ωab [J ab , Γμ ]

on noting that

[J ab , Γμ ] = [J ab , J cd ]Vcν Vνd,μ

and using the Lie algebra commutation rules for the Lorentz group generators ((J ab )). Finally, we should replace ordinary partial derivatives of the tetrad field by covariant derivatives, ie, Γμ (x) = J ab Vaν (x)Vνb:μ (x)

[35] The scattering matrix for the interaction between photons, electrons, positrons and gravitons. Calculating the scattering matrix using the Feynman path integral and also using the operator formalism with the Feynman diagrammatic rules. [36] Atom interacting with a Laser; The general theory based on quantum electrodynamics. Representing the quantum electromagnetic field using finite sets of creation and annihilation operators. Representing any density operator for the quantum electromagnetic field via the diagonal Glauber-Sudarshan representation. Representing any state of the laser interacting with the spin of an atom using the Galuber-Sudarshan representation having matrix coefficients. Expressing the evolution equation for the density operator of the laser-atom system using the Glauber-Sudarshan representation. The Hamiltonian of the field is given by HF =

p 

ωk a∗k ak , a = (a1 , ..., ap ), a∗ = (a∗1 , ..., a∗p ),

k=1

[ak , a∗m ] = δkm and all the other commutators vanish. The Hamiltonian of the atom is HA an N × N Hermitian matrix and finally, the interaction Hamiltonian between the atom and the field is given by HI (t) =

p 

(Ak (t)Fk (a, a∗ ) + Ak (t)∗ Fk (a, a∗ )∗ )

k=1

where Ak (t) is a time varying N ×N matrix and Fk� s are ordinary complex valued functions which become field operators when their complex arguments are replaced by the field operators a, a∗ . The evolution of the joint state ρ(t) of the atom and field follows the Schrodinger equation iρ� (t) = [H(t), ρ(t)], H(t) = HF + HA + HI (t)

Quantum Mechanics for Scientists and Engineers 40

41

Note that HA and HF commute. For obtaining the interaction representation, we define ρ˜(t) = U0 (t)∗ ρ(t)U (t) where

U0 (t) = U0A (t)U0F (t), U0A (t) = exp(−itHA ), U0F (t) = exp(−itHF )

We note that

U0F (t)∗ ak U0F (t) = exp(−iωk t)ak = ak (t), U0F (t)∗ a∗k U0F (t) = exp(iωk t)a∗k = ak (t)∗

Thus, U0F (t)∗ HI (t)U0F (t) =



(Ak (t)Fk (a(t), a(t)∗ ) + Ak (t)∗ Fk (a(t), a(t)∗ ))

k

and defining the N × N complex matrices

Bk (t) = U0A (t)∗ Ak (t)U0A (t) we get the interaction picture Schrodinger equation for the atom and field ˜ I (t), ρ˜(t)] i˜ ρ� (t) = [H where

˜ I (t) = U0 (t)∗ HI (t)U0 (t) = H



Bk (t)Fk (a(t), a(t)∗ ) + Bk (t)∗ Fk (a(t), a(t)∗ )∗

k

We write so

Fk (a(t), a(t)∗ ) = F ({ak exp(−iωk t)}, {a∗k exp(iωk t)}) = Fk (t, a, a∗ ) ˜ I (t) = H



(Bk (t)Fk (t, a, a∗ ) + Bk (t)∗ Fk (t, a, a∗ )∗ )

k

and the Schrodinger equation in the interaction picture becomes ˜ I (t), ρ˜(t)] i˜ ρ� (t) = [H To solve this differential equation, we adopt the Galuber-Sudarshan diagonal represention: Let a(z) =  ∗ p k zk ak where z = (zk ) ∈ C . Then write  z n a∗n |0 > /n! = exp(a(z)∗ )|0 > |e(z) >= n

with the obvious p-tuple notation. The normalized energy eigenstates of the field are √ |n >= a∗n |0 > / n! and hence |e(z) >= Thus,

 n

√ z n |n > / n!

< e(u)|e(z) >= exp(< u|z >) We can normalize |e(z) > by multiplying it by a function φ(z) of z so that  I = |e(z) > φ(z) < e(z)|d2n z We have Also,

a(u)|e(z) >=< u|z > |e(z) >, ak |e(z) >= zk |e(z) > a∗k |e(z) >=

 n

z n a∗k a∗n|0 > /n! =

∂ |e(z) > ∂z



¯k ak , a(z)∗ kz

=

41 Quantum Mechanics for Scientists and Engineers

42

We can evaluate

ak |e(z) >< e(z)| = z¯k |e(z) >< e(z)|, a∗k |e(z) >< e(z)| =

∂ |e(z) >< e(z)| ∂zk

Thus, F (t, a, a∗ )|e(z) >< e(z)| = F (t, z,

∂ )|e(z) >< e(z)| ∂z 

assuming that in the expression F (t, a, a∗ ), all the a� s appear to the left of all the a∗ s. Such a representation is possible  in view of the commutation relations between the a� s and the a∗ s. Likewise, we have |e(z) >< e(z)|F (t, a, a∗ ) and

= |e(z) > (F (t, a, a∗ )∗ |e(z) >)∗ F (t, a, a∗ )∗ |e(z) >= F¯ (t, a∗ , a)|e(z) >

where now in the expression for F¯ (t, a∗ , a) all the a∗ s will appear to the left of all the a� s. We thus get |e(z) >< e(z)|F (t, a, a∗ ) = F¯ (t,

∂ , z¯)|e(z) >< e(z)| ∂ z¯

Here, F¯ (t, u, v) is the function obtained by conjugating all the coefficients in the Taylor series expansion of F (t, .). Now, with this understanding, we note that  ˜ I (t) = (Bk (t)Fk (t, a, a∗ ) + Bk (t)∗ Fk (t, a, a∗ )∗ ) H k

can be written as

˜ I (t) = H



Ck (t)Gk (t, a, a∗ )

k



where in the G�k s all the a� s appear to the left of all the a∗ s. Then, by the above observation, [Gk (t, a, a∗ ), |e(z) >< e(z)|] = [Gk (t, z,

∂ ¯ k (t, z¯, ∂ )]|e(z) >< e(z)| )−G ∂z ∂ z¯

∂ acts only on the factor |e(z) > while ∂∂z¯ acts only on the factor < e(z)| in the term It should be noted that ∂z |e(z) >< e(z)|. This is because z → |e(z) > is an analytic Hilbert space valued function of the complex variable z and z¯ →< e(z)| is therefore an analytic function of the complex variable z¯. Using

< e(u), e(z) >= exp(< u, z >) it is easy to show that

φ(z) = π −p exp(� z �2 )

Indeed, then a simple Gaussian integral evaluation gives   < e(u)| φ(z)|e(z) >< e(z)|d2p z|e(v) >= φ(z)exp(< u|z > + < z|v >)d2p z = 1 We can express the joint state of the atom and the field in the interaction representation as a Glauber-Sudarshan integral:  ρ˜(t) = ψ(t, z, z¯) ⊗ |e(z) >< e(z)|d2p z where

˜ z, z¯) ∈ CN ×N ψ(t,

We then get ρ˜� (t) =



∂ψ(t, z, z¯) ⊗ |e(z) >< e(z)|d2p z ∂t

Quantum Mechanics for Scientists and Engineers 42

43

and further,  k

 k

˜ I (t), ρ˜(t)] = [H [Ck (t)ψ(t, z, z¯) ⊗ Gk (t, a, a∗ )|e(z) >< e(z)| − ψ(t, z, z¯)Ck (t) ⊗ |e(z) >< e(z)|Gk (t, z, z ∗ )]d2p z

¯ k (t, z¯.∂/∂ z¯)|e(z) >< e(z)|]d2p z [Ck (t)ψ(t, z, z¯) ⊗ Gk (t, z, ∂/∂z)|e(z) >< e(z)| − ψ(t, z, z¯)Ck (t) ⊗ G =

 k

¯ k (t, z¯, ∂/∂ z¯)T )ψ(t, z, z¯) ⊗ |e(z) >< e(z)|d2p z Ck (t)(Gk (t, z, ∂/∂z)T − G

where we have used integration by parts. Here for example, (z1m1 z2m2 ...zpmp = (−1)n1 +...+np

∂ n1 +...+np T n ) ∂z1n1 ...∂zp p

∂ n1 +...+np m1 mp ...zp n z ∂z1n1 ...∂zp p 1

It follows that ψ(t, z, z¯) satisfies the pde

∂ψ(t, z, z¯) = ∂t  ¯ k (t, z¯, ∂/∂ z¯)T )ψ(t, z, z¯) Ck (t)(Gk (t, z, ∂/∂z)T − G i

k

=

 k

¯ k (t, z¯, ∂/∂ z¯)T )Ck (t)ψ(t, z, z¯) (Gk (t, z, ∂/∂z)T − G

Remark: The following notation has been used here: If G(t, z, u) is a polynomial in variables z, u which may even be ¯ z, u), we mean the polnomial obatined from G(t, z, u) by replacing its complex noncommuting operators, then by G(t, coefficients by their respective conjugates without making any change in the variables z, u. [37] The classical and quantum Boltzmann equations. Quantum Boltzmann equation: Let Hi , i = 1, 2, ..., N be Hilbert spaces and let H=

N  k=1

Hk

Let ρ(t) be a density operator in H. Let the Hamiltonian according to which ρ evolves have the form H=

N 

Hk +

k=1



Vkj

1≤k 0, the second marginal state ρ12 (t)will be a small perturbation of ρ1 (t) ⊗ ρ2 (t) (Note that, we are assuming that ρ2 (t) is an identical copy of ρ1 (t)). Thus, if the interaction potential V12 is small, then in (3), we can replace [V12 , ρ12 (t)] by [V12 , ρ1 (t) ⊗ ρ2 (t)]. Then, the approximate solution to (3) is given by  t ρ12 (t) = exp(−itad(H1 + H2 ))(ρ12 (0)) + exp(−i(t − s)ad(H1 + H2 ))ad(V12 )(ρ1 (s) ⊗ ρ1 (s))ds − − − (4) 0

and this can be substituted into (2) to get the following nonlinear integro-differential equation for the first marginal ρ1 (t):  t iρ�1 (t) = [H1 , ρ1 (t)]+(N −1)T r2 [ad(V12 )T12 (t)(ρ12 (0))]+(N −1)T r2 [ ad(V12 )T12 (t−s)ad(V12 )(ρ1 (s)⊗ρ1 (s))ds]−−−(5) 0

where T12 (t) = exp(−itad(H1 + H2 )) = exp(−itad(H1 )).exp(−itad(H2 )) = exp(−itad(H2 )).exp(−itad(H1 )) since H1 and H2 commute. (5) may be termed as the quantum Boltzmann equation. Other versions of this equation exist like we can solve (3) to get ρ12 (t) = exp(−itad(H12 ))(ρ12 (0)) where (2) then gives

H12 = H1 + H2 + V12 iρ�1 (t) = ad(H1 )(ρ1 (t)) + (N − 1)T r2 ad(V12 )exp(−itad(H12 )(ρ12 (0)) − − − (6)

The third version is to note that the last term in (4) already contains a multiplicative factor ad(V12 ) and hence since we are interested in terms only upto linear orders in V12 , we can replace ρ1 (s) in (4) by exp(−is.ad(H1 ))(ρ1 (0)) (4)then becomes ρ12 (t) = exp(−itad(H1 + H2 ))(ρ12 (0)) +



t

0

exp(−i(t − s)ad(H1 + H2 ))ad(V12 )exp(−isad(H1 + H2 ))(ρ1 (0) ⊗ ρ1 (0))ds

Thus, (2) can be approximated by iρ�1 (t) = [H1 , ρ1 (t)]+(N −1)T r2 [ad(V12 exp(−it.ad(H1 +H2 ))(ρ12 (0))]+(N −1)T r2 [ad(V12 )



t

exp(−i(t−s)ad(H1 +H2 ))ad(V12 )exp

0

[38] Bands in a semiconductor: Derivation using the Bloch wave functions in a 3-D periodic lattice. V : R3 → R is the potential of the periodic lattice produced by nuclei located at different sites of the crystal. The Lattice vectors are a1 , a2 , a3 and the periodicity gives V (r + n1 a1 + n2 a2 + n3 a3 ) = V (r + n.a) = V (r), n1 , n2 , n3 ∈ Z V (r) can be expressed as a Fourier series V (r) =



c(m)exp(2πimT M r)

m∈Z3

where the reciprocal lattice matrix M is calculated as M [a1 , a2 , a3 ] = I ie

M = A−1 , A = [a1 , a2 , a3 ]

Quantum Mechanics for Scientists and Engineers 44

It follows that

45

mT M (n1 a1 + n2 a2 + n3 a3 ) = mT M An = mT n ∈ Z, n ∈ Z3

This ensures that the above Fourier series for V (r) is periodic along the a1 , a2 , a3 directions. The wave function ψ(r) satisfies [−∇2 /2m + V (r)]ψ(r) = Eψ(r) Changing r to r + An (An = n1 a1 + n2 a2 + n3 a3 ) and using V (r + An) = V (r) gives Hψ(r + An) = Eψ(r + An) so by uniqueness (assuming non-degeneracy), ψ(r + An) = C(n)ψ(r), C(n) ∈ C, |C(n)| = 1 We write

Ck = C(ak ), k = 1, 2, 3

Then if Nk nuclei are along the ak direction, k = 1, 2, 3, then by imposing periodicity relations for the wave function at the crystal boundaries, we get CkNk = 1, k = 1, 2, 3 and hence

Ck = exp(2πilk /Nk ), k = 1, 2, 3

for some lk = 0, 1, ..., Nk − 1. We define the Bloch wave function corresponding to l = (l1 , l2 , l3 ) by ψ(r) = ul (r)exp(2πilT Kr) = where K is some 3 × 3 matrix. Then, for ul to be periodic with periods a1 , a2 , a3 , we require that ψ(r + ak )exp(−2πilT Kak ) = ψ(r), k = 1, 2, 3 or equivalently, Thus we require that and so we can take We note that

Ck = exp(2πilT Kak ) exp(2πilk /Nk ) = ex[(2πilT Kak ) K = N −1 M = N −1 A−1 , N = diag[N1 , N2 , N3 ] lT N −1 A−1 ak = lT N −1 ek = lk /Nk , k = 1, 2, 3

We write Then,

b = b(l) = 2πK T l ψ(r) = u(r)exp(ibT r), u = ul

We substitute this into the Schrodinger equation and derive the pde satisfied by u(r). After that since u is periodic with period A, we can expand it in a Fourier series as we did for V (r):  u(r) = ul (r) = d(m)exp(2πimT M r) m∈Z3

and then derive a difference equation for the Fourier coefficients d(m) of u(r). For each l = (l1 , l2 , l3 ) ∈ ×3k=1 {0, 1, ..., Nk − 1} we thus obtain a sequence of solutions ulk (r), k = 1, 2, ... with energy eigenvalues E = E(l, k), k = 1, 2, .... We say that each l defines an energy band. [39] The Hartree-Fock apporoximate method for computing the wave functions of a many electron atom. The Hamiltonian of the system comprising N particles has the form H=

N 

k=1

Hk +



1≤k= ⊗N k=1 |ψk >

where |ψk >∈ Hk . We substitute this into the expression < ψ|H|ψ > for the average energy and extremize this w.r.t. the component wave functions |ψk >, k = 1, 2, ..., N subject to the constraints < ψk |ψj >= δkj . Incorporating these constraints using Lagrange multiplier λ(k, j) gives us the functional to be extremized as  λ(k, j)(< ψk |ψj > −δkj ) S[{ψk , λ(k, j)}] =< ψ|H|ψ > − 1≤k≤j≤N

We observe that < ψ|H|ψ >=

 k

< ψk |Hk |psik > + 

1≤k≤j≤N

The variational equations gives Hk |ψk > +

and

1≤k

λ(k, j)(< ψk |ψj > −δkj ) δS/δψk∗ = 0



j:j>k

< Ik ⊗ ψj |Vkj |ψk ⊗ ψj > + −λ(k, k)|ψk > −

For example, if Hk = L2 (R3 ) and







j:j

λ(k, j)|ψj >= 0

j:k ⊗...|psiσn > σ∈Sn

Carry out for this trial wave function the above extremization of < ψ|H|ψ > subject to the constraints < ψk |psij >= δkj and specialize to the position representation. Next, take into account the spin of each electron so that the component wave function |ψk > depends on both the position rk and spin variable sk = ±1/2. The trial wave function is then  ψσ1,sσ1 (r1 ) ⊗ ... ⊗ ψσn,sσn (rn ) ψ(r1 , s1 , ..., rn , sn ) = (n!)−1/2 σ∈Sn

Note that the constraints are < ψk,s |ψk ,s >= δkk δss [40] The Born-Oppenheimer approximate method for computing the wave functions of electrons and nuclei in a lattice.

Quantum Mechanics for Scientists and Engineers 46

47

Nucleon positions are Rk , k = 1, 2, ..., N . electron positions associated with the k th nucleus are rkl , l = 1, 2, ..., Z. Nucleon mass is M , electron mass is m. Total Hamiltonian of the system is H = TN + Te + Vee + VN N + VeN where

TN = TN (R) = −

is the total kinetic energy operator of all the nucleons. Te = Te (r) = −

 k

 k,l

∇2Rk /2M

∇2rkl /2m

is the total kinetic energy of all the electrons. Vee =



(k,l)�=(m,j)

e2 /2|rkl − rmj |

is the total electron-electron interaction potential energy.  Z 2 e2 /|Rk − Rm | VN N = k�=m

is the total nucleon-nucleon interaction potential energy.  VeN = − Ze2 /|Rk − rml | k,m,l

is the total electron-nucleon interaction potential energy. We first solve (Te + Vee + VeN )Φ(r, R) = Ee (R)Φ(r, R) ie, the eigenfunctions for the electrons with fixed values of the nuclear positions. The energy levels of the electrons then depend on the nucleon positions R. We then Assume that the total wave function of the electrons and nucleons is Ψ(r, R) = Φ(r, R)χ(R) Substituting this into the complete electron-nucleon eigenvalue equation gives (Te + TN + Vee + VN N + VeN )(Φ(r, R)χ(R)) = EΦ(r, R)χ(R) or using the above electron eigenvalue equation, (TN + VN N + Ee (R))(Φ(r, R)χ(R)) = E(Φ(r, R)χ(R)) or equivalently,

Φ(r, R)−1 TN (Φ(r, R)χ(R)) + (VN N + Ee (R))χ(R) = Eχ(R)

Now,

= −(2M )−1

 k

TN (Φ(r, R)χ(R)) =  ∇2Rk /2M )(Φ(r, R)χ(R)) (− k

[χ(R)(∇2Rk Φ(r, R)) + Φ(r, R)∇2Rk χ(R) + 2(∇Rk Φ(r, R), ∇Rk χ(R))]

[41] The performance of quantum gates in the presence of classical and quantum noise.  Suppose we design a quantum gate by perturbing the Hamiltonian H0 of a quantum system to H0 + δ k fk (t)Vk and 2 running the unitary evolution for a duration of T seconds. Upto O(δ ), the evolution gate at time T is given by U (T ) = U0 (T )W (T ), U0 (T ) = exp(−iT H0 ),

Chapter

5

Quantum Gates, Scattering THeory, Noise in Schrodinger and Dirac’s Equations k

[41] The performance of quantum gates in the presence of classical and quantum noise.  Suppose we design a quantum gate by perturbing the Hamiltonian H0 of a quantum system to H0 + δ k fk (t)Vk and Quantum Mechanics 47 2 running the unitary evolution for a duration of T seconds. Upto O(δ ), the evolution gate at time T is given by U (T ) = U0 (T )W (T ), U0 (T ) = exp(−iT H0 ),  T  W (T ) = I − iδ fk (t)V˜k (t)dt − δ 2 fk (t)fk (s)V˜k (t)V˜m (s)dtds 0

k

where

k,m

0 −exp(−itH0 )|φi >= 0 so that

|ψi >= Ω− |φi >, Ω− = s.limt→∞ exp(itH).exp(−itH0 )

Quantum Mechanics for Scientists and Engineers

56 54

Likewise |φo > is the out state ie, the free projectile state at t = +∞ projected to time t = 0. It also evolves according to the free particle Hamiltonian H0 . The out scattered state |ψo > from which it evolves, evolves according to exp(−itH). Thus, limt→∞ exp(−itH)|ψo > −exp(−itH0 )|phio >= 0

or equivalently,

|ψo >= Ω+ |φo >, Ω+ = s.limt→∞ exp(itH).exp(−itH0 )

We define the scattering operator by

S = Ω∗+ Ω−

It follows that < φo |S|φi >=< ψo |ψi >

ie, the probability of scattering from an in scattered state |ψi > to an out scattered state |ψo > equals | < ψo |ψi > |2 = | < φo |S|φi > |2 where |φi > is the input free state of the projectile and |φo > is the output free state of the projectile. The domain of Ω+ is not the whole of H but rather contained in the absolutely continuous spectrum of H0 . We denote the spectral measure of H0 by E0 and that of H by E. We then have  ∞ d (exp(itH).exp(−itH0 )dt Ω+ = I + dt 0  ∞ =I +i exp(itH)V.exp(−itH0 )dt 0

and hence

Ω∗+ = I − i Likewise, we have Ω− = I − =I −i





∞

0

d (exp(itH).exp(−itH0 ))dt dt

−∞



exp(itH0 )V.exp(−itH)dt

0

0

exp(itH)V.exp(−itH0 )dt

−∞



=I −i

∞

exp(−itH)V.exp(itH0 )dt

0

and hence

Ω∗− = I + i It is clear that Ω+ and Ω− are isometries, ie,



∞

exp(−itH0 )V.exp(itH)dt

0

Ω∗+ Ω+ = I, Ω∗− Ω− = I defined respectively on the domains of Ω+ and Ω− . We have R = S − I = Ω∗+ Ω− − Ω − −∗ Ω− Now, (I − i



0

= Ω− − i where we have used the identity

∞



0

S = Ω∗+ Ω− = exp(itH0 )V.exp(−itH)dt).Ω−

∞

exp(itH0 )V.Ω− .exp(−itH0 )dt

exp(−itH)Ω− = lims→∞ exp(−itH).exp(isH).exp(−isH0 ) = lims→∞ exp(i(s − t)H).exp(−i(s − t)H0 ).exp(−itH0 ) Ω− .exp(−itH0 )

Quantum Mechanics for Scientists and Engineers

Note that likewise, we have

57 55

exp(−itH)Ω+ = Ω+ exp(−itH0 )

We now also have I = Ω∗− Ω− = (I + i = Ω− + i Hence, 

−i[

∞

0

= −i



−





= −i R3 ×R+

−2π

0

0

exp(−itH0 )V.exp(itH)dt)Ω−

exp(−itH0 )V.Ω− .exp(itH0 )dt



0



∞

−∞

R3

E0 (dλ)V (I − i

R3

∞

∞

R=S−I =  ∞ exp(itH0 )V.Ω− .exp(−itH0 )dt + exp(−itH0 )V.Ω− exp(itH0 )dt] = −i

= −i





E0 (dλ)V Ω− E0 (dμ).exp(it(λ − μ))dt





∞

0

R3

E0 (dλ)V exp(−isH)V exp(isH0 )E0 (dμ).exp(it(λ − μ))dtds

R3 ×R+

+2πi

exp(−isH)V.exp(isH0 )ds)E0 (dμ).exp(it(λ − μ))

E0 (dλ)V.E0 (dμ)exp(it(λ − μ))

= −2πi



exp(itH0 )V.Ω− .exp(−itH0 )dt

= −2πi

E0 (dλ)V.E0 (dμ)δ(λ − μ)

E0 (dλ)V.exp(−is(H − x))V E0 (dx)E0 (dμ)δ(λ − μ)dxds = −2πi

R3

 

E0 (dλ)V.E0 (dμ)δ(λ − μ)

E0 (dλ)V.(H − x − i0)−1 V E0 (dx)E0 (dμ)δ(λ − μ)dxds 

R2

E0 (dλ)(V − V (H − λ − i0)−1 V )E0 (dμ)δ(λ − μ)

From this it follows that the initial projectile energy μ is conserved after the scattering process, ie, λ = μ and if |φi > is a state of the incident projectile having energy in the range λ + dλ and likewise for |φo >, then < φo |S − I|φi >≈ −2πi < φo |V − V (H − λ − i0)−1 V |φi > In other words, if R(λ) = (S − I)(λ) denotes the scattering matrix elements for the energy λ between two free particle states having momenta along the directions ω and ω � (∈ S 2 ) respectively, then this matrix element is given by −2πi times the diagonal matrix D√V minus the√product of √ the matrices DV , (K +DV −λI)−1 and DV where DV is the √ diagonal � ω � )δ( 2mλ� ω � − 2mλ�� ω �� ). K is the non-diagonal matrix (−1/2m)∇2 δ( 2mλ� ω � − matrix having elements V ( 2mλ √ 2mλ�� ω �� ) The final matrix DV − DV (K + DV − λI)−1 DV is to be evaluated at the row index (λ, ω � ) and column index (λ, ω) with λ being fixed. [48] Design of quantum gates using time dependent scattering theory. H(t) = H0 + V (t) Ω(+) = limt→∞ T {exp(−i



t

H(s)ds)}−1 .exp(−itH0 )

0

Ω(−) = limt→∞ T {exp(−i



0

−t

H(s)ds)}.exp(itH0 )

Quantum Mechanics for Scientists and Engineers

58 56

S = Ω(+)∗ Ω(−) = limt→∞ exp(itH0 ).T {exp(−i



t

0

= limt→∞ exp(itH0 ).T {exp(−i

H(s)ds)}.T {exp(−i 

t



0

H(s)ds)}.exp(itH0 )

−t

H(s)ds)}.exp(itH0 )

−t

For large t, the approximate value of the transition probability from an initial state |i > to a final state |f > is given by (assuming V (t) is an operator valued stochastic process), P2t (|i >→ |f >) =  t E[| < f |exp(itH0 ).T {exp(−i H(s)ds)}.exp(itH0 )|i > |2 ] −t

We have upto linear orders in V , the truncated Dyson series   t H(s)ds)} = exp(−itH0 )(I − i T {exp(−i −t

where

t

V˜ (s)ds).exp(−itH0 )

−t

V˜ (s) = exp(isH0 )V (s).exp(−isH0 )

Thus in this approximation, the scattering operator is approximately given by  t S =I −i V˜ (s)ds −t

and the transition probability from the state |i > to the state |f > is given by  t E| < f | V˜ (s)ds|i > |2 −t

The exact expression for the matrix elements of the scattering operator is given by  ∞  (−i)n V˜ (s1 )...V˜ (sn )ds1 ...dsn |i > |2 E| < f | −∞ is the exponential vector in the Boson Fock space. Then we claim that E(jt (X)|ηt ) = U (t)∗ Et (X|ηti )U (t) Note that ηti commutes with X and hence ηt commutes with jt (X). To prove the above formula, we observe that if if Z ∈ ηti , then U (t)∗ ZU (t) ∈ ηt and hence E[(jt (X) − U (t)∗ Et (X|ηti )U (t))U (t)∗ ZU (t)] = E[U (t)∗ XZU (t) − U (t)∗ Et (X|ηti )ZU (t)]

= E[U (t)∗ (XZ − Et (X|ηti )Z)U (t)] = Et (XZ − Et (XZ|ηti )) = 0

by the definition of conditional expectation. This proves the claim. Now suppose F (t) is a system operator, ie, in L(h) such that Et (X) = E(U (t)∗ XU (t)) = E(F (t)∗ XF (t)) = EF (t) (X) Note that [F (t), ηti ] = 0. It follows clearly that E(F (t)∗ F (t)) = E(U (t)∗ U (t)) = 1 Then, we claim that

EF (t) (X|ηti ) = E(F (t)∗ XF (t)|ηti )/E(F (t)∗ F (t)|ηti )

To prove this, we observe that the lhs and the numerator and denominator of the rhs all commute with each other since ηti is an Abelian algebra. Further, we have for any operator Z ∈ ηti that EF (t) [(X − (E(F (t)∗ XF (t)|ηti )/BbbE(F (t)∗ F (t)|ηti ))Z] = E[F (t)∗ XZF (t) − F (t)∗ (E(F (t)∗ XF (t)Z|ηti )/E(F (t)∗ F (t)|ηti ))F (t)]

= E[F (t)∗ XZF (t)] − E[E(F (t)∗ XZF (t)|ηti )F (t)∗ F (t)/E(F (t)∗ F (t)|ηti )]

Quantum Mechanics for Scientists and Engineers F (t)∗ F (t)|ηti ))F (t)]

62

since F (t) commutes with ηti and Z ∈ ηti . Further, conditioning the second term above on ηti and then taking the expectation gives E[E(F (t)∗ XZF (t)|ηti )F (t)∗ F (t)/E(F (t)∗ F (t)|ηti )] = E[E(F (t)∗ XZF (t)|ηti )E(F (t)∗ F (t)|ηti )/E(F (t)∗ F (t)|ηti )] = E(F (t)∗ XZF (t)) We have thus proved that EF (t) [(X − (E(F (t)∗ XF (t)|ηti )/BbbE(F (t)∗ F (t)|ηti ))Z] = 0 and hence the claim. The Hudson-Parthasarathy noisy Schrodinger evolution is described by the qsde β dU (t) = (Lα β dΛα (t))U (t)

where summation over the repeated indices α, β ≥ 0 is understood and the basic processes Λα β satisfy quantum Ito’s formula μ μ α dΛα β .dΛν = �ν dΛβ where �μν is zero if either μ = 0 or ν = 0 or μ �= ν and is one otherwise. The system operators Lα β satisfy the following conditions for U (t) to describe a unitary evolution: 60

0 = d(U ∗ U ) = dU ∗ .U + U ∗ .dU + dU ∗ .dU = β α β β∗ μ β ν U ∗ (Lβ∗ α dΛα + Lβ dΛα + Lα Lν dΛα dΛμ )U

so that

β α β β∗ μ β ν (Lβ∗ α dΛα + Lβ dΛα + Lα Lν dΛα dΛμ )U

which gives

α ν∗ μ ν Lβ∗ α + Lβ + Lα Lβ �μ = 0

The Evans-Hudson flow corresponding to this HP equation is obtained by taking a self-adjoint operator X in the system Hilbert space h and setting jt (X) = U (t)∗ XU (t) Then application of quantum Ito’s formula gives djt (X) = dU ∗ XU + U ∗ XdU + dU ∗ XdU = μ α ν ν∗ β U ∗ (Lβ∗ α X + XLβ + �μ Lα XLβ )U.dΛα

= jt (θβα (X))dΛβα where the structure maps θβα are given by

θβα (X) = μ α ν ν∗ Lβ∗ α X + XLβ + �μ Lα XLβ

We note that they satisfy the structure equations since jt as defined is a ∗ unital algebra homomorphism. The Belavkin input measurement processes are taken as β Yin,k (t) = cα β [k]Λα , k = 1, 2, ..., r, t ≥ 0

where the α = β = 0 term is omitted. These processes jointly generate an Abelian family of Von-Neumann algebras.The corresponding output processes are Yout,k (t) = Yk (t) = U (t)∗ Yin,k (t)U (t) and since U (t) is unitary, and Yin,k (t) commute with the system operators, it follows that Yk (t) = U (T )∗ Yin,k (t)U (T ), T ≥ t and hence

[Yk (t), js (X)] = 0, s ≥ t

ie the output measurements are non-demolition processes. Let ηt = σ(Yk (s) : s ≤ t, k = 1, 2, ..., r). Then ηt is an Abelian Von-Neumann algebra and belongs to the commutant of the algebra generated by jt (X) as X ranges over the system operators. We write πt (X) = E(jt (X)|ηt )

Quantum Mechanics for Scientists and Engineers

63

where the expectation is taken in the state |f φ(u) > with |f >∈ h, < f, f >= 1 and φ(u) = exp(− � u �2 /2)|e(u) > where u ∈ L2 (R+ ) ⊗ Cd and |e(u) > is the corresponding exponential vector in Γs (L2 (R+ ) ⊗ Cd ). Now we can write dπt (X) = Ft (X)dt + Gkmt (X)(dYm (t))k where the summation in the last term is over m ≥ 1, k ≥ 1 and Ft (X), Gkmt (X) are all ηt measurable operators, ie, they can be regarded as commutative stochastic processes. We can write dYm (t) = dYin,m (t) + dU (t)∗ .dYin,m (t).U (t) + U (t)∗ dYin,m (t).dU (t)∗ = +jt (Sβα [m])dΛβα and hence for k ≥ 1

(dYm (t))k = jt (Sβα [m, k])dΛβα (t)

where Sβα [k] are system operators, ie, operators in h and S00 [k] = 0, k ≥ 2. The equations E[(djt (X) − dπt (X))(dYm (t))k |ηt ] + E[(jt (X) − πt (X))(dYm (t))k ] = 0, k ≥ 1 and

E[(djt (X) − dπt (X))|ηt ] = 0

follow by taking the differential of the expression E[(jt (X) − πt (X))C(t)] = 0 where C(t) satisfies the qsde dC(t) =



fm,k (t)C(t)(dYm (t))k , C(0) = 1

m,k≥1

and using the arbitrariness of the complex valued functions fm,k (t). We have E[djt (X)(dYm (t))k |ηt ] = E[jt (θβα (X))jt (Sνμ [m, k])dΛβα (t).dΛνμ (t)|ηt ] = �βμ πt (θβα (X).Sνμ [m, k])uν (t)¯ uα (t)dt E[dπt (X)(dYm (t))k |ηt ] =

Grst (X)E[(dYs (t))r (dYm (t))k |ηt ]

= Grst (X)jt (Sβα [s, r]Sνμ [m, k])E[dΛβα (t).dΛνμ (t)|ηt ] = Grst (X)πt (Sβα [s, r]Sνμ [m, k])�βμ uν (t)¯ uα (t)dt [53] Classical control of a stochastic dynamical system by error feedback based on a state observer derived from the EKF. [54] Quantum control using error feedback based on Belavkin quantum filters for the quantum state observer. Let Y (t) be a measurement noise process and V (t) a system operator process say like A(t) + A(t)∗ . Consider a qsde dUc (t) = (−iV (t)dY (t) − V (t)2 dt/2)Uc (t)

We are assuming that V (t) is Hermitian. If V (t) = V does not vary with time, we can write the solution to the above qsde as Uc (t) = exp(−iV Y (t)) Uc (t) is a unitary matrix and its application to the state evolved from a qsde can remove the effect of noise if Y (t) is present as a noise in the qsde. For example, consider the following qsde dU (t) = (−(iH + V 2 /2)dt − iV dY (t))U (t)

We can write this approximately as U (t + dt) = (I − dt(iH + V 2 /2) − iV dY (t))U (t) We now apply the control unitary (I − iV (t)dY (t) − V 2 (t)dt/2)−1 = I + iV (t)dY (t) − V 2 (t)dt/2 to U (t + dt) to get (I + iV (t)dY (t) − V 2 (t)dt/2)(I − iHdt − iV (t)dY (t) − V 2 (t)dt/2)U (t) = (I − iHdt)U (t)

∗

to 64

Quantum Mechanics for Scientists and Engineers − 2

ie, the effect of the noise is cancelled out. We note that the output measurement is given by Yo (t) = U (t)∗ Y (t)U (t) and it is this process which follows the non-demolition property. We can measure only Yo without disturbing the dynamics generated by U (t) on the system Hilbert space h. We cannot measure the input measurement Y without disturbing the system dynamics. Now Belavkin’s quantum filtering equation can be expressed as dπt (X) = Ft (X)dt + Gt (X)dYt 62 where

jt (X) = U (t)∗ XU (t), πt (X) = E[jt (X)|ηt ] ηt = σ{Ys : s ≤ t}

is the output measurement Abelian algebra at time t. U (t) is generated by the Hudson-Parthasarathy noisy Schrodinger equation and expectations are calculated in the pure state |f φ(u) > where f is a normalized system vector and φ(u) is a normalized coherent vector (See the paper by John Gough and Kostler). Ft (X), Gt (X) are linear functions of the system observable and belong to the measurement algebra ηt . More precisely, we can express the above filtering equation as dπt (X) = πt (θ0 (X))dt + [πt (L1 X + XL∗1 ) − π(L2 )π(X)](dYt − πt (L3 X + XL∗3 )dt)

t where L2 is a Hermitian matrix. For quadrature measurements, Zt = Yt − 0 πs (L3 X + XL∗3 )ds is a Brownian motion process. θ0 is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator on the system operator space. More generally, for general non-demolition measurements, like photon counting and a combination of quadrature and photon counting measurements, the Belavkin equation has the form   dπt (X) = πt (θ0 (X))dt + [ fk (πt (Mk ))πt (θk (X))](dYt − gk (πt (Nk ))πt (φk (X))dt) k≥1

k≥1

where θk , φk , k ≥ 1 are linear operators on the Banach space of system operators. If we write ρt for the density operator on the system space conditioned on the measurements upto time t, we have πt (X) = T r(ρt X) ρt should be viewed as a random density matrix on the system operator algebra where the randomness comes from conditioning on the measurements upto time t. Thus, the above Belavkin equation reads   dρt = θ0∗ (ρt )dt + [ fk (T r(ρt Mk ))θk∗ (ρt )][dYt − gk (T r(ρt Nk ))φ∗k (ρt )dt] k≥1

k≥1

It is to be noted that the process Mt = Yt −

 t

0 k≥1

gk (T r(ρs Nk ))φ∗k (ρs )ds

is a Martingale (Ph.d thesis of Luc Bouten). The above equation for ρt is called a Stochastic Schrodinger Equation. Now, we wish to remove the noise from the above Belavkin equation by an appropriate control so that the evolution equation of the density has just the first term θ0∗ (ρt ), ie, we wish to recover the GKSL equation. Consider now the control unitary Uc (t) = exp(iW Yt ) where W is a system observable and Yt the above output measurement. Assume for simplicity, that we take quadrature measurements, ie, Mt is a Wiener process. We apply Uc (t) to ρt to get ρc,t = Uc (t)ρt Uc (t)∗ Then, by Quantum Ito’s formula, dρc,t = dUc (t)ρt Uc (t)∗ + Uc (t)dρt Uc (t)∗ + Uc (t)ρt dUc (t)∗ + dUc (t)dρt Uc (t)∗ + Uc (t)dρt dUc (t)∗ [55] Lyapunov’s stability theory with application to classical and quantum dynamical systems. [56] Imprimitivity systems as a description of covariant observables under a group action. Construction of imprimitivity systems, Wigner’s theorem on the automorphisms of the orthogonal projection lattice. (Ω, F, pμ) is a measure space. P is a spectral measure on this space, ie, for each E ∈ F, P (E) is an orthogonal projection operator in a Hilbert space H. The set of all orthogonal projections on H is denoted by P(H). It is also called the projection lattice.  Let τ be an  automorphism of P(H), ie, if τ : P(H) → P(H) is such that if P1 , P2 , ... are mutually orthogonal, then τ ( j Pj ) = j τ (Pj ). More generally, we require that τ (max(P1 , P2 )) = max(τ (P1 ), τ (P2 )) and τ (min(P1 , P2 )) = min(τ (P1 ), τ (P2 )) for any two P1 , P2 ∈ P(H). Then, Wigner proved that there exists a unitary or antiunitary operator U in H such that τ (P ) = U P U ∗ , P ∈ P(H)

= min(τ (Pand (P2 and τ (min(P 1 , P2 )) for 1 ), τEngineers Quantum Mechanics Scientists

65

It follows that if G group that acts on the measure space (Ω, F, P ) and g ∈ G → τg is a homomorphism from G into aut(P(H)) such that τg (P (E)) = P (g.E), then there exists a projective unitary-antiunitary representation g → Ug of G Quantum Mechanics 63 into U A(H) such that ∗ P (gE) = Ug P (E)Ug , g ∈ G, E ∈ F We note that

τg1 τg2 (P (E)) = P (g1 g2 E) = τg1 g2 (P (E))

so the requirement that g → τg be a homomorphism is natural from the viewpoint of covariant transformation of observables. Let now H be a subgroup of G. Consider the homogeneous space X = G/H. G acts transitively on X. Let μ be a quasi-invariant measure on X, ie, for any g ∈ G, the measures μ.g −1 and μ are absolutely continuous with respect to each other. Consider f ∈ L2 (X, μ). Consider for f ∈ L2 (X, μ), Ug f (x) = (dμ.g −1 /dμ)1/2 f (g −1 x), x ∈ X Then,



� Ug f (x) �2 dμ(x) ==



X

|f (g −1 x)|2 dμ.g −1 (x) =



X

|f (x)|2 dμ(x)

This proves that Ug is a unitary operator. It is easy to see that U is also a representation: Ug1 g2 f (x) = (dμ.(g1 g2 )−1 (x)/dμ)f (g2−1 g1−1 x) On the other hand,

Ug1 (Ug2 f )(x) = (dμ.g1−1 (x)/dμ)1/2 (Ug2 f )(g1−1 x) = (dμ.g1−1 (x)/dμ)1/2 (dμ.g2−1 g1−1 (x)/dμ.g1−1 x)1/2 f (g2−1 g1−1 x) = (dμ.g2−1 g1−1 (x)/dμ)f (g2−1 g1−1 x)

Thus,

Ug1 g2 = Ug1 .Ug2 , g1 , g2 ∈ G

Let A(g, x) be a map from G × X into the algebra of linear operators in a Hilbert space � such that A(g1 g2 , x) = A(g1 , g2 x)A(g2 , x) Then consider the operator Ug defined on L2 (μ, h) Ug f (x) = (dμ.g −1 (x)/dμ)1/2 A(g, g −1 x)f (g −1 x) We see that

Ug1 (Ug2 f )(x) = (dμ.g1−1 (x)/dμ)1/2 A(g1 , g1−1 x)(Ug2 f )(g1−1 x) −1.g1−1

= (dμ.g1−1 (x)/dμ)1/2 A(g1 , g1−1 x)(dμ.g2

(x)/dμ.g1−1 )1/2 A(g2 , g2−1 g1−1 x)f (g2−1 g1−1 x)

= (dμ.g2−1 g1−1 (x)/dμ)1/2 A(g1 g2 , g2−1 g1−1 x)f ((g1 g2 )−1 x) = Ug1 g2 f (x) Thus, g → Ug is a representation of G in L (X, h). 2

[57] Schwinger’s analysis of the interaction between the electron and a quantum electromagnetic field. Let A(t, r) denote the vector potential corresponding to a quantum electromagnetic field. The dynamical variables q, p of the electron bound to the nucleus commute with A. Let Φ(t, r) denote the quantum scalar potential of the quantum t electromagnetic field. Note that if we adopt the Lorentz gauge, Φ = −c2 0 divAdt while if we adopt the Coulomb gauge, then divA = 0, Φ = 0 where for the latter, we are assuming that there is no externally charged matter to generate the scalar potential. The Hamiltonian of the atom interacting with the quantum electromagnetic field is given by H(t) = (p + eA)2 /2m + V (q) − eΦ + Hem where the field Hamiltonian Hem has also been added. H(t) is thus the total Hamiltonian of the atom interacting with the quantum em field. We have H(t) = p2 /2m + V (q) + ((p, A) + (A, p))/2m − eΦ + Hem We note that

(p, A) + (A, p) = 2(A, p) − i.div(A) = −2i(A, ∇) − idiv(A)

Schrodinger’s equation for the wave function ψ(t) of the atom and field is given by iψ � (t) = H(t)ψ(t)

Quantum Mechanics for Scientists and Engineers

66 64

Making the transformation

ψ(t) = exp(−itHem ))φ1 (t)

and assuming the Coulomb gauge gives ˜ p)/m)φ1 (t) iφ�1 (t) = (p2 /2m + V (q) + (A, where

A˜ = exp(itHem ).A.exp(−itHem )

Making another transformation

φ1 (t) = exp(−iS(t))φ2 (t)

where S(t) is linear in the creation and annihilation operators of the em field, we get φ�1 (t) = exp(−iS(t))(−iS � (t) + [S(t), S � (t)]/2)φ2 (t) + exp(−iS(t))φ�2 (t) since [S(t), S � (t)] is a c-number function of time owing to the commutation relations between the creation and annihilation t t ˜ ˜ operators of the em field. Taking S(t) = (p, 0 Adt)/m = (p, Z(t)) where Z(t) = 0 Adt/m gives iφ�2 (t) = exp(iS(t))(p2 /2m − S � (t) − i[S(t), S � (t)]/2 + V (q) + S � (t))exp(−iS(t))φ2 (t) = (p2 /2m + V (q + Z(t)))φ2 (t) where the c-number function −i[S(t), S � (t)] has been neglected as it only gives an additional phase factor to the wave function. (1)

[58] Quantum control: The system observable X evolves to jt (X) after time t. The desired system observable Xd (2) evolves to jt (Xd ). Here, (k) jt = (Hk (t), Lk ), k = 1, 2 which means that

(k)

jt (Z) = Uk (t)∗ ZUk (t), k = 1, 2, Z = X, Xd

where Uk (t) satisfies the HP equation dUk (t) = (−(iHk (t) + Qk )dt + Lk dA(t) − L∗k dA(t)∗ )Uk (t), Qk = Lk L∗k Define ˜= X ˜ = jt (X)





X 0

(1)

0 Xd



,

jt (X) 0 (2) 0 jt (Xd )



Let h denote the system Hilbert space and Γs (L2 (R+ )) the Boson Fock space. jt is a ∗ unital homomorhism from the algebra B(h) ⊕ B(h) into the algebra (B(h ⊗ Γs (L2 (R+ ))) ⊕ B(h × Γs (L2 (R+ ))). Define the operators H(t) = diag[H1 (t), H2 (t)], L = diag[L1 , L2 ], U (t) = diag[U1 (t), U2 (t)], Q = diag[Q1 , Q2 ] Then, the HP equations for Uk , k = 1, 2 can be expressed as a single qsde: dU (t) = [−(iH(t) + Q)dt + LdA(t) − L∗ dA(t)∗ ]U (t) ˜ A non demolition measurement for the ˜ in B(h) ⊕ B(h) evolve to U (t)∗ XU ˜ (t) = jt (X). The corresponding observables X ˜ in the sense of Belavkin is given by process jt (X) Y o (t) = U (t)∗ Y i (t)U (t), Y i (t) = A(t) + A(t)∗ It satisfies the sde

dY o (t) = dY i (t) + dU (t)∗ dY i (t)U (t) + U (t)∗ dY i (t).dU (t) = dY i (t) − U (t)∗ (L + L∗ )U (t)dt = dY i (t) + jt (S)dt = dA(t) + dA(t)∗ + jt (S)dt

Quantum Mechanics Mechanics for Scientists and Engineers Quantum

6567

where

S = −(L + L∗ )

Let ηt = σ(Yso : s ≤ t). Then ηt is an Abelian Von-Neumann algebra and we define the conditional expectation ˜ t ) = πt (X) ˜ Ejt (X)|η We can assume that

dπt (Z) = Ft (Z)dt + Gt (Z)dY o (t)

where Ft (Z) and Gt (Z) are in ηt . Then, its evident [See the paper by Gough and Koestler] that E[(jt (Z) − πt (Z))dY o (t)|ηt ] + E[(djt (Z) − dπt (Z))dY o (t)|ηt ] = 0 − − − (1) E[(djt (Z) − dπt (Z))|ηt ] = 0 − − − (2)

We can write

djt (Z) = jt (θ0 (Z))dt + jt (θ1 (Z))dA(t) + jt (θ2 (Z))dA(t)∗

where for Z = diag[Z1 , Z2 ], we have (1)

(2)

jt (Z) = diag[jt (Z1 ), jt (Z2 )] and

(1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

djt (Z1 ) = jt (θ10 (Z1 ))dt + jt (θ11 (Z1 ))dA(t) + jt (θ12 (Z1 ))dA(t)∗ djt (Z2 ) = jt (θ20 (Z2 ))dt + jt (θ21 (Z1 ))dA(t) + jt (θ22 (Z2 ))dA(t)∗

Thus,

θk (Z) = diag[θ1k (Z1 ), θ2k (Z2 )], k = 0, 1, 2

[59] Quantum error correcting codes: Let H = CN be the Hilbert space of the quantum system and let C be a subspace of H. C is called the code subspace. If ρ is a density operator in H, we say that ρ ∈ C if Range(ρ) ⊂ C. If N is a code, if there exist subspace of the space CN ×N of all linear operators in H, then we say C is an N error correcting  operators R0 , ..., Rr ∈ CN ×N such that k Rk∗ Rk = I and whenever E1 , ..., Er ∈ N are such that k Ek Ek∗ = I, and ρ ∈ C, then  Rk E(ρ)Rk∗ = cρ k

where c ∈ C and

E(ρ) =



Ek ρEk∗

k

is the output state of the noisy channel {Ek }. {Rk } are called recovery operators for the noise subspace N and the code subspace C. If there exist such recovery operators, then we say that C is an N error correcting code. Theorem (Knill-Laflamme): C is an N error correcting code iff P N2∗ N1 P = λ(N2 ∗ N1 )P for all N1 , N2 ∈ N and some complex scalar λ(N2∗ N1 ) (dependent on N2∗ N1 ). Here, P is the orthogonal projection onto C. Proof: Suppose first that C is an N error correcting code with recovery operators R0 , ..., Rr . Let Ek , k = 1, 2, ..., m be a quantum channel in N . Then we have for all |ψ >∈ C the relation  Rk Es |ψ >< ψ|Es∗ Rk∗ = λ(ψ)|ψ >< ψ| k,s

where λ(ψ) is a complex scalar possibly dependent on |ψ >. It follows that for all |ψ >⊥ |ψ >, ie, < φ|ψ >= 0, we have  | < φ|Rk Es |ψ > |2 = λ| < φ|ψ > |2 = 0 k,s

Thus, ie,

Rk Es |ψ >⊥ |φ > ∀|φ >⊥ |ψ Rk Es |ψ >= βks (ψ)|ψ >, ∀|ψ >∈ C

Quantum Mechanics for Scientists and Engineers

68 66

and some complex numbers βks (ψ). By linearity of the operators, it is clear that βks cannot depend on |ψ >. We get therefore, Rk Es P = βks P, ∀k, s

Thus,

P Eq∗ Rk∗ Rk Es P = βks β¯kq P  ∗ and summing this equation over k and using k Rk Rk = I, we get P Eq∗ Es P = aqs P, aqs ∈ C

We have thus proved that

P N2∗ N1 P =∝ P, ∀N1 , N2 ∈ N

Conversely, suppose that this relation holds. Then, let

N0 = {N ∈ N : λ(N ∗ N ) = 0} It is clear that N0 is a subspace of N . Indeed, suppose N1 , N2 ∈ N0 . Then, P Nk∗ Nk P = λ(Nk∗ Nk )P = 0, k = 1, 2 and hence

Nk P = 0, k = 1, 2

Thus,

P Nj∗ Nk P = 0, j, k = 1, 2

and hence,

P (c1 N1 + c2 N2 )∗ (c1 N1 + c2 N2 )P = 0, .c1 , c2 ∈ C

Now, define for N ∈ N ,

[N ] = N + N0 ∈ N /N0

We define

< [N1 ], [N2 ] >= λ(N1∗ N2 ), N1 , N2 ∈ N

This definition is valid since for N0 ∈ N0 and N ∈ N we have

λ(N ∗ N0 ) = λ(N0∗ N ) = 0 This is because,

P N0∗ N0 P = λ(N0∗ N0 )P = 0

implies

P N0∗ = N0 P = 0

and hence,

0 = P N0∗ N P = λ(N0∗ N )P, 0 = P N ∗ N0 P = λ(N ∗ N0 )P

It is clear therefore that < ., . > defines an inner product on N /N0 . So, we can choose an onb {[Nk ], k = 1, 2, ..., r} for N /N0 w.r.t. this inner product. Now, define Rk = P Nk∗ , Pk = Nk P Nk∗ , k = 1, 2, ..., r, R0 = I − Then,

r 

Pk

k=1

Pk Pj = Nk P Nk∗ Nj P Nj∗ = λ(Nk∗ Nj )Nj P Nj∗ = δkj Pj

proving that Pj , j = 1, 2, ..., r for a set of mutually orthogonal orthogonal projections in H. Thus R0 is also an orthogonal projection. Now, suppose ρ ∈ C. Then ρ = P ρ = ρP = P ρP . Then, for N ∈ N ,   Rk N ρN ∗ Rk∗ = P Nk∗ N P ρP N ∗ Nk P + R0 N ρN ∗ R0 k

k≥1

Quantum Mechanics for Scientists and Engineers

69 67

=



k≥1

|λ(Nk∗ N )|2 ρ + R0 N ρN ∗ R0

= λ(N ∗ N )ρ + R0 N ρN ∗ R0

Here, we have used the Bessel equality/Parseval relation for orthonormal bases for a Hilbert space. Further,   R0 N P = N P − Pk N P = N P − Nk P Nk∗ N P k

= NP −

k



λ(Nk∗ N )Nk P

k

= NP − NP = 0

by the generalized Fourier series in a Hilbert space and the relation N0 P = 0 for all N0 ∈ N . Thus, we finally get r 

Rk∗ N ρN ∗ Rk∗ = λ(N ∗ N )ρ

k=0

and further,

r 

k=0

Rk∗ Rk =



Nk P Nk∗ + R0 =

k≥1



k≥1

Pk + I −



Pk = I

k≥1

This completes the proof of the Knill-Laflamme theorem. Construction of quantum error correcting codes using imprimitivity systems: Let A be a finite Abelian group and define the onb |x >, x ∈ A for L2 (A) so that |x >= [δx,0 , ..., δx,N −1 ]T where we are assuming

A = {0, 1, ..., N − 1}

with addition modulo N . with each n ∈ A, define the character

χn (x) = exp(2πinx/N ), x ∈ A We may identify this character with n ∈ A. In this way, we get an isomorphism of Aˆ with A and we write < n, x >=< x, n >= χn (x) Define the unitary operators Ux , Vx on L2 (A) by Ua |x >= |x + a >, Va |x >=< a, x > |x > Then define the Weyl operator

W (x, y) = Ux Vy , x, y ∈ A

on L (A). These are N unitary operators and in fact, as we shall soon see, they form an orthogonal basis for L2 (A × A), for the space of all linear operators in L2 (A). We derive the Weyl commutation relations: 2

2

W (x, y)|a >= Ux Vy |a >= Ux < y, a > |a >=< y, a > |a + x > Thus, Also,

Vy Ux |a >= Vy |a + x >=< y, a + x > |a + x >=< y, a >< y, x > |a + x > Vy Ux =< y, x > Ux Vy =< x, y > Ux Vy W (x, y)W (u, v) = Ux Vy Uu Vv =< y, u > Uu+x Vv+y =< y, u > W (u + x, v + y)

So (x, y) → W (x, y) is a projective unitary representation of the Abelian group G = A × A into the space of operators in L2 (A). Now, W (x, y)|a >=< y, a > |a + x >

Quantum Mechanics for Scientists Engineers Quantumand Mechanics

70 68

Hence, T r(W (x, y)) =



< a|W (x, y)|a >=

a



< y, a > δa+x,a

a

It follows that T r(W (x, y)) = 0 if x �= 0 and if x = 0, then T r(W (x, y)) = T r(W (0, y)) = for all x, y ∈ A, T r(W (x, y)) = N δx,0 δy,0



a

< y, a >= N δy,0 Thus,

It follows that W (x, y)∗ W (u, v) = V−y U−x Uu Vv = V−y Uu−x Vv =< y, x − u > Uu−x Vv−y =< y, x − u > W (u − x, v − y) Thus,

T r(W (x, y)∗ W (u, v)) =< y, x − u > T r(W (u − x, v − y)) = 0, (u, v) �= (x, y)

and further,

T r(W (x, y)∗ W (x, y)) = T r(I) = N

√

Thus, {W (x, y)/ N : x, y ∈ A} forms an orthonormal basis for L2 (A × A) and in particular, this is a set of N 2 linearly independent operators. We can express this orthonormality relations as T r(W (x, y)∗ W (u, v)) = N δ[x − u]δ[y − v] Now let H be a Gottesman subgroup of G = A × A. This means that for each pair (u, v), (x, y) ∈ H, the identity < u, y >∗ < v, x >= 1 holds. For arbitrary (u, v) ∈ G, we define the map ωH (u, v) : H → C such that ωH (u, v)(x, y) =< u, y >∗ < v, x > Then we have

ωH (u, v) = 1, (u, v) ∈ H

In other words, H ⊂ Ker(ωH ) = K say. Now we have that for (x, y), (u, v) ∈ H, W (x, y)W (u, v) =< y, u > W (x + u, y + v), W (u, v)W (x, y) =< v, x > W (u + x, v + y) and since

< y, u > / < v, x >=< y, u >< v, x >∗ = 1

it follows that

[W (x, y), W (u, v)] = 0∀(x, y), (u, v) ∈ H

Thus, there exists an onb for L2 (A) such that relative to this basis, W (x, y) is diagonal for all (x, y) ∈ H. W (x, y) = diag[αk (x, y), k = 1, 2, ..., N ], (x, y) ∈ H

˜ (x, y), (x, y) ∈ G by the equation Define W

˜ (x, y) W (x, y) = α1 (x, y)W

Note that |αk (x, y)| = 1∀k and also for (x, y), (u, v) ∈ H we have

˜ (x, y)W ˜ (u, v) = (α1 (x, y)α1 (u, v))−1 W (x, y)W (u, v) = W (α1 (x, y)α1 (u, v))−1 < y, u > W (x + u, y + v) =

Now the relation

< y, u > α1 (x + u, y + v) ˜ W (x + u, y + v) α1 (x, y)α1 (u, v)

W (x, y)W (u, v) =< y, u > W (x + u, y + v)

implies using the diagonal representation when restricted to H that α1 (x, y)α1 (u, v) =< y, u > α1 (x + u, y + v)

Quantum Quantum Mechanics Mechanics for Scientists and Engineers

and hence

6971

˜ (x, y)W ˜ (u, v) = W ˜ (x + u, y + v), (x, y), (u, v) ∈ H W

ie, the projective unitary representation W of G reduces to an ordinary unitary representation when restricted to H. An example: Let F be a finite field and consider the vector spaces V = Fp . Let L : V → V and M : V → V be two linear transformations such that LT M : V → V is symmetric. Define N : V → V by LT M = N + N T . For x, y ∈ V , we define the Weyl operator W (x, y) : L2 (V ) → L2 (V ) in the usual way. We note that V is a finite vector space over F and if F consists of a elements, then V will consist of ap elements and dimL2 (V ) = ap . Now, let χ0 be a character of the field F viewed as an Abelian group under addition. Consider for a fixed a ∈ V ˜ (u) = χ0 (aT u + uT N u)W (Lu, M u), u ∈ V W

Now

˜ (u + v) = χ0 (aT (u + v) + uT N u + v T N v)W (Lu, M u)W (Lv, M v) W χ0 (aT (u + v) + uT N u + v T N v)χ0 (uT M T Lv)W (L(u + v), M (u + v)) = χ0 (aT (u + v) + uT N u + v T N v + uT (N + N T )v)W (L(u + v), M (u + v)) = χ0 (aT (u + v) + (u + v)T N (u + v))W (L(u + v), M (u + v)) ˜ (u + v) =W

provided that we assume that the Weyl operator W (x, y) has been defined so that in the expression W (x, y)W (u, v) =< y, u > W (x + u, y + v), < y, u >= χ0 (y T u) or in other words, the character of V corresponding to any y ∈ V when V is viewed as an Abelian group under addition, ˜ (u) is a unitary representation of V . We note that H = {(Lu, M u) : u ∈ V } is a is given by u → χ0 (y T u). Thus u → W Gottesman subgroup of V × V . This is because, for u, v ∈ V , ¯0 (uT LT M v)χ0 (uT M T Lv) = χ0 (uT (M T L − LT M )v) = 0 < Lu, M v >∗ < M u, Lv >= χ since

χ ¯0 (x) = χ0 (x)−1

Construction of quantum error correcting codes using the Imprimitivity theorem of Mackey: Let G be a finite group acting on a finite  set X. For each x ∈ X let Px be an orthogonal projection onto a Hilbert space H such that Px Py = δx,y I, x∈X Px = I. Assume that g → Ug is a unitary representation of G in H satisfying the Imprimitivity condition Ug Px Ug∗ = Pgx , g ∈ G, x ∈ X

Let E ⊂ X and then we have

Ug P (E)Ug∗ = P (gE), g ∈ G

Now let N be a linear space of linear operators in H spanned by {Ug P (E) : g ∈ G} where E ⊂ X is fixed. We choose x ∈ X and ask the question when does the quantum code Px detect N . This happens iff Px Ug P (E)P (E)Ug∗ Px is proportional to Px for all g, h ∈ G. This is the same as saying that Px P (gE)Px = λPx This happens iff either x ∈ / gE in which case, the lhs is zero, or if gE = x for all g ∈ G which is impossible. Thus, Px corrects N iff g −1 x ∈ / E. Now we take F ⊂ X and derive the conditions for P (F ) to detect N . This happens iff P (F )Ug P (E)P (E)Ug∗ P (F ) is proportional to P (F ), ie, iff P (F ∩ gE) is proportional to P (F ). This happens iff either gE ∩ F = φ or gE ⊂ F . Now we consider the correction problem. P (F ) corrects N iff for all g, h ∈ G, P (F )Ug P (E)P (E)Uh∗ P (F ) = λP (F )

Quantum Mechanics for Scientists and Engineers

72

where λ may depend on g, h. This happens iff P (F )P (gE)P (gh−1 F )Uhg−1 = λP (F ) or equivalently, iff

P (F ∩ gE ∩ gh−1 F )Uhg−1 = λP (F )

Post-multiplying both sides of this equation by their adjoints gives us

P (F ∩ gE ∩ gh−1 F ) = |λ|2 P (F ) This is a necessary condition for P (F ) to correct N and happens only if for each g, h either F ∩ gE ∩ gh−1 F = φ or / gE or F ⊂ gE ∩ gh−1 F . As a special case, choosing F = {x}, we see that Px corrects N only if for each g, h either x ∈ or x = gh−1 x. We also derive the following conclusions from the above discussion. Px corrects N iff for each g, h, either x∈ / gE or x = gh−1 x and Ugh−1 Px = λPx We also observe that if Ua Px = λPa for some a ∈ G, then, Ua Px Ua∗ = |λ|2 Px or equivalently,

Pax = |λ|2 Px

which implies that |λ| = 1 and ax = x. Thus if we define

Gxeig = {g ∈ G : Ug Px = λPx } and then we find that

Gxiso = {g ∈ G : gx = x} Gxeig ⊂ Gxiso

and that Px corrects N = span{Ug P (E) : g ∈ G} iff for each g, h ∈ G (either x ∈ / gE or gh−1 ∈ Gxeig }) In particular, Px corrects span{Ug Py : g ∈ G} iff for each g, h ∈ H, (either x �= gy or gh−1 ∈ Gxeig ). As a special case, Px corrects span {Ug Px : g ∈ G} if for each g, h ∈ G either g ∈ / Gxiso or gh−1 ∈ Gxeig . Let C be a cross section of G/Gxiso . Let c1 , c2 ∈ C, g, h ∈ Gxeig . Then Px Uc∗1 g Uc2 h Px = Px Ug−1 c−1 c2 h Px = λPx 1

since ⊂ . Hence, Px corrects span{Ug : g ∈ for each x ∈ G. A linear algebraic example of a quantum error correcting code. Let the code subspace projection P be given by   Ir 0 P = ∈ Cn×n 0 0 Gxeig

Gxiso

CGxeig }

Let U be a fixed n × n unitary matrix and let the noise subspace of operators N be the span of the adjoints of all n × n matrices Nk having the block structure   λkj Ak λkj Bk Nkj = , k = 0, 1, ..., n/k − 1 Ckj Dkj where [Ak |Bk ] ∈ Cr×(n−r) is obtained as the kr+1 to (k+1)r rows of U arranged one below the other (k = 0, 1, ..., (n/k)− 1) and the λkj are arbitrary complex numbers. We find that since ∗ = δkm Ir Ak A∗m + Bk Bm

(obtained using U U ∗ = In ), we have ∗ Nkj Nml =



¯ kl δkm Ir λkj λ F2

F1 F3



Quantum Mechanics for Scientists and Engineers

73

where F1 , F2 , F3 are matrices of size r × (n − r), (n − r) × r and (n − r) × (n − r) respectively. Thus, ∗ ¯ kl P P = δkm λkj λ P Nkj Nml

and hence the quantum code P corrects N . [60] Quantum hypothesis testing: Let A, B be two density matrices and let 0 ≤ T ≤ I. The probability of making a correct decision when the density is A and the measurement is T is given according to quantum mechanical rules by T r(AT ). The probability of making a correct decision when the density is B is given by T r(B(I − T )). Now we wish to choose T such that T r(AT ) is a maximum subject to the constraint T r(BT ) ≤ α where α ∈ [0, 1) is given. Such a test corresponds to minimizing the error probability under A subject to the constraint that the error probability under B is smaller than a prescribed threshold. The function c → T (c) = {A ≥ cB} is assumed to be continuous on [0, ∞). We note that T (0) = I, T (∞) = 0 (By {U ≥ V } for Hermitian operators U, V , we mean the orthogonal projection onto the space spanned by all vectors v for which (U − V )v = λv for some λ ≥ 0.) Hence, T r(BT (c)) takes all values in [0, 1) as c varies over [0, ∞). Let α ∈ [0, 1) and choose c such that T r(BT (c)) = α. Then, we claim that T (c) is an optimal test. To see this, suppose 0 ≤ T ≤ I is any other test (ie measurement or positive operator) such that T r(BT ) ≤ α. Then T r(AT ) = T r((A − cB)T ) + cT r(BT ) ≤ T r((A − cB)T ) + cα = T r((A − cB)T (c)T ) + T r((A − cB){A < cB}T ) + cα ≤ T r((A − cB)T (c)) + cα = T r(AT (c))

Here, we make use of the fact that since A − cB commutes with T (c) and T (c)2 = T (c), we have T r((A−cB)T (c)T ) = T r(T (c)(A−cB)T (c)T ) = T r(T 1/2 T (c)(A−cB)T (c)T 1/2 ) ≤ T r(T (c)(A−cB)T (c)) = T r((A−cB)T (c))

since 0 ≤ T 1/2 ≤ I and T (c)(A − cB)T (c) ≥ 0. Now, we can derive bounds on the error probabilities. We have for s ≥ 0, P1 (c) = T r(A(I − T (c)) = T r(A{A < cB}) ≤ cs T r(A1−s B s ) Thus,

log(P1 (c)) ≤ s.log(c) + log(T r(A1−s B s ) = s(R + log(A1−s B s )/s)

where

R = log(c)

Consider now the function

f (s) = sR + log(T r(A1−s B s ), s ≥ 0

We wish to select R so that f (s) has a minimum at s = 0. For this, we require that f � (0) = 0, f �� (0) ≥ 0. Now, f � (0) = R − D(A|B) where

D(A|B) = T r(A(logA − logB))

is the relative entropy between A and B. Thus, the first condition gives R = D(A|B). The second condition f �� (0) ≥ 0 gives d [T r(−A1−s log(A)B s + A1−s B s log(B))/T r(A1−s B s )]|s=0 ≥ 0 ds or equivalently, T r(A(log(A))2 − 2Alog(A)log(B) + A(log(B)2 ) + (T r(Alog(A) − Alog(B)))2 ≥ 0 which is true since by the Schwarz inequality, |T r(Alog(A)log(B))|2 ≤ T r(A(log(A))2 ).T r(A.(log(B))2 ) It follows that

P1 (eR ) = 0

if R = D(A|B) and for the second kind of error probability, we get P2 (c) = T r(BT (c)) = T r(B{A < cB}) = 1 − T r(B{A > cB})

Quantum Mechanics for Scientists and Engineers

74

Also, for s ≤ 1

T r(B{A > cB}) ≤ cs−1 T r(A1−s B s )

so We now find on letting s = 0 that

log(1 − P2 (c)) ≥ (s − 1)log(c) + log(T r(A1−s B s )) log(1 − P2 (c)) ≥ −log(c) + 1

so if we put R = log(c), we get

log(1 − P2 (eR )) ≥ 1 − R

or equivalently,

P2 (eR ) ≤ 1 − exp(1 − R)

We also note that So, taking log(c) = D(A|B) − δ gives

lims→0 log(T r(A1−s B s )/s = −D(A|B)

log(1 − P2 (c)) ≥ −log(c) + s(log(c) + log(T r(A1−s B s ))/s) ≥ −D(A|B) + δ + s(D(A|B) − δ − D(A|B)) = −D(A|B) + (1 − s)δ

for s ≤ 1. We also recall that

log(P1 (c)) ≤ s(log(c) + log(T r(A1−s B s ))/s)

= s(D(A|B) − δ + D(A|B) + �(s)) = −s(δ − �(s))

where

�(s) → 0, s → 0

Hence, if s is sufficiently small, say s ≤ s0 , then �(s) < δ/2 and for all such s, we have log(P1 (c)) ≤ −sδ/2 ≤ −s0 δ/2 Note: Consider the function We have

g(s) = log(T r(A1−s B s ))/s

g � (s) = T r(−A1−s log(A)B s + A1−s B s log(B))/(s.T r(A1−s B s )) − log(T r(A1−s B s ))/s2

The limit of this as s → 0 is the same as the limit of

−D(A|B)/s + D(A|B)/s as s → 0 which is zero. We also note that the limit of g �� (s) as s → 0 is the same as the limit of T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + D(A|B)/s2 + (D(A|B)2 /s) + (D(A|B)/s2 ) + (2/s3 )log(T r(A1−s B s ) which is same as the limit of T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + D(A|B)/s2 + (D(A|B)2 /s) + (D(A|B)/s2 ) − 2D(A|B)/s2 The above equals

T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + (D(A|B)2 /s

which is positive. Hence, g(s) for s ≥ 0 attains its minimum value of D(A|B) at s = 0.

[61] The Sudarshan-Lindblad equation for observables on the quantum system has the following general form: X � = i[H, X] −

1 ∗ (Lk Lk X + XL∗k Lk − 2L∗k XLk ) 2

Assume that H=

k

1 T 1 p F1 p + q T F2 q 2 2

Chapter

)/s2 ) − 2D(A|B)/s2

7

Master Equations, Non-Abelian Gauge Fields T r(A(log(A))2 + B(log(B))2 − 2Alog(A)log(B))/s + (D(A|B)2 /s

[61] The Sudarshan-Lindblad equation for observables on the quantum system has the following general form: X � = i[H, X] −

1 ∗ (Lk Lk X + XL∗k Lk − 2L∗k XLk ) 2 k

Assume that H= and

1 T 1 p F1 p + q T F2 q 2 2

Lk = λTk p + μTk q, k ≥ 1

We have

p = ((pn )), q = ((qn )), [qn , pm ] = iδn,m L∗k Lk X

say. We thus have

+ XL∗k Lk − 2L∗k XLk = L∗k [Lk , X] + [X, L∗k ]Lk = θk (X) [Lk , qn ] = [λTk p, qn ] = −iλk [n], ¯ T p] = iλ ¯ k [n] [qn , L∗k ] = [qn , λ k

so so

¯T p + μ ¯ k [n](λT p + μT q) ¯Tk q) + iλ θk (qn ) = −iλk [n](λ k k k 

θk (q) =

k



¯ k λT )p + (−iλμ∗ + iλ ¯ k μT )q] [(−iλk λ∗k + iλ k k k

k

=

 [2Im(λk λ∗k )p + 2Im(λk μ∗k )q] k

Likewise, so so

¯Tk q] = −i¯ μk [n] [Lk , pn ] = [μTk q, pn ] = iμk [n], [pn , L∗k ] = [pn , μ ¯T p + μ ¯Tk q) − i¯ μk [n](λTk p + μTk q) θk (pn ) = iμk [n](λ k θk (p) = −2Im(μk λ∗k )p − 2Im(μk μ∗k )q

We write

θ=



θk

k

Then,

[H, qn ] = [pT F1 p/2, qn ] = −i ie, Likewise,



F1 [n, m]pm

m

[H, q] = −iF2 p [H, p] = iF1 q

[62] The Yang-Mills field and its quantization using path integrals: Let G be a finite dimensional Lie group and g its Lie algebra. For simplicity, assume that G is a subgroup of U (n, C). Then g consists of n × n complex skew-Hermitian matrices. We can thus choose a basis {iτa : a = 1, 2, ..., n} for g with the τa� s Hermitian matrices. Thus, the commutation relations of these basis elements has the form [iτa , iτb ] = −C(abc)iτc

where the C(abc) are real constants and summation over the repeated index c is understood. Equivalently, these commutation relations can be expressed as [τa , τb ] = iC(abc)τc We let Aμ : R4 → g be the gauge fields. Thus, the covariant derivatives in terms of these gauge fields are defined by where e is a real constant. We can write

∇μ = ∂μ − eAμ

Aμ (x) = iAaμ (x)τa

Quantum Mechanics for Scientists and Engineers

76

where the Aaμ (x)� s are now real valued fields. There are in all 4n of such fields. We assume that the wave function ψ(x) ∈ R4n satisfies an equation of the form [γ μ i∂μ ⊗ In + eAaμ (x)γ μ ⊗ τa − mI4n ]ψ(x) = 0 where γ μ are the four Dirac γ matrices forming a basis for the Clifford algebra in C4×4 . Formally, we can write the above wave equation as γ μ (i∂μ − ieAμ ) − m)ψ = 0 or equivalently as

[γ μ (i∂μ + eAaμ τa ) − m]ψ = 0

This eqations can be derived from the variational principle

δψ S = 0 where S[ψ] =



ψ¯∗ γ 0 (γ μ (i∂μ + eAaμ τa ) − m)ψd4 x =



Ld4 x

Note that γ 0 γ μ are Hermitian matrices and so is γ 0 γ μ ⊗ τa . Hence using integration by parts, it follows that S[ψ] is real. In terms of the gauge covariant derivative defined above, we have L = ψ ∗ γ 0 (iγ μ ∇μ − m)ψ We consider the following transformation of the wave function ψ(x) → ψ � (x) = g(x)ψ(x) where g(x) ∈ G is to be intepreted as I4 ⊗ g(x). This is called a local gauge transformation of the wave function. Then we consider a corresponding transformation of the gauge field Aμ (x): Aμ (x) → A�μ (x) so that if then

∇�μ = ∂μ − eA�μ ∇�μ ψ � = g(x)∇μ ψ

If this happens, the the above Lagrangian density L will be invariant under a local gauge transformations of the wave function and the gauge fields. To get this, we must satisfy g(x)(∂μ − eAμ (x))ψ(x) = (∂μ − eA�μ (x))ψ � (x) which is equivalent to

= (∂μ − eA�μ (x))g(x)ψ(x) A�μ = gAμ g −1 + e−1 (∂μ g)g −1

Note that we get a gauge invariant Lagrangian by considering ψ ∗ γ 0 γ μ (∂μ −eAμ )ψ which is the same as ψ ∗ gamma0 γ μ ∇μ ψ or more precisely, ψ ∗ (γ 0 γ μ ⊗ ∇μ )ψ

we note that the gauge transformation g(x) is actually to be interpreted as I4 ⊗ g(x) and it acts only on the second component in the tensor product of C4 ⊗ Cn . [63] A general remark on path integral computations for gauge invariant actions. Suppose that we have an action integral I(φ) of the fields φ(x) and we compute a path integral of the form  S = exp(iI(φ))B[φ]dφ

Quantum Mechanics for Scientists and Engineers

77

Suppose that the action integral I(φ) is invariant under a Gauge transformation φ → φΛ . Suppose also that the path integral measure dφ = Πx∈R4 dφ(x) is invariant under the same Gauge transformation. Then, we can write  S = exp(iI(φΛ )B[φΛ ]dφΛ 

=

exp(iI(φ))B[φΛ ]dφ

If follows that for an function C(Λ) on the gauge group (Λ ∈ G), we have    S C(Λ)dΛ = exp(iI(φ))dφ B[φΛ ]C(Λ)dΛ Now let Λ, λ be two gauge group transformations. Then, we have  B[φΛ ]C(Λ)dΛ =



B[φΛoλ ]C(Λoλ)dλ

assuming that the measure dλ on the gauge group is a left invariant Haar measure so that dΛoλ = dλ. Define for any functional F of the fields φ F(φ, x) = (δF (φΛ) )/δΛ)(x)|Λ=Id We have

(δF (φΛoλ )/δλ)(x) = (δF (φΛoλ )/δ(Λoλ))(x)(δ(Λoλ)/δλ)(x)

It follows that on evaluating both sides at λ = Id, the identity gauge transformation, we get F˜ (φ, Λ, x) = F (φ, Λ, x)G(Λ) where

F˜ (φ, Λ, x) = (δF (φΛoλ )/δλ)(x)|λ=Id ,

and

G(Λ) = (δ(Λoλ)/δλ)|λ=Id F (φ, Λ, x) = (δF (φΛ )/δΛ)(x)

Now consider S= =





exp(iI(φ))B[F [φ]]F (φ, x)Πdφ(x)

exp(iI(φ))B[F [φΛ ]]Πx F (φΛ , x)dφ(x)

provided that we assume invariance of the path measure exp(iI(φ))dφ under gauge transformations Λ, ie, exp(iI(φΛ ))dφΛ = exp(iI(φ))dφ It follows that S 

= So by choosing C = G, we get





C(Λ)dΛ =

exp(iI(φ))B[F [φΛ ]]Πx F (φΛ , x)dφ(x)C(Λ)dΛ exp(iI(φ))B[F [φΛ ]](δF [φΛ ]/δΛ)G(Λ)−1 C(Λ)dΛ

S



C(Λ)dΛ =

Quantum Mechanics for Scientists and Engineers

78 76



exp(iI(φ))B[F [φ]]dF [φ] =



exp(iI(φ))B[φ]dφ

establishing the invariance of the scattering matrix S under the gauge fixing functional F . [64] Calculation of the normalized spherical harmonics. Ylm (θ, φ) = Plm (cos(θ))exp(imφ) where f (x) = Plm (x) satisfies the modified Legendre equation ((1 − x2 )f � )� + (l(l + 1) − m2 /(1 − x2 ))f = 0 or equivalently, or

(1 − x2 )2 f �� − 2x(1 − x2 )f � + (l(l + 1)(1 − x2 ) − m2 )f = 0 (1 − x2 )2 f �� − 2x(1 − x2 )f � + (l(l + 1) − m2 − l(l + 1)x2 )f = 0

For simplicity of notation, let

λ = l(l + 1)

Then, the above equation is the same as (1 + x4 − 2x2 )f �� − 2x(1 − x2 )f � + (λ − m2 − λx2 )f = 0 Let

f (x) =



c(n)xn

n≥0

Then, we get on substituting this into the above differential equation,     c(n)n(n − 1)(xn−2 + xn+2 − 2xn ) − 2 nc(n)(xn − xn+2 ) + (λ − m2 ) c(n)xn − λ c(n)xn+2 = 0 n

n

n

n

Equating coefficients of xn gives (n + 2)(n + 1)c(n + 2) + (n − 2)(n − 3)c(n − 2) − 2n(n − 1)c(n) − 2nc(n) + 2(n − 2)c(n − 2) + (λ − m2 )c(n) − λc(n − 2) = 0 or or

(n + 2)(n + 1)c(n + 2) + ((n − 1)(n − 2) − λ)c(n − 2) + (λ − m2 − 2n2 )c(n) = 0 (n + 4)(n + 3)c(n + 4) + (n(n + 1) − l(l + 1))c(n) + (l(l + 1) − m2 − 2(n + 2)2 )c(n + 2) = 0

If l is not a non-negative integer, then there will be nonzero c(n)� s for arbitrarily large n and as n → ∞, we would get c(n + 4) + c(n) − 2c(n + 2) ≈ 0, n → ∞ or equivalently,

c(n + 4) − c(n + 2) − (c(n + 2) − c(n)) ≈ 0

which would imply that c(n + 2) − c(n) converges to a constant, say K. Then c(2n) or c(2n + 1) for large n behaves as Kn and since the series n≥0 nx2n behaves as x/(1 − x2 )2 which is not integrable over x ∈ [0, 1] since it has a singularity at x = 1 (ie, at θ = 0), it follows that the series has to terminate at some finite N , ie, there must exist a finite integer N such that c(n) = 0 for all n > N which is equivalent to saying that f (x) must be a polynomial. This can happen only if we impose the condition l = N and c(N + 2) = 0. Assume first that N = 2r is even. Then, we may assume that c(2n + 1) = 0 for all n and (2n + 4)(2n + 3)c(2n + 4) + (2n(2n + 1) − 2r(2r + 1))c(2n) + (2r(2r + 1) − m2 − 4(n + 1)2 )c(2n + 2) = 0 for n = 0, 1, ..., r − 1. Putting n = r we then get c(2r + 2) = 0 since we are assuming that c(2r + 4) = 0. Thus, c(2n) = 0 for all n > r and we get that r  c(2n)x2n f (x) = n=0

Quantum Mechanics for Scientists and Engineers

79

where c(0) is arbitrary, c(2) is obtained by putting n = −1 in the above difference equation and using c(k) = 0, k < 0: c(2) = (m2 − 2r(2r + 1))c(0)/2 and (2n + 4)(2n + 3)c(2n + 4) = −[(2n(2n + 1) − 2r(2r + 1))c(2n) + (2r(2r + 1) − m2 − 4(n + 1)2 )c(2n + 2)]/(2n + 4)(2n + 3) for n = 0, 1, ..., r − 2. In a similar way, we can describe the polynomials Plm (x) for l odd, ie l = 2r + 1. We assume that c(2n) = 0 for all n and get using the recursion (n + 4)(n + 3)c(n + 4) + (n(n + 1) − (2r + 1)(2r + 3))c(n) + ((2r + 1)(2r + 3) − m2 − 2(n + 2)2 )c(n + 2) = 0 at n = −1,

6c(3) + ((2r + 1)(2r + 3) − m2 − 2)c(1) = 0

so that

c(3) = (m2 + 2 − (2r + 1)(2r + 3))c(1)/6

and then replacing n by 2n + 1 in the above recursion,

c(2n+5) = −[(2n+5)(2n+4)]−1 [((2n+1)(2n+3)−(2r+1)(2r+3))c(2n+1)+((2r+1)(2r+3)−m2 −2(2n+3)2 )c(2n+3)] for n = 0, 1, ..., r − 2 and we may assume that c(2n + 1) = 0 for n = r + 1, r + 2, .... The values of c(0) and c(1) for the two cases are determined by the normalization condition  1 Plm (x)2 dx = 1 2π −1

This guarantees that



0

π



2π

0

|Ylm (θ, φ)|2 sin(θ)dθdφ = 1

A technique for calculating the representation matrix πl (R) for R ∈ SO(3). Here, πl (R) is defined by l 

Ylm (R−1 n ˆ) =

[πl (R)]m m Ylm (ˆ n)

m =−l

From this equation, it is clear that [πl (R)]m m =



For example, taking we have

Ylm (R−1 n ˆ )Y¯lm (ˆ n)dΩ(ˆ n)

R = Rx (β) R−1 [cos(φ)sin(θ), sin(φ)sin(θ), cos(θ)]T =

[cos(φ)sin(θ), sin(φ)sin(θ)cos(β) − cos(θ)sin(β), sin(φ)sin(θ)sin(β) + cos(θ)cos(β)]T

Denoting, this new point by (θ� , φ� ), we get

θ� = cos−1 (sin(φ)sin(θ)sin(β) + cos(θ)cos(β)),

We then have

φ� = tan−1 [(sin(φ)sin(θ)cos(β) − cos(θ)sin(β))/cos(φ)sin(θ)] [πl (Rx (β))]m m =



0

π



0

2π

Y¯lm (θ, φ)Ylm (θ� , φ� )sin(θ)dθdφ

A MATLAB programme for tabulating the matrix elements [πl (Rx (−β))]m m would then proceed along the following lines: We store [πl (Rx (−2πk/N ))]m m with k = 0, 1, ..., N − 1, m� , m = −l, −l + 1, ..., l as a two dimensional array A with matrix elements A[k + 1, (2l + 1)(m� + l) + m + 1] of size N × (2l + 1)2 . Thus,

Quantum Mechanics for Scientists and Engineers

80

for k = 0 : N − 1 for m� = −l : l for m = −l : l sum = 0; for r = 0 : N − 1 for s = 0 : N − 1 θ = πr/N φ = 2πs/N β = 2πk/N θ� = cos−1 (sin(φ)sin(θ)sin(β) + cos(θ)cos(β)), φ = tan−1 [(sin(φ)sin(θ)cos(β) − cos(θ)sin(β))/cos(φ)sin(θ)] �

sum = sum + conj(Ylm (θ, φ)) ∗ Ylm (θ� , φ� ) ∗ (π/N ) ∗ (2π/N ); end; end; A[k + 1, (2l + 1)(m� + l) + m + 1] = sum; end; end; end;

[65] Volterra systems in quantum mechanics: The Hammiltonian has the form H(t) = H0 + f (t)V0 Let U (t) be the Schrodinger evolution operator: U � (t) = −iH(t)U (t), U (0) = I Then,

U (t) = U0 (t)W (t), U0 (t) = exp(−itH0 )

and W (t) has the Dyson series W (t) = I +



(−i)n



f (t1 )...f (tn )V (t1 )...V (tn )dt1 ...dtn

0 ua (t) =< f φ(u)|jt (θ˜ba (X)θ˜dc (X))|f φ(u) > �bc ud (t)¯ uk (t) =< f φ(u)|jt (θjk (X)θpj (X)|f (φ(u) > up (t)¯ ¯k (t) + < f φ(u)|jt (θjk (X)θ0j (X))|f φ(u) > u + < f φ(u)|jt (θj0 (X)θkj (X))|f φ(u) > uk (t)

[66] Quantum scattering theory, the wave operators and the scattering matrix. A is the free particle Hamiltonian and B = A + V is the Hamiltonian after the free particle starts interacting with the scatterer. Let |φi > be the ”in state” ie the free particle state at time t = 0 after it arrives from an infinite distance from the scatterer at time t = −∞. Then, we see that if |ψi > is the scattered state at time t = 0 ie after the particle interacts with the scatterer, we must have limt→−inf ty exp(−itB)|ψi > −exp(−itA)|φi >= 0 so where

|ψi >= Ω− |φi > Ω− = limt→−∞ exp(itB)exp(−itA)

Likewise let |φo > be the ”out-state”, ie the free particle state at time t = 0 which evolves under the free particle Hamiltonian A to the scattered state at time t = ∞. Let |ψo > be the corresponding scattered state at time t = 0. Then we must have limt→∞ exp(−itB)|ψo > −exp(−itA)|φo >= 0

Quantum Quantum Mechanics Mechanics for Scientists and Engineers

8385

and hence

|ψo >= Ω+ |φo >

where

Ω+ = limt→∞ exp(itB)exp(−itA)

It should be noted that the wave operators Ω± must be defined on appropriate domains. We have < ψo |ψi >=< φo |Ω∗+ Ω− |φi > and the magnitude square of this quantity gives the probability that the free particle starting at t = −∞ in the state |phio > interacts with the scatterer and then gets scattered at time t = ∞ to the state |φo > . This proability can be expressed as P (φi → φo ) = | < φo |S|φi > |2 where

S = Ω∗+ Ω−

is the scattering matrix or scattering operator. We have the following interesting relations:  ∞  ∞ d (exp(itB)exp(−itA))dt = i exp(itB)V.exp(−itA)dt Ω+ − I = dt 0 0  0 d Ω− − I = − (exp(itB)exp(−itA))dt −∞ dt  0 = −i exp(itB)V.exp(−itA)dt −∞



= −i

∞

exp(−itB)V.exp(itA)dt

0

We can derive some formal expressoins for the wave operators based on the above formulas: Let   A= λEA (dλ), B = λEB (dλ) R

R

be the spectral representations of the self adjoint operators A and B respectively. Then, we get from the above  exp(it(B − lambda + i�))V EA (dλ)dt Ω+ − I = i [0,∞)×R



=− It follows that Ω+ = =



R

R



R

(B − λ + i�)−1 V EA (dλ)

(I − (B − λ + i�)−1 V EA (dλ)

(B − λ + i�)−1 ((A − λ + i�)−1 EA (dλ)

= (i�)−1



R

(B − λ + i�)−1 EA (dλ)

Example: Let A be self-adjoint and B = A + V where V is a bounded operator. Consider W (t) = exp(−itA).exp(itB) = exp(−itA).exp(it(A + V )) We know that W (t) has the Dyson series  W (t) = I + U0 (−t1 )V U0 (t2 − t1 )V U0 (t3 − t2 )...U0 (tn − tn−1 )V U0 (−tn )dt1 ...dtn n≥1

where

0 ∀y ∈ D(S) where we have used D(S) ⊂ D(S ∗ ). Now, we show that zn is a convergent sequence. We have that lim(S−i)zn = (S ∗ −i)x exists and hence (S − i)zn is a Cauchy sequence, ie (S − i)(zn − zm ) → 0, n, m → ∞ and hence or equivalently, Now

� (S − i)(zn − zm ) �2 → 0 � S(zn − zm ) �2 + � (zn − zm ) �2 +2Im(< S(zn − zm ), zn − zm >) → 0 < Sz, z >=< z, Sz > ∀z ∈ D(S)

Quantum Mechanics for Scientists and Engineers 90

93

since S ⊂ S ∗ . Thus, Im(< Sz, z >) = 0∀z ∈ D(S). Thus, Im(< S(zn − zm ), zn − zm >) = 0. This proves that � S(zn − zm ) �2 + � (zn − zm ) �2 → 0 and hence

� zn − zm �→ 0

Thus, zn is Cauchy and hence convergent. Let zn → z. Then since (S +i)zn converges, it follows that Szn also converges. ¯ and we get Thus, z ∈ D(S) < x, (S + i)y >=< z, (S + i)y > ∀y ∈ D(S)

Thus,

x − z ⊥ R(S + i)

¯ Thus, we have proved that and since R(S + i) is dense in H, it follows that x = z ∈ D(S). ¯ D(S ∗ ) ⊂ D(S)

¯ ⊂ D(S ∗ ) ⊂ D(S) ¯ from which we deduce that D(S ∗ ) = D(S) ¯ Now S ⊂ S ∗ implies S¯ ⊂ S ∗ since S ∗ is closed. Thus, D(S) and therefore S¯ = S ∗ . Further, S ⊂ S ∗ impliesS ∗∗ ⊂ S ∗ = S¯ and since S ∗∗ is a closed extension of S, it follows that ¯ Thus, S ∗ = S¯ = S ∗∗ which proves that S¯ is self-adjoint, ie, (S) ¯ ∗ = S. ¯ S ∗∗ = S. Remarks: (a) If A, B are operators in H such that A ⊂ B, then B ∗ ⊂ A∗ . Indeed, let x ∈ D(B ∗ ). Then < B ∗ x, y >=< x, By >=< x, Ay >, ∀y ∈ D(A) ⊂ D(B). Hence, x ∈ D(A∗ ) and < B ∗ x − A∗ x, y >= 0∀y ∈ D(A). This proves that B ∗ x − A∗ x ⊥ D(A) and since A is densely defined, it follows that B ∗ x − A∗ x = 0, ie, B ∗ x = A∗ x, proving the claim. (Note that we are assuming without any loss in generality that all operators are densely defined). (b) If A is any operator, then A∗ is closed. Indeed, let xn ∈ D(A∗ ), xn → x, A∗ xn → y. Then for all z ∈ D(A), we have < A∗ xn , z >→< y, z >, < A∗ xn , z >=< xn , Az >→< x, Az > and hence, < y, z >=< x, Az > This proves that y ∈ D(A ) and A x = y, proving that A∗ is closed. ¯ then B = A. ¯ Indeed, it suffices (c) If A is any closable operator and B is a closed operator such that A ⊂ B ⊂ A, ¯ and ¯ Then there exists a sequence xn ∈ D(A) such that xn → x, Axn → Ax to show that A¯ ⊂ B. So let x ∈ D(A). ¯ and since B is closed, it follows that x ∈ D(B), Bx = Ax. ¯ since A ⊂ B, we have xn ∈ D(B), xn → x, Bxn = Axn → Ax ¯ then A¯ ⊂ B. ¯ This This proves the claim. A related statement is that if B is any closed operator such that A ⊂ B, ¯ implies Gr(A) ⊂ Gr(B) ¯ ¯ ¯ follows from the implications A ⊂ B and hence Gr(A) ⊂ Gr(∗B). Another related remark is that if A is any closable operator then A∗∗ is a closed extension of A and hence by the above A¯ ⊂ A∗∗ . For suppose ¯ y = Ax ¯ and we have for z ∈ D(A∗ ), xn ∈ D(A), xn → x, Axn → y. Then, x ∈ D(A), ∗

∗

< y, z >= lim < Axn , z >= lim < xn , A∗ z >=< x, A∗ z > Hence, x ∈ D(A∗∗ ) and y = A∗∗ x. This proves that A¯ ⊂ A∗∗ . Now we come to the last statement. Let S ⊂ S ∗ and R(S ± i) = H. Then, we have to show that S ∗ = S. Indeed, let x ∈ D(S ∗ ). Then there exists a y ∈ D(S) such that (S ∗ − i)x = (S − i)y since R(S − i) = H. Thus for any z ∈ D(S) ⊂ D(S ∗ ), it follows that < (S ∗ − i)x, z >=< x, (S + i)z >=< (S − i)y, z >=< y, (S ∗ + i)z >=< y, (S + i)z > and hence, ie and therefore, This proves that and hence

< x − y, (S + i)z >= 0, z ∈ D(S) x−y ⊥H x = y ∈ D(S) D(S ∗ ) ⊂ D(S) ⊂ D(S ∗ ) D(S) = D(S ∗ )

Quantum Mechanics for Scientists and Engineers 91

94

so that

S = S∗

The proof is complete. A.3.Stochastic processes Here we discuss various kinds of noise processes that arise in perturbed quantum systems and explain how to compute transition probabilities of quantum systems in the presence of such noise processes. [1] Brownian motion [2] Poisson process [3] Compound Poisson processes [4] Levy processes [5] Reflected and absorbed Brownian motion Let B(t), t ≥ 0 be a standard Brownian motion starting at any given point and let T0 = inf (t ≥ 0 : B(t) = 0) Obviously, the statistics of T0 will depend on B(0). Absorbed Brownian motion is the Brownian motion process upto time T0 and after time T0 it is set equal to zero. We denote this process by X(t), t ≥ 0. We can show that X is a Markov process by the following simple intuitive reasoning: If at time t0 the process X(t0 ) = 0, then obviously X(t) = 0 for all t > t0 . On the other hand, if at time t0 , X(t0 ) = x > 0, then obviously B(t0 ) = x and by the Markovian property of B(.), the statistics of X(t) for all t > t0 will depend only on (t0 , x). It remains to compute the transition probability of X(.). For x, y > 0, we have P (X(t) > x|X(0) = y) = Py (B(t) > x, T0 > t) = Py (B(t) > x, mins≤t B(s) > 0) = P0 (B(t) > x − y, mins≤t B(s) > −y)

= P0 (−B(t) > x − y, mins≤t (−B(s)) > −y)

= P0 (B(t) < y − x, −maxs≤t (B(s)) > −y) = P0 (B(t) < y − x, St < y) = P0 (B(t) < y − x) − P0 (B(t) < y − x, St > y)

where

St = maxs≤t B(s)

By the reflection principle, Hence,

P0 (B(t) < y − x, St > y) = P0 (B(t) > y + x) P (X(t) > x|X(0) = y) = P0 (B(t) < y − x) − P0 (B(t) > y + x)

and hence the transition density of X(t) (taking values in [0, ∞)) is given by qt (x|y) = − Also

d P (X(t) > x|X(0) = y) = (2πt)−1/2 (exp(−(x − y)2 /2t) − exp(−(x + y)2 /2t)), x, y > 0 dx P (X(t) = 0|X(0) = y) = Py (T0 < t) = Py (mins≤t B(s) ≤ 0) = P0 (mins≤t B(s) ≤ −y)

= P0 (St > y) = 2P0 (B(t) > y) the last step following from the reflection principle. We thus verify that  ∞ qt (x|y)dx + 2P (B(t) > y) P (X(t) ≥ 0|X(0) = y) = 0

= P (B(t) > −y) − P (B(t) > y) + 2P (B(t) > y) = P (B(t) > −y) + P (B(t) > y) = P (B(t) < y) + P (B(t) > y) = 1

[6] Bessel processes [7] The Brownian local time process

Quantum Mechanics for Scientists and Engineers 92

95

[1] Let θ(x) be the unit step function. Let B(t) be Brownian motion and L(t) its local time at zero, ie, dL(t) = δ(B(t))dt Let

τ (s) = inf {t > 0 : L(t) ≥ s}

then, clearly τ (s) is a stopping time for each s. By Ito’s formula,

1 d(B(t)θ(B(t)) = θ(B(t))dB(t) + δ(B(t))dt + B(t)δ � (B(t))dt 2 Now, Hence, and hence

xδ � (x) = (xδ(x))� − δ(x) = −δ(x) 1 θ(B(t))dB(t) = d(B(t)θ(B(t)) − dL(t) 2 

t

0

1 θ(B(s))dB(s) = B(t)θ(B(t)) − L(t) 2

Therefore applying Doob’s optional stopping theorem to the exponential martingale  t  t θ(B(s))dB(s) − (a2 /2) θ(B(s))ds) M (t) = exp(−a 0

0

and the stopping time τ (t) gives 1 = E[M (τ (t))] = exp(at − (a2 /2)



τ (t)

θ(B(s))ds)

0

(Note that B(τ (t)) = 0, L(τ (t)) = t). Thus, E[exp(−s



τ (t)

√ θ(B(s))ds)] = exp(− 2st)

0

(Reference: Marc Yor, Some aspects of Brownian motion, Birkhauser). [8] Stochastic integration with respect to continuous semi-Martingales. Let Xt be continuous semimartingale. By the Doob-Meyer theorem, it can be decomposed as Xt = At + Mt where At is a process of bounded variation and Mt is a martingale. We can write − At = A+ t − At − where A+ t and At are increasing processes. The integral of a bounded adapted process Yt w.r.t At is defined as an ordinary Riemann-Stieltjes integral or equivalently as a Lebesgue integral. Since Mt is not a process of bounded variation but has a well defined finite quadratic variation, its integral must be defined in the Ito sense. Specifically, defining T n Yt dMt as the mean square limit of partial sums of the form Sn = k=1 Ytn,k (Mtn,k+1 −Mtn,k ) as maxk |tn,k+1 −tn,k | → 0 0. We have for n > m,  E(Sn − Sm )2 = E(Yt2n,k )E(Mtn,k+1 − Mtn,k )2

+

 k

k

E(Yt2m,k )(E(Mtm,k+1 − Mtm,k )2 ) −2E(Sn Sm )

Suppose we assume that the partition {tn,k } is a refinement of the partition {tm,k }. Consider for example a term like 3 Ys1 ((Ms2 − Ms1 ) in Sm and a term like k=1 Ytk (Mtk+1 − Mtk ) in Sn where s1 = t1 < t2 < t3 < t4 = s2 . The expected value of their product is an example of a term in E(Sn Sm ). This term can be expressed as E(Yt21 (Mt2 − Mt1 )2 ) + E(Yt1 Yt2 (Mt3 − Mt2 )2 )

Quantum Mechanics for Scientists and Engineers 93

96

+E(Yt1 Yt3 (Mt4 − Mt3 )2 )

From this observation, it is clear that if the quantity  T E Yt2 d < M >t < ∞ 0

then,

E((Sn − Sm )2 ) → 0, n, m → ∞

and hence by the fact that L2 (Ω, F, P ) is a Hilbert space, it follows that there exits a random variable S∞ which we T denote as 0 Yt dMt and call it the stochastic integral of Y w.r.t M over the interval [0, T ]. We are assuming that t the increasing function t →< M >t = 0 (dMs )2 defines a finite measure on a bounded interval like [0, T ] and that the adapted process Yt is Riemann integrable w.r.t this measure. More precisely, the stochastic integral w.r.t a martinagle can be defined for almost surely progressively measurable processes Yt that are integable w.r.t the random measure d < M >t over finite intervals. For a detailed discussion of this construction see the book by Karatzas and Shreve on ”Brownian motion and stochastic calculus” or the book by Revuz and Yor on ”Continuous martingales and Brownian motion. Remark: Consider the quantum system dU (t) = [−(iH(t)dt +

p p  1  Vk (t)Vm (t)d < Mk , Mm > (t)) − i Vk (t)dMk (t)]U (t) 2 k,m=1

k=1

where Mk (t), k = 1, 2, ..., p are Martingales. It is easily verified using Ito’s formula that U (t) is unitary for all t if U (0) = I. We introduce a perturbation parameter � into the martingale terms and obtain dU (t) = [−(iH(t)dt + �2

p p  1  Vk (t)Vm (t)d < Mk , Mm > (t)) − i� Vk (t)dMk (t)]U (t) 2 k,m=1

k=1

We solve for U (t) using perturbation theory: U (t) =



�m Um (t)

m≥0

Equating same powers of � gives successively

dU0 (t) = −iH(t)U0 (t)dt,  Vk (t)U0 (t)dMk (t), dU1 (t) = −iH(t)U1 (t) − i k

 1 Vk (t)Vm (t)U0 (t)d < Mk , Mm > (t) − i Vk (t)U1 (t)dMk (t) dU2 (t) = −iH(t)U2 (t) − 2 k,m

k

and in general,

dUr (t) = −iH(t)Ur (t) −

 1 Vk (t)Vm (t)Ur−2 (t)d < Mk , Mm > (t) − i Vk (t)Ur−1 (t)dMk (t), r ≥ 2 2 k,m

k

After calculating U0 (t) +

N 

�r Ur (t)

r=1

we calculate the transition probablity from an initial state |i > to a final state |f > in time T : E[| < f |U0 (T ) +

N  r=1

�r Ur (T )|i > |2 ]

where the average is taken over the probability distribution of the martingale processes over [0, T ]. Examples of Martingales and the Ito formula for them (a)  t  M (t) = f (s, ω)dB(s, ω) + g(s, x, ω)(N (ds, dx, ω) − λ(s)dF (x)ds) 0

s≤t,x∈E

Quantum Mechanics for Scientists and Engineers 94

97

where f, g are progressively measurable functions with B being Brownian motion and N (t, ., ω) a Poisson field with EN (ds, E) = λ(s)dsF (E). Ito’s formula for this martingale is  g(t, x)2 N (dt, dx) (dM (t))2 = d < M > (t) = f (t)2 dt + x∈E

A.4.Syllabus for a short course on Linear algebra and its application to classical and quantum signal processing. [1] Vector space over a field, linear transformations on a vector space. [2] Finite dimensional vector spaces, basis for a vector space, matrix of a linear transformation relative to a basis, Similarity transformation of the matrix of a linear transformation under a basis change. [3] Examples of finite and infinite dimensional vector spaces. [4] range, nullspace, rank and nullity of a linear transformation. [5] Subspace, direct sum decompositions of vector spaces, projection operators. [6] Linear estimation theory in the language of orthogonal projection operators. [7] Statistics of the estimation error of a vector under a small random perturbation of the data matrix, statistics of the perturbation of the orthogonal projection operator under a small random perturbation of the data matrix. [8] Primary decomposition theorem, Jordan decomposition theorem, functions of matrices. [9] Cauchy’s residue theorem in complex analysis and its approach to the computation of functions of a matrix. [10] Norms on a vector space, norms on the space of matrices, Frobenius norm, spectral norm. [11] Notions of convergence in a vector space, calculating the exponential function and inverse of a perturbed matrix using a power series. [12] Recursive least squares lattice algorithms for time and order updates of prediction error filters based on appending rows and columns to matrices and computing functions of the appended matrices, RLS lattice for second order Volterra systems, Statistical properties of the prediction filter coefficients under the addition of a small noise process to the signal process: A statistical perturbation theory based approach. [13] The MUSIC and ESPRIT algorithms for direction of arrival estimation based on properties of signal and noise eigensubspaces. [14] Computing the solution to time varying linear state variable systems using the Dyson series. Convergence of the Dyson series. [15] Computing the approximate solution to nonlinear state varable systems using Dyson series applied to the linearized system. [16] Dyson series in quantum mechanics. [17] Computing transition probabilities for quantum systems with random time varying potentials using the Dyson series. [18] Approximate solution to stochastic differential equations driven by Brownian motion and Poisson fields using linearization combined with Dyson series. Mean and variance propagation equations based on linearization. [19] The spectral theorem for finite dimensional normal operators and infinite dimensional unbounded self-adjoint operators in a Hilbert space. [20] Properties of spectral families in finite and infinite dimensional Hilbert spaces. [21] The general theory of estimating parameters in linear models for Gaussian and non-Gaussian noise. [22] The quantum stochastic calculus of Hudson and Parthasarathy and its application to the modeling of a quantum system coupled to a photon bath. [23] Kushner nonlinear filter and its linearized EKF version. [24] The Belavkin quantum filter based on non-demolition measurements and its application to quantum control. Application to quantum control: The Belavkin equation can be expressed as dρt = Lt (ρt )dt + Mt (ρt )dWt where Wt is a classical Wiener process arising from the innovations process of the measurement dWt = dYt − πi (St + St∗ )dt = dYt − T r(ρt (St + St∗ ))dt Note that ρt can be viewed as a classical random process with values in the space of signal space density matrices. St is a system space linear operator. The Belavkin equation is a commutative equation since all the terms appearing in it like ρt , Lt (ρt ), Wt etc. are signal space operator valued functionals of the commutative noise processs {Yy }. Now let Uc (t) be the control unitary satisfying the sde dUc (t) = (−(iH1 (t) + Q1 (t))dt − iK(t)dYt )Uc (t)

Quantum Mechanics for Scientists and Engineers 95

98

We have

Y (t) = U (t)∗ Yi (t)U (t) = U (T )∗ Yi (t)U (T ), T ≥ t

Here Yi (t) is an operator on the Boson Fock space and is thus independent of the system Hilbert space operators. We have taking Yi (t) = A(t) + A(t)∗ , dY (t) = dYi (t) + jt (Zt )dt where Zt is a system space operator and jt (Z) = U (t)∗ ZU (t). Thus, dUc (t) = (−(iH1 (t) + Q1 (t))dt − iK(t)(jt (Zt )dt + dYi (t)))Uc (t) We have

d(Uc∗ Uc ) = dUc∗ Uc + Uc∗ dUc + dUc∗ dUc = Uc (t)∗ (−(Q∗1 + Q1 )dt + idYt K(t) − iK(t)dYt + dY (t)K(t)2 dY (t))Uc (t)

If K(t) commutes with dY (t) and Q1 + Q∗1 = K(t)2 , then we would get d(Uc∗ Uc ) = 0 and Uc will be a control unitary operator. Taking K(t) = jt (Pt ) where Pt is a system operator, we have K(t)dY (t) = jt (Pt )(jt (Zt )dt + dYi (t)) = jt (Pt Zt )dt + jt (Pt )dYi (t) and

dY (t)K(t) = jt (Zt Pt )dt + jt (Pt )dYi (t)

So for Uc (t) to be unitary, we require that [Zt , Pt ] = 0 for all t. Note that Zt , Pt are system Hilbert space operators. We can now define ρc (t) = Uc (t)ρ(t)Uc (t)∗ Now,

dρc (t) = dUc (t)ρ(t)Uc (t)∗ + Uc (t)dρ(t).Uc (t)∗ + Uc (t)dρ(t)dUc (t)∗ + dUc (t)dρ(t)Uc (t)∗ +Uc (t)dρ(t).dUc (t)∗ + dUc (t)ρ(t)dUc (t)∗

[25] Linear algebra applied to the study of the linearized Einstein field equations in the presence of matter and radiation for the study of galactic evolution as the propagation of small non-uniformities in matter and radiation propagating in an expanding universe. (0) gμν (x) is the background Robertson Walker (RW) metric. Its perturbation is (0) + δgμν gμν = gμν

The coordinate system can be chosen so that Then, the linearized Ricci tensor is

δg0μ = 0 α δRμν = δΓα μα,ν − δΓμν,α β(0)

β α(0) −(δΓα μν )Γαβ − Γμν δΓαβ α(0)

β(0)

+Γμβ δΓβνα + (δΓα μβ )Γαβ This expression can be expressed as

α δRμν = (δΓα μα ):ν − (δΓμν ):α

where the covariant derivatives are computed using the unperturbed RW metric. The energy momentum tensor of the matter field is Tμν = (ρ + p)vμ vν − pgμν and that of the radiation field is

Sμν =

1 Fαβ F αβ gμν − Fμα Fνα 4

For example, computing in a flat space-time S00 =

1 Fαβ F αβ − F0α F0α 4

Quantum Mechanics for Scientists and Engineers 96

Now

99

Fαβ F αβ = −2F0r F0r + Frs Frs = −2|E|2 + 2|B|2 F0α F0α = −F0r F0r = −|E|2

so S00 =

1 1 (−|E|2 + |B|2 ) + |E|2 = (|E|2 + |B|2 ) 2 2

which is the correct expression for the energy density of the electromagnetic field. Likewise, S 0r = S r0 defines the energy flux as well as the momentum density and S rs = S sr defines the momentum flux. We can write the expression for δRμν in the general form δRμν = C1 (μναβρ, x)δgαβ,ρ (x) + C2 (μναβρσ, x)δgβρ,σ (x) (0)

where C1 , C2 are functions of x determined completely from the background gravitational field gμν (x) The perturbation to the energy momentum tensor of matter is given by δTμν = (δρ + δp)Vμ(0) Vν(0) + (ρ(0) + p(0) )(Vμ(0) δvμ + Vν(0) δvμ ) (0) −δpgμν + p(0) δgμν

We note that

Vμ(0) = (1, 0, 0, 0),

and p(0) , ρ(0) are functions of time only. Further, (0)

(0)

(0)

(0)

g00 = 1, g11 = −S 2 (t)f (r), g22 = −S 2 (t)r2 , g33 = −S 2 (t)r2 sin2 (θ), 1 1 − kr2 with k = 1 for a spherical universe, k = 0 for a flat universe and k = −1 for a hyperbolic universe. We now consider a flat unperturbed universe for which the metric has the form f (r) =

dτ 2 = dt2 − S 2 (t)(dx2 + dy 2 + dz 2 ) Thus,

g00 = 1, grr = −S 2 (t), r = 1, 2, 3

The Ricci tensor components are:

α α β R00 = Γα 0α,o − Γ00,α − Γ00 Γαβ β +Γα 0β Γ0α

Now, r Γα 0α = Γ0r =

So

1 rr g grr,0 = S  /S, r = 1, 2, 3 2   Γα 0α,0 = (S /S)

Γα 00,α = 0 β Γα 00 Γαβ = 0

 r

So

β Γα 0β Γ0α =   (Γr0r )2 = (g rr grr,0 /2)2 = 3S 2 /S 2 r





R00 = (S  /S) + 3S 2 /S 2 = S  /S + 2S 2 /S 2 β α α β α Rkk = Γα kα,k − Γkk,α − Γkk Γαβ + Γkβ Γkα  Γ0kk,0 − Γ0kk Γr0r + 2Γ0kk Γkk0 r

Quantum Mechanics for Scientists and Engineers 97

100

(No summation over k)

= −gkk,00 /2 + (gkk,0 /2)(S  /S) + 2(−gkk,0 /2)(gkk,0 /2gkk ) 

We have in fact

= S  /2 − S 2 /2S + S  (S  /2S) = S  /2 Rkm = (S  /2)δkm

Now let us study the perturbed Einstein field equations w.r.t. the above flat space-time metric. First note that we can choose our coordinate system so that δg0μ = 0 and hence δg 0μ = 0. Raising and lowering of indices are carried out w.r.t. the above flat space time metric which is diagonal. The unperturbed space-time is comoving, ie, v μ(0) = (1, 0, 0, 0) define geodesics in the unperturbed space-time. The unperturbed pressure and density p(0) (t), ρ(0) (t) are functions of t only. The unperturbed energy momentum tensor of matter is T μν(0) = (ρ(0) + p(0) )v μ(0) v ν(0) − p(0) g μν(0) so that

T 00(0) = ρ(0) , T kk(0) = p(0) /S 2

with the other components of the energy momentum tensor being zero. The unperturbed Einstein field equations 1 (0) Rμν = K.(Tμν(0) − T (0) gμν(0) ), K = −8πG 2 thus give after noting that

(0) μν(0) T (0) = gμν T =

(0)

and hence the unperturbed field equations are

ρ(0) − 3p(0) ,

Tkk = −S 2 T kk(0) = −p(0)

 1 S  /S + 2S 2 /S 2 = K.(ρ(0) − (ρ(0) − 3p(0) ), 2

S  /2 = K(−p(0) +

S 2 (0) (ρ − 3p(0) )) 2

These two equations along with an equation of state: p(0) = f (ρ(0) ) determing the three functions of time S(t), ρ(0) (t), p(0) (t). The perturbed equations are δRμν = KδTμν where

Now,

δTμν = (ρ(0) + p(0) )(vμ(0) δvν + vν(0) δvμ ) + (δρ + δp)vμ(0) vν(0) (0) −δpgμν − p(0) δgμν α(0)

β α δRkm = δΓα kα,m − δΓkm,α − Γkm δΓαβ β(0)

α(0)

β −Γαβ δΓα km + Γkβ δΓmα α +Γβ(0) mα δΓkβ

The perturbed field equations taking into account contributions from the electromagneti field are expressible in the form C1 (μναβγ, x)δgαβ,γ (x) + C2 (μναβγσ, x)δgαβ,γσ (x) = C3 (μνα, x)δvα (x) + C4 (μν, x)δρ(x) + C5 (μν, x)δp(x) +C6 (μναβ, x)δAα,β (x) = 0

Quantum Mechanics for Scientists and Engineers 98

101

The equations implied by the Bianchi identity are (T μν + S μν ):ν = 0 This is the same as

(T μν + S μν ),ν + Γμαν (T αν + S αν ) + Γναν (T μα + S μα ) = 0

We calculate is first order perturbed version: αν + δS αν )+ (δT μν + δS μν ),ν + Γμ(0) αν (δT μα Γν(0) + δS μα )+ αν (δT

δΓμαν (T αν(0) + S αν(0) )+ δΓναν (T μα(0) + δS μα(0) ) = 0 This can be put in the form C7 (μαβ, x)δvα,β (x) + C8 (μα, x)δvα (x) + C9 (μαβγ, x)δAα,βγ (x) +C10 (μαβγ, x)δgαβ,γ (x) + C11 (μ, x)δρ(x) + C12 (μ, x)δp(x) = 0 We note that this last equation contains 3 equations for the velocity components δvr , r = 1, 2, 3 and one equation for δρ(x). δp(x) is determined from δρ(x) using the equation of state. Also δv0 is determined from δvr using 0 = δ(gμν v μ v ν ) (0) μ(0) = (δgμν )v μ(0) v ν(0) + gμν (v δv ν + v ν(0) δv μ

using that the unperturbed dynamics is comoving,ie, v μ(0) = (1, 0, 0, 0) we get from this equation δg00 + 2δv 0 = 0 so

1 δv 0 = − δg00 2

and hence,

δv0 = δ(g0μ v μ ) = δg00 (0)

since v r(0) = 0 and g0r = 0. [26] Perturbation theory applied to electromagnetic problems. Here, we discuss the rudiments of the theory of time independent perturbation theory in quantum mechanics and explain how the same techniques can be used to solve waveguide and cavity resonator problems having almost aribtrary cross sections by transforming the boundary into a simpler boundary using the theory of analytic functions of a complex variable. Let D be a flat connected region parallel to the xy plane. D represents the cross section of a waveguide or a cavity resonator. The boundary ∂D is a closed curve that represents the boundary of the guide or resonator. The z direction is orthogonal to the D plane and for the guide, it represents the direction along which the em waves propagate. For the resonator case, we assume that 0 ≤ z ≤ d with the surfaces z = 0 and z = d being perfectly conducting surfaces as is the case with the side walls. We choose a system (q1 , q2 , z) of coordinates so that (q1 , q2 ) are functions of (x, y) alone. Being more specific, we assume that w = q1 (x, y) + jq2 (x, y) = f (z) = f (x + jy) with an inverse

z = x + jy = g(w) = g(q1 + jq2 )

and assume that f (z) is an analytic function of the complex variable z and its inverse g(w) is also an analytic function of the complex variable w. We can regard (z, z¯) as independent variables just as (x, y) are. The relation between the two pairs of variables is z = x + jy, z¯ = x − jy, x = (z + z¯)/2, y = (z − z¯)/2j We have

∂ ∂ ∂ = z,x + z¯,x ∂x ∂z ∂ z¯

Quantum Mechanics for Scientists and Engineers 99

102

= and likewise,

∂ ∂ + ∂z ∂ z¯

∂ ∂ ∂ = j( − ) ∂y ∂z ∂ z¯

Thus, ∇2⊥ = =(

∂2 ∂2 + 2 2 ∂x ∂y

∂ 2 ∂ 2 ∂ ∂ + ) −( − ) ∂z ∂ z¯ ∂z ∂ z¯ ∂2 =4 ∂z∂ z¯

Now,

∂ ∂ = w� (z) ∂z ∂w ∂ ∂ =w ¯ � (z) ∂ z¯ ∂w ¯ also, we clearly have since w� (z) is an analytic function of z and w ¯ � (z) is an analytic function of z¯ and hence that ¯ � (z) is an analytic function of w ¯ that 1/w� (z) = dz/dw is an analytic function of w and 1/w ∇2⊥ = 4

∂2 ∂2 = 4|z � (w)|−2 ∂z∂ z¯ ∂w∂ w ¯

= |z � (w)|−2 (

∂2 ∂q 2 + 2 ∂q12 )

Thus, the eigenvalue problem (∇2⊥ + h2 )ψ(x, y) = 0 with the Dirichlet boundary condition ψ = 0 on ∂D becomes (∇2⊥,q + h2 F (q1 , q2 ))ψ(q1 , q2 ) = 0 with

ψ(a, q2 ) = 0

where the boundary ∂D is the same as q1 = a. For example, if we take w = log(z) = log(ρ) + jφ, then q1 = a is the circle q1 = ρ = ea which is a circle and (q1 , q2 ) = |z � (w)|2 = |g � (q1 + jq2 )|2 Also, we have defined ∇2⊥,q =

∂2 ∂2 + 2 =L 2 ∂q1 ∂q2

say. Solution by perturbation theory: Let h2 = −λ. Then, we have to solve (L − λ(1 + �G(q1 , q2 )))ψ(q1 , q2 ) = 0 where we are assuming that

F (q) = 1 + �G(q)

so that the boundary is a small perturbation of the rectangular boundary. [27] Perturbation theory applied to general nonlinear partial differential equations with noisy terms. The gravitational wave equations, fluid dynamical equations, Klein-Gordon and Dirac equations are special cases of this. The field φ : Rn → Rp satisfies a nonlinear pde n 

bk (x)φl,k (x) +

k=1

p 

akm (x)φl,km (x) + δ.Fl (φl (x), φl,k (x), φl,km (x), x)

k,m=1

+δ.

 m

Glm (φl (x), φl,k (x), φl,km (x), x)wm (x)

Quantum Mechanics for Scientists and Engineers 100

103

where wm (x) are Gaussian noise processes. [28] The Knill-Laflamme theorem for quantum error correcting codes: Explicit construction of the recovery operators for a noisy quantum channel in terms of the code subspace and the noise subspace. [29] Post-Newtonian equations of hydrodynamics. The perturbations are carried out in powers of the velocity. The mass parameter is of the order of the square of the velocity (v 2 = GM/r for the orbital velocity) and the following expansions are valid ρ = ρ2 + ρ4 + ..., p = p4 + p6 + ..., v r = v1r + v3r + ..., v 0 = 1 + v20 + v40 + ... g00 = 1 + g00(2) + g00(4) + ..., g0r = g0r(3) + g(0r(5) + ... grs = −δrs + grs(2) + grs(4) + ... g 00 = 1 + g200 + g400 + ... g 0r = g30r + g50r + ..., g rs = −δ rs + g2rs + g4rs + ... The equation can be expressed as so the O(1) equation is

T μν = (ρ + p)v μ v ν − pg μν gμν v μ v ν = 1

g00 v 02 + 2g0r v 0 v r + grs v r v s = 1 v00 = 1

The O(v 2 ) equation is v20 + g00(2) − or equivalently,



v20 = −g00(2) + We have

v1r2 = 0

r



v1r2

r

T 00 = T200 + T400 + ... T 0r = T30r + T50r + ... T rs = T2rs + T4rs + ...

where

T200 = ρ2 , T400 = 2ρ2 v20 , T600 = ρ2 v202 + 2ρ4 v20 + 2p4 v20 − p4 g200 T30r = 2ρ2 v20 v1r , T50r = 2ρ2 v20 v1r + 2ρ2 v3r + 2ρ4 v1r + 2p4 v1r , T2rs =

[30] Lab problems on linear algebra based signal processing: [1] If X is an m × n matrix, then calculate δPX upto O(� δX) �) where PX is the orthogonal projection onto R(X) and X gets perturbed to X + δX where δX is a small random perturbation of X. Calculate using this formula, the second order statistics of δPX , ie, E(δPX ⊗ δPX ) in terms of E(δX ⊗ δX). Calculate δPX upto O(||δX||2 ) [2] Take an n × n matrix A. Add a row and a column to this matrix at the end and express the inverse of this matrix B in terms of the inverse of A. Assume now that A gets perturbed to A + δA and correspondingly, the appended row and column get perturbed by small amounts. Then calculate the inverse of the perturbed appended matrix in terms of A−1 and the appended rows and columns and their perturbations upto linear orders in the perturbations.

Quantum Mechanics for Scientists and Engineers 101

104

[3] Generate some functionals of the Brownian motion process B(t) like M (t) = max(B(s) : s ≤ t), m(t) = min(B(s) : s ≤ t), Ta = min(t > 0 : B(t) = a), |B(t)| (reflected Brownian motion), Absorbed Brownian motion (B(min(t, T0 ) : t ≥ 0 where B(0) = a > 0 and T0 = min(t > 0 : B(t) = 0, Local time process La (t) of the Brownian motion process at the t t level a. This is defined as La (t) = 0 δ(B(s) − a)ds and is approximated by (2�)−1 0 χ[a−�,a+�] (B(s))ds where � is a very small positive number. For a bivariate standard Brownian motion process (B1 (t), B2 (t)), t ≥ 0, simulate the area t process A(t) = 0 B1 (s)dB2 (s) − B2 (s)dB1 (s) and calculate its statistics. Simulate the Bessel process of order d, ie d X(t) = ( k=1 Bk (t)2 )1/2 , t ≥ 0 and verify by numerical simulations that it satisfies its standard stochastic differential equation. [4] Verification of the Knill-Laflamme theorem for quantum error correcting codes. Generate a set of r < n linearly independent column vectors {f1 , ..., fr } in H = Cn . Denote the subspace spanned by these r vectors by C. Calculate the orthogonal projection P onto C using the standard formula P = A(A∗ A)−1 A∗ where A = [f1 , ..., fr ] = Cn×r . Generate K n × n matrices having the block structure   C1k C2k Nk = , k = 1, 2, ..., K C3k C4k where C1k is an r × r matrix for each k such that for 1 ≤ k, j ≤ K, we have ∗ ∗ C1j C1k + C3j C3k = λjk Ir ∗ ∗ for some complex numbers λjk . This can be achieved by choosing the r × n matrices [C1j |C3j ], 1 ≤ j ≤ K as nonoverlapping rows of an n × n unitary matrix U and then multiply the resultant matrices by complex constants. Thus, we must have K ≤ [n/r]. The matrices C2k , C3k , C4k can be chosen arbitrarily. Now take the orthogonal projection P onto C as   Ir 0 P = 0 0

It is then easily seen that

P Nk∗ Nj P = λkj P

and hence the quantum code P can correct the noise subspace {Nk : 1 ≤ k ≤ K}. [5] (a) Study waves in a plasma influenced by a strong gravitational field and electromagnetic fields by the method of linearization: The Boltzmann particle distribution function f (t, r, v) where v = (v r = dxr /dt)r = 13 satisfies (after approximating the collision term by a linear relaxation term k f,vk (t, r, v) = (f0 (r, v) − f (t, r, v))/τ (v) f,t (t, r, v) + v k f,xk (t, r, v) + v,0

Here the velocity v k satisfies the geodesic equation in an electromagnetic field: k + γγ,0 v k dxk /dτ = γv k , dv k /dτ = γd/dt(dxk /dτ ) = γd/dt(γv k ) = γ 2 v,0

where

γ = dt/dτ = (g00 + 2g0r v r + grs v r v s )−1/2

or equivalently,

k γ 2 v,0 + γγ,0 v k + γ 2 Γk00 + 2γ 2 Γk0m v m + γ 2 Γkmp v m v p = eγ(F0m + Fsm v s ) − − − (a) k v,0 + (γ,0 /γ)v k + Γk00 + 2Γk0m v m + Γkmp v m v p = eγ −1 (F0m + Fsm v s )

k is substituted into the above Boltzmann equation to get This value of v,0

f,t (t, r, v)+v k f,xk (t, r, v)−(γ,0 /γ)v k +Γk00 +2Γk0m v m +Γkmp v m v p −eγ −1 (F0m +Fsm v s ))f,vk (t, r, v)−(f0 (r, v)−f (t, r, v))/τ (v) = 0 k k We note that γ,0 is a function of v,0 , v k , x. We need to get a Boltzmann equation that does not involve v,0 . For this purpose, we go back to the equation of motion of the charged particle (a). First observe that −1/2

γ,0 = (g00 + 2g0r v r + grs v r v s ),0

r = (−γ 3 /2)(g00,0 + g00,k v k + 2g0r,0 v r + 2g0r,s v r v s + 2g0r v,0

s +grs,0 v r v s + grs,m v r v s v m + 2grs v,0 ) − − − (b)

k 3 k )k=1 which is inverted to get v,0 as a function (v k ), Substituting (b)into (a) gives us a linear algebraic equation for (v,0 xμ , the electromagnetic field F μν (x) and of course the metric gμν (x) and its first order partial derivatives gμν,α . Thus,

Quantum Mechanics for Scientists and Engineers 102

105

we get a well defined Boltzmann equation. Now given the particle distribution function f (t, r, v), we need to calculate the energy momentum tensor of matter. We have T μν = (ρ + p)V μ V ν − pg μν where ρ=m Ur = V 0 is calculated using



v r f d3 v/





f d3 v,

f d3 v, V r = γ(U )U r

gμν V μ V ν = 1

Equivalently,

γ(U )2 g00 + 2γ(U )2 U r g0r + γ(U )2 grs U r U s = 1

or

γ(U ) = (g00 + 2g0r U r + grs U r U s )−1/2 , V 0 = γ(U )

The average internal kinetic energy per particle is K(t, r) = (m/2)

  ( (v r − U r )2 )f (t, r, v)d3 v r

and the pressure field is given by p(t, r) = nm < |v − U |2 > /3 = (m/3)

  (v r − U r )2 f (t, r, v)d3 v r

where m is the mass of a plasma particle and n is the number of plasma particles per unit volume. This energy momentum tensor of the plasma can be added to the energy momentum tensor of the electromagnetic field and substituted into the right side of the Einstein field equations. Thus, we get a couple system of pde’s for f, gμν , Aμ . An alternate way to define the energy momentum tensor is as  T μν = m f (t, r, v)γ(t, r, v)v μ v ν d3 v − p(t, r)g μν (t, r) where

v μ = dxμ /dt

so that

v 0 = 1, v r = dxr /dt

and γ = γ(t, r, v) is defined by the equation γ = (g00 + 2g0k v k + gkm v k v m )−1/2 The pressure p is as defined earlier. [31] Quantum image processing: The image field is obtained by passing a quantum em field through a spatio temporal filter having impulse response h(t, τ, r, r� ). Assume the Coulomb gauge with zero charge density. Then the scalar potential A0 = 0 and the vector potential is given by  Ar = [(2|K|)−1/2 a(K, σ)er (K, σ)exp(−i(|K|t − K.r)) + (2|K|)−1/2 a∗ (K, σ)¯ er (K, σ)exp(i(|K|t − K.r))]d3 K The Coulomb gauge condition divA = Ar,r = 0 implies K r er (K, σ) = 0 or equivalently, (K, e(K, σ)) = 0 which means that there are only two degrees of polarization which are indexed by σ = 1, 2. The electric field is  er (K, σ)exp(i(|K|t − K.r))]d3 K Er = −Ar,0 = i [(|K|/2)1/2 a(K, σ)er (K, σ)exp(−i(|K|t − K.r)) − (|K|/2)1/2 a∗ (K, σ)¯

Quantum Mechanics for Scientists and Engineers 103

106

or equivalently in three vector notation,  e(K, σ)exp(i(|K|t − K.r))]d3 K E = i [|K|/2)1/2 a(K, σ)e(K, σ).exp(−i(|K|t − K.r)) − (|K|/2)1/2 a∗ (K, σ)¯ The magnetic field is given by  B = curlA = i [(2|K|)−1/2 a(K, σ)K×e(K, σ)exp(−i(|K|t−K.r))−(2|K|)−1/2 a∗ (K, σ)K×e(K, σ)exp(i(|K|t−K.r))]d3 K The em field of the image can be regarded as the output of a spatio-temporal filter with this free em field as input. Thus, the output field is   E o (t, r) = h(t, τ, r, r� )E(τ, r� )dτ d3 r� , B o (t, r) = h(t, τ, r, r� )B(τ, r� )dτ d3 r� and hence the output field can be expressed as  ¯ E (t, r, K, σ)a∗ (K, σ))d3 K E o (t, r) = (HE (t, r, K, σ)a(K, σ) + H B o (t, r) =



¯ B (t, r, K, σ)a∗ (K, σ)]d3 K [HB (t, r, K, σ)a(K, σ) + H

where HE , HB are 3 × 1 functions constructed from the image impulse response h(t, τ, r, r� ). The energy of the em field coming from the image is  HF = (|E o |2 + |B o |2 )d3 r/2 and this can be expressed in the form after discretization of the spatial frequencies HF (θ) = (1/2)

p 

¯ 2 (k, m, θ)a∗ a∗ ) (Q1 (k, m, θ)a∗k am + Q2 (k, m, θ)ak am + Q k m

k,m=1

where

¯ 1 (k, m, θ) = Q1 (m, k, θ) Q

Here θ is a parameter vector upon which the image impulse response h(t, τ, r, r� ) depends. θ is the parameter which contains all information about the image field and is to be estimated by exciting an atom with the output image field and and taking measurements on the state of the atom at different times. We can simplify the form of the image field Hamiltonian as p  HF (θ) = Q(k, m, θ)a∗k am k,m=1

Here,

[ak , a∗m ] = δkm , [ak , am ] = 0, [a∗k , a∗m ] = 0

The Hamiltonian of the atom assumed to be an N state system, is an N × N Hermitian matrix HA and the interaction Hamiltonian between the image field and the atom has the form Hint (t) =

p 

k=1

(Fk (t, θ) ⊗ ak + Fk (t, θ)∗ ⊗ a∗k )

[32] Performance analysis of the MUSIC algorithm. X = AS + W, X ∈ CN ×K , A ∈ CN ×p , S ∈ Cp×K , W ∈ CN ×K 2 Rxx = K −1 E(XX ∗ ), Rss = K −1 E(SS ∗ ), σw I = K −1 E(W W ∗ ), E(S ⊗ W ) = E(S ⊗ W ∗ ) = 0

All signals are complex Gaussian. Thus,

E(X ⊗ X) = 0, E(S ⊗ S) = 0, E(W ⊗ W ) = 0

Quantum Mechanics for Scientists and Engineers 104

107

The stochastic perturbation in the array signal correlation matrix is given by δRxx = K −1 XX ∗ − Rxx = 2 K −1 ASS ∗ A∗ + K −1 W W ∗ + K −1 ASW ∗ + K −1 W S ∗ A∗ − ARss A∗ − σw I

= AδRss A∗ + δRww + K −1 (ASW ∗ + W S ∗ A∗ )

where

2 δRss = K −1 SS ∗ − Rss , δRww = K −1 W W ∗ − σw I

To calculate the mean and covariance of the DOA estimates, we need the mean and covariance of the statistical perturbation δRxx of Rxx . Now, E(δRss ⊗ δRss ) = K −2 E(SS ∗ ⊗ SS ∗ ) − Rss ⊗ Rss Now,

E(SS ∗ ⊗ SS ∗ ) = E[(S ⊗ S)(S ∗ ⊗ S ∗ )]

This is equivalent to calculating

E(si sj s¯k s¯m ) = E(si s¯k )E(sj s¯m ) + E(si s¯m )E(sj s¯k ) = [Rss ]ik [Rss ]jm + [Rss ]im [Rss ]jk Also since S and W are independent random matrices, it follows that δRss and δRww are independent zero mean random matrices. The second order moments of δRxx are thus computed as E(δRxx ⊗ δRxx ) = E[AδRss A∗ + δRww + K −1 (ASW ∗ + W S ∗ A∗ ))⊗2 ] = (A ⊗ A)E(δRss ⊗ δRss )(A∗ ⊗ A∗ ) +K

−2

where F denotes the flip operator:

+E(δRww ⊗ δRww )

(I + F )E(ASW ∗ ⊗ W S ∗ A∗ ) F (x ⊗ y) = y ⊗ x

Computing the last expectation is equivalent to computing

E(aij sjk w ¯lk wi j  s¯k j  a ¯ k  l ) = aij a ¯lk ) ¯k l E(sjk s¯k j  )E(wi j  w Thus the second order moments of δRxx are easily computed and this can be combined with matrix perturbation theory ˆ xx = Rxx + δRxx . These covariances to obtain the covariance of the signal and noise eigenvalues and eigenvectors of R can in turn be used to calculate the error covariances in the DOA estimates using the MUSIC pseudospectrum. [33] Estimating the quantum image parameters from measurements on the state of an atom excited by the quantum em field coming from the image in the interaction representation. The image em field interacts with an atom described by an N × N Hamiltonian matrix HA . This interaction Hamiltonian can be expressed as Hint (t|θ) =

p 

k=1

(Gk (t|θ) ⊗ ak + Gk (t|θ)∗ ⊗ a∗k )

where Gk (t|θ) ∈ C , θ is the image parameter vector. The joint density of the atom and the image em field at time t can be expressed using the Glauber-Sudarshan representation:  ρ(t) = C exp(−|z|2 )A(t, z) ⊗ |e(z) >< e(z)|d2p z N ×N

where |e(z) >=



n≥0

z n a∗n |0 > /n! =



n≥0

√ z n |n > / n!

Quantum Mechanics for Scientists and Engineers 105

108

since

√ |n >= a∗n |0 > / n!

is the normalized state of the field in which a∗k ak has the eigenvalue nk and n = (nk ). Further C = π −p We have ak |e(z) >= zk |e(z) >, a∗k |e(z) >=

∂ |e(z) > ∂zk

Thus, ak |e(z) >< e(z) >= zk |e(z) >< e(z)|, a∗k |e(z) >< e(z)| = |e(z) >< e(z)|ak =

∂ |e(z) >< e(z)|, ∂zk

∂ |e(z) >, e(z)|, ∂ z¯k

|e(z) >< e(z)|a∗k = z¯k |e(z) >< e(z)|

Note that z → |e(z) > is an analytic function of z and so z¯ →< e(z)| is an analytic function of z¯. Now, the joint density ρ(t) satifies Schrodinger’s equation in the interaction picture: iρ� (t) = [Hint (t|θ), ρ(t)] and this translates to i



A,t (t, z) ⊗ |e(z) >< e(z)|exp(−|z|2 )d2p z =

 ( (Gk (t|θ)A(t, z) ⊗ ak |e(z) >< e(z)| + Gk (t|θ)∗ A(t, z) ⊗ a∗k |e(z) >< e(z)| k

−A(t, z)Gk (t|θ) ⊗ |e(z) >< e(z)|ak − A(t, z)Gk (t|θ)∗ ⊗ |e(z) >< e(z)|a∗k )exp(−|z|2 )d2p z)  ∂ ( (Gk (t|θ)A(t, z)zk ⊗ |e(z) >< e(z)| + Gk (t|θ)∗ A(t, z) ⊗ ( |e(z) >< e(z)|) = ∂zk k

∂ |e(z) >< e(z)|) − A(t, z)Gk (t|θ)∗ z¯k ⊗ |e(z) >< e(z)|)exp(−|z|2 )d2p z) ∂ z¯k  ∂ ( (zk Gk (t|θ)A(t, z) − (Gk (t|θ)∗ ( − z¯k )A(t, z) = ∂zk

−A(t, z)Gk (t|θ) ⊗ (

k

∂ +( − zk )A(t, z)Gk (t|θ) − z¯k A(t, z)Gk (t|θ)) ⊗ |e(z) >< e(z)|d2p z) ∂ z¯k Thus, we get

iA,t (t, z) = (T (t|θ)A)(t, z)

where T (t|θ) is a differential operator acting on the space of matrix valued functions of the complex variable z ∈ Cp defined by  ∂ T (t|θ)X(z) = (zk Gk (t|θ)X(z) − z¯k X(z)Gk (t|θ) − Gk (t|θ)∗ ( − z¯k )X(z) ∂zk k

+(

∂ − zk )X(z)Gk (t|θ)) ∂ z¯k

The formal solution to this partial differential equation is A(t, z) = τ {exp(−i



t

T (s|θ)ds)}(A(0, z))

0

where τ is the time ordering operator. The atomic (system) density at time t is given by  ρA (t) = T r2 (ρ(t)) = π −p A(t, z)d2p z A(t, z) and ρA (t) are N × N matrices.

Quantum Mechanics for Scientists and Engineers 106

109

[34] Existence and uniqueness of solutions to stochastic differential equations. (a) Kolmogorov’s inequality for discrete time sub-martingales. Let Mk , k = 0, 1, 2, ... be a non-negative sub-martingale. Let τa = min(k ≥ 0 : Mk ≥ a), a > 0 Then,

{τa = m} = {M0 < a, M1 < a, ..., Mm−1 < a, Mm ≥ 0}

Thus, if Fn is the underlying filtration for {Mn }, it follows that

{τa = m} ∈ Fm In particular, τa is a stop-time. Thus, by the submartinagle property, for n ≥ m, E(Mn χτa =m ) ≥ E(Mm χτa =m ) ≥ a.P (τa = m) and summing over m gives or

E(Mn ) ≥ E(Mn χτa ≤n ) ≥ a.P (τa ≤ n) P (τa ≤ n) = P (max0≤k≤n Mk ≥ a) ≤ E(Mn )/a

This can be generalized to the continuous time scenario. Specifically, if Mt , t ≥ 0 is a continuous martingale, then P (max0≤s≤t |Ms | ≥ a) ≤ E|Mt |/a since |Mt | is a submartingale by Jensen’s inequality. Using this, we can deduce a version Doob’s inequality: E(max0≤t≤T |Mt |2 ) ≤ C1 (E(MT2 ) Now consider the sde

dXt = f (t, Xt )dt + g(t, Xt )dMt , X0 = x

where f, g satisfy appropriate Lipshitz conditions which will be specified later. We wish to prove the existence and (n) uniqueness of the solution to this sde. We define the processes Xt , n = 1, 2, ... recursively as  t  t (n+1) =x+ f (s, Xs(n) )ds + g(s, Xs(n) )dMs Xt 0

0

These processes are all adapted to the underlying filtration on which the process M is defined and we have by Doob’s in  T  T (n+1) (n) (n) (n−1) 2 − Xt |2 ) ≤ KT E |Xs(n) − Xs(n−1) |2 ds + KC1 (E |Xt − Xt | d < M >t ) E(max0≤t≤T |Xt 0

0

If we assume that the measure < M > is absolutely continuous w.r.t. the Lebesgue measure, with bounded RadonNikodym derivative, then we derive from the above that  T (n) (n) |Xt − X (n−1 )t |2 dt E(max0≤t≤T |X (n+1) − Xt |2 ) ≤ (K0 T + K1 ) 0

from which the existence of a solution to the sde can be inferred using standard arguments based on Gronwall’s inequality. [35] Statistical analysis of the RLS lattice algorithm. Let x[n] be a random process and we form the vector ξn = [x[n], x[n − 1], ..., x[0]]T ∈ Rn+1×1 z −p ξn = [x[n − p], x[n − p − 1], ..., x[−p]]T ∈ Rn+1×1

where x[k] = 0f ork < 0. Define the data matrix

Xn,p = [z −1 ξn , ..., z −p ξn ] ∈ Rn+1×p Let Pn,p = PR(Xn,p ) . The forward and backward prediction error sequences or order p at times n and n − 1 are respectively defined by ⊥ ⊥ −p−1 ξn , eb [n − 1|p] = Pn,p z ξn ef [n|p] = Pn,p

Quantum Mechanics for Scientists and Engineers 107

110

We have the obvious formula based on orthogonal direct sum decompositions: ⊥ ⊥ = Pn,p − Psp{eb [n−1|p]} Pn,p+1

and hence

ef [n|p + 1] = ef [n|p] − Kf [n|p]eb [n − 1|p] − − − (1)

and likewise,

e˜b [n|p + 1] = eb [n − 1|p] − Kb [n|p]ef [n|p − − − (2)]

where

⊥ −p−1 e˜b [n|p + 1] = Psp{ξ ξn −1 ξ ,...,z −p ξ } z n ,z n n

It is easy to see that

eb [n|p + 1] = Here, the forward reflection coefficient is



e˜b [n|p + 1] 0



Kf [n|p] =< eb [n − 1|p], ef [n|p > /Eb [n − 1|p] and the backward reflection coefficient is Kb [n|p] =< eb [n − 1|p], ef [n|p] > /Ef [n|p] where We thus have

Ef [n|p] =� ef [n|p] �2 , Eb [n − 1|p] =� eb [n − 1|p] �2 0 ≤ Kf [n|p]Kb [n|p] = | < eb [n − 1|p], ef [n|p] > |2 /Ef [n|p]Eb [n − 1|p] ≤ 1

From (1) and (2), we get

Ef [n|p + 1] = Ef [n|p] + Kf2 [n|p]Eb [n − 1|p] − 2Kf [n|p]2 Eb [n − 1|p] = Ef [n|p] − Kf2 [n|p]Eb [n − 1|p] and likewise,

= (1 − Kf [n|p]Kb [n|p])Ef [n|p]

Eb [n|p + 1] = (1 − Kf [n|p]Kb [n|p])Eb [n − 1|p]

The time update formulas for Kf and Kp and of Ef , Ep require time update formulas for Pn,p . We have T T Pn,p = Xn,p (Xn,p Xn,p )−1 Xn,p

Let

T Rn,p = Xn,p Xn,p

Then since Xn+1,p = where we get



T ηn,p Xn,p



ηn,p = x[n], x[n − 1], ..., x[n − p + 1] T Rn+1,p = ηn,p ηn,p + Rn,p

and hence −1 −1 Rn+1,p = Rn,p −

−1 T −1 ηn,p ηn,p Rn,p Rn,p T R−1 η 1 + ηn,p n,p n,p

Thus, T T −1 ] (Rn,p − Pn+1,p = [ηn,p |Xn,p T ] .[ηn,p |Xn,p

−1 T −1 ηn,p ηn,p Rn,p Rn,p T R−1 η 1 + ηn,p n,p n,p

).

Quantum Mechanics for Scientists and Engineers 108

⎛

111

T −1 ηn,p Rn,p ηn,p

⎜ ⎝

T −1 T ηn,p Rn,p Xn,p

−1 1+etaT n,p Rn,p ηn,p −1 Xn,p Rn,p n,p T R−1 η 1+ηn,p n,p n,p

η

Pn,p −

T R−1 η 1+ηn,p n,p n,p −1 T −1 T Xn,p Rn,p ηn,p ηn,p Rn,p Xn,p −1 T 1+ηn,p Rn,p ηn,p

⎞ ⎟ ⎠

[36] Electric dipole moment and magnetic dipole moment of an atom with an electron in a constant electromagnetic field. The unperturbed Hamiltonian of the atom is given by H0 = p2 /2m − eV (r) The perturbing Hamiltonian is −eV1 + e2 V2 where V1 = −(r, E) + (Lz + gσz )B0 /2m V2 = (B0 zˆ × r)2 /2m = B02 (x2 + y 2 )/2m

To calculate the eigenfunctions of H0 − eV1 + e2 V2 upto O(e2 ), we need to develop second order time independent perturbation theory for degenerate unperturbed systems. Consider therefore a Hamiltonian H = H0 + δH1 + δ 2 H2 (0)

with the eigenvalues of H0 being En , n = 1, 2, ... and an orthonormal basis for the eigenspace N (H0 − En ) being (0) |ψnk >, k = 1, 2, ..., dn . Let dn � (0) c(n, k)|ψnk > |ψn(0) >= k=1

be the unperturbed state of the system. We note that this state has an eigenvalue En for H0 . The constants c(n, k) are yet to be determined. We write for the perturbed state |ψn >= |ψn(0) > +δ|ψn(1) > +δ 2 |ψn(2) + O(δ 3 ) and correspondingly for the perturbed energy level En = En(0) + δEn(1) + δ 2 En(2) + O(δ 3 ) Substituting these expansions into the eigenequation and equating coefficients of δ m , m = 0, 1, 2 successively gives (H0 − En(0) )|ψn(0) >= 0 − − − (1) which is already known,

(H0 − En(0) )|ψn(1) > +H1 |ψn(0) > −En(1) |ψn(0) >= 0 − − − (2)

(1) (0) (1) (1) (2) (0) (H0 − En(0) )|psi(2) n > +H1 |ψn > +H2 |ψn > −En |ψn > −En |ψn >= 0 − − − (3) (0)

From (2), we get on formijng the bracket with < ψmk | from the left, (0)

(0) (Em − En(0) ) < ψmk |ψn(1) > +

� l

(0)

(0)

< ψmk |H1 |ψnl c(n, l)

−En(1) c(n, k)δmn = 0 − − − (4) (1)

Setting m = n gives us the secular equation for the possible values of En that lift the degeneracy of the unperturbed state: � (0) (0) < ψnk |H1 |ψnl > c(n, l) = En(1) c(n, k), 1 ≤ k ≤ dn l

(1) Thus, En assumes the (0) (0) ψnk |H1 |ψnl >))1≤k,l≤dn

1 values Enj , k = 1, 2, ..., dn which are the eigenvalues of the dn × dn secular matrix ((< (1)

n with the eigenvector corresponding to the eigenvalue Enj being denoted by ((cj (n, k)))dk=1 . dn We may assume that these dn eigenvectors form an orthonormal basis for C . From (4) with m �= n, we get

(0)

< ψmk |ψn(1) >=

� < ψ (0) |H1 |ψ (0) > c(n, l) mk nl (0)

l

(0)

E n − Em

Quantum Mechanics for Scientists and Engineers 109

112

(0)

and hence the first orde perturbation to the eigenvector |ψn

eigenvalue

(0) En

+

(1) δ.Enj

(1)

>, namely δ.|ψnj > corresponding to the perturbed

is given by (1)

|ψnj >=



(0)

mkl,m�=n

(1)

(0)

(0)

(0) |ψmk >< ψmk |H1 |ψnl > cj (n, l)/(En(0) − Em )

(1)

(1)

(1)

(0)

Turning now to (3), we assume En = Enj and |ψn >= |ψnj >. Taking the bracket of this equation with < ψmk |, we get (0)

(0)

(1)

(0)

(1)

(0)

(1)

(1)

(0)

(1)

(0) (0) (2) (Em − En(0) ) < ψmk |psi(2) n > + < ψmk |H1 |ψnj > + < ψmk |H2 |ψn > −Enj < ψmk |ψnj > −En cj (n, k)δmn = 0

For m = n, this gives (0)

(1)

< ψnk |H1 |ψnj > +

 l

(0)

< ψnk |H2 |ψnl > cj (n, l) − Enj < ψnk |ψnj >

−En(2) cj (n, k) = 0 − − − (5)

(2)

Actually, these equations are not all linearly independent. The only linearly independent equation for En which  (0) (0) emerges from this is obtained by forming the bracket of (3) with + < ψn | = k c¯j (n, k) < ψnk |. In fact, if we extend (2) dn the conjugate of this vector to an onb for C and form the bracket with these vectors, then the term involving En disappears. Thus we infer from from (5) that (2) En(2) = Enj =    (0) (1) (0) (0) ( |cj (n, k)|2 )−1 [ c¯j (n, k) < ψnk |H1 |ψnj > + < ψnk |H2 |ψnl > c¯j (n, k)cj (n, l) k

k

(1) −Enj



k,l

c¯j (n, k)
]

[37] Induced characters: Let G be a finite group and H  a subgroup of G. Select  one element x in each left coset of H. Let I denote the set of all such x� s. Thus, we have x∈I xH = G and xH yH = φ, x �= y, x, y ∈ I. Let L be a representation of H acting in the vector space V . We shall formally write x.V for the vector space V attached to the element x. The representation space for the representation π = IndG H L (ie π is the representation of G induced by the  x.V . π(g) acts on this space by mapping x.v to [gx].v where representation L of H) may be denoted by U = x∈I g ∈ G, x ∈ I, v ∈ V and [gx] ∈ I is the element for which [gx]H = gxH. We may use the notation gxV = [gx]V . Let χU denote the character of π and χV that of L. It is clear that g ∈ G will map x.V onto itself iff gxV = xV iff x−1 gxV = V iff x−1 gx ∈ H. For a given g ∈ G, let Xg denote the set of all such x� s. In other words, Xg = {x ∈ I : x−1 gx ∈ H} It follows that

χU (g) =



χV (x−1 gx)

x∈Xg

We note that for any x, g ∈ G, x−1 gx ∈ H iff y −1 gy ∈ H for all y ∈ G(g).x and in this case, x−1 gx = y −1 gy, where G(g) is the centralizer of g in G. Thus, we can write for x ∈ Xg ,  χV (y −1 gy) χV (x−1 gx) = μ(G(g))−1 y∈G(g)x

where μ(G(g)) denotes the number of elements in G(g). It follows that  χV (y −1 gy) χU (g) = μ(G(g))−1 x∈Xg ,y∈G(g)x

Further, it is clear that h ∈ H and y gy ∈ H implies (yh)−1 gyh = h−1 y −1 gyh ∈ H and hence χV ((yh)−1 gyh) = χV (y −1 gy). Thus, the above formula can also be expressed as  χV (y −1 gy) χU (g) = μ(G(g))−1 μ(H)−1 −1

x∈Xg H,y∈G(g)x

Quantum Mechanics for Scientists and Engineers 110

113

= (μ(G(g))μ(H))−1



χV (y −1 gy)

y∈G(g)Xg H

Now, consider the set of conjugacy classes of g ∈ G:

C(g) = {xgx−1 : x ∈ G} C(g) contains

μ(C(g)) = μ(G)/μ(G(g))

distinct elements. Reference: Claudio Procesi, ”Lie groups, an approach through invariants and representations”, Springer. A.5. Some more problems in group theory and quantum mechanics. [1] The basic observables of non-relativistic quantum mechanics obtained from the unitary representations of the Galilean group based on Mackey’s theory of semidirect products. Suppose N is a vector space regarded as an Abelian group under addition. Let H be a group that acts on N as n → τh (n). Let G = N ⊗s H. Thus, any element g ∈ G can be uniquely expressed uniquely as g = nh, n ∈ N, h ∈ H and the composition law in G is given by n1 h1 n2 h2 = n1 τh1 (n2 )h2 . We have τh1 h2 = τh1 oτh2 , τh (n + n� ) = τh (n) + τh (n� ) In other words, h → τh is an homomorphism of H into aut(N ) with aut(N ) being the same as End(N ) (aut(N ) is a group theoretic notation while End(N ) is a vector space theoretic notation. The composition of n1 h1 and n2 h2 : Then we may regard N as being normalized by the action τ of H. Equivalently, via the construction of a group isomorphism, we can regard τh as begin given by τh (n) = hnh−1 , so that n1 h1 n2 h2 = (n1 + τh1 (n2 )).h2 Now let B(n1 , n2 ) be a skew symmetric real bilinear form on N that is H-invariant, ie, B(τh (n1 ), τh (n2 )) = B(n1 , n2 ) Then consider

σ(n1 h1 , n2 h2 ) = exp(iB(n1 , τh1 (n2 )))

We claim that σ satisfies the conditions for a multiplier on G, ie, for a projective unitary (pu) representation U of G, ie, σ(g1 , g2 )σ(g1 g2 , g3 ) = σ(g1 , g2 g3 )σ(g2 , g3 ) − − − (1) We leave this verification to the reader. This follows from the fact that if U satsifies by the definition of a pu representation U (g1 )U (g2 ) = σ(g1 , g2 )U (g1 g2 ) and hence by associativity of linear operator multiplication, (U (g1 )U (g2 ))U (g3 ) = U (g1 )(U (g2 )U (g3 )) we get or

σ(g1 , g2 )U (g1 g2 )U (g3 ) = U (g1 )U (g2 g3 )σ(g2 , g3 ) σ(g1 , g2 )σ(g1 g2 , g3 )U (g1 g2 g3 ) = σ(g1 , g2 g3 )σ(g2 , g3 )U (g1 g2 g3 )

ie (1). [2] Let f (r1 , ..., rN ) and g(r1 , ..., rN ) be two functions on (R3 )N . Let R ∈ SO(3) and σ, τ, ρ ∈ SN . Let χλ (σ) be an irreducible character of Sn corresponding to the Young tableaux (ie a partition of n) λ and consider  I(f, g, r1 , ..., rN ) = f (rσ1 , ..., rσn )¯ g (rτ 1 , ..., rτ n )χλ (στ −1 ) σ,τ ∈Sn

We have

I(f, g, rρ1 , ..., rρN ) =

 σ,τ

f (rρσ1 , ..., rρσn )¯ g (rρτ 1 , ..., rρτ n )χλ (στ −1 )

Quantum Mechanics for Scientists and Engineers 111

114

=



f (rσ1 , ..., rσn )¯ g (rτ1 , ..., rτ n )χλ (ρστ −1 ρ−1 )

σ,τ

= I(f, g, r1 ‘, ..., rN )

since

χ(ρσρ−1 ) = χ(σ)

for any character χ of Sn . More generally, we can define  � )= f (rσ1 , ..., rσn )¯ g (rτ� 1 , ...., rτ� n )χλ (στ −1 ) I(f, g, r1 , ..., rN , r1� , ..., rN σ,τ ∈Sn

Then, we get � � I(f, g, rρ1 , ..., rρn , rρ1 , ..., rρn )=



� � f (rρσ1 , ..., rρσn )¯ g (rρτ 1 , ..., rρτ n )

σ,τ

=



.χλ (στ −1 ) = f (rσ1 , ..., rσn )¯ g (rτ� 1 , ..., rτ� n )

σ,τ

� ) χλ (στ −1 ) = I(f, g, r1 , ..., rN , r1� , ..., rN � ∈ R3 . Now,let R ∈ SO(3). The rotated and permuted image fields obtained from f and g are ∀r1 , ..., rN , r1� , ..., rN � � � f1 (r1 , ..., rN ) = f (R−1 rρ1 , ..., R−1 rρN ), g1 (r1� , ..., rN ) = g(R−1 rρ1 , ..., R−1 rρN )

and we get

� � ) = I(f, g, R−1 r1 , ..., R−1 rN , R−1 r1� , ..., R−1 rN )= I(f1 , g1 , r1 , ..., rN , r1� , ..., rN

and hence if χl denotes an irreducible character of SO(3), we get  � I(f1 , g1 , S1 r1 , ...S1 rN , S2 r1� , ..., S2 rN )χl (S1 S2−1 )dS1 dS2 SO(3)×SO(3)

=



SO(3)×SO(3)

=



� I(f, g, R−1 S1 r1 , ..., R−1 S1 rN , R−1 S2 r1� , ..., R−1 S2 rN )χl (S1 S2−1 )dS1 dS2

SO(3)×SO(3)

=



� I(f, g, S1 r1 , ..., S1 rN , S2 r1� , ..., S2 rN )χl (RS1 S2−1 R−1 )dS1 dS2

SO(3)×SO(3)

It follows that

� I(f, g, S1 r1 , ..., S1 rN , S2 r1� , ..., S2 rN )χl (S1 S2−1 )dS1 dS2

� I0 (f, g, r1 , ..., rN , r1� , ..., rN )=



SO(3)×SO(3)

� I(f, g, S1 r1 , ...S1 rN , S2 r1� , ..., S2 rN )χl (S1 S2−1 )dS1 dS2

is invariant under permutations and rotations. References for A.4 and A.5.: [1] Hoffman and Kunze, Linear Algebra, Prentice Hall. [2] T.Kato, Perturbation theory for linear operators, Springer. [3] K.R.Parthasarathy, ”An introduction to quantum stochastic calculus”, Birkhauser. [4] S.J.Orfanidis, ”Optimum signal processing”, Prentice Hall. [5] C.R.Rao, ”Linear statistical inference and its applications, Wiley. [6] J.Gough and Koestler, ”Quantum filtering in coherent states”. [7] Lec Bouten, ”Filtering and control in quantum optics”, Ph.D thesis. [8] Leonard Schiff, ”Quantum mechanics”. [9] W.O.Amrein, ”Hilbert space methods in quantum mechanics”. [10] K.R.Parthasarathy, ”Coding theorems of classical and quantum information theory”. Hindustan Book Agency. [11] Naman Garg, H.Parthasarathy and D.K.Upadhyay, ”Estimating parameters of an image field modeled as a quantum electromagnetic field using its interaction with a finite state atomic system”, Technical Report, NSIT, 2016.

Quantum Mechanics for Scientists and Engineers 112

115

[12] Naman Garg, H.Parthasarathy and D.K.Upadhyay, ”MATLAB implementation of Hudson-Parthasarathy noisy Schrodinger equation and Belavakin’s quantum filtering equation with analysis of entropy evolution”, Technical Report, NSIT, 2016. A.6. New Syllabus for a short course on transmission lines and waveguides. [1] Study of non-uniform transmission lines by expanding the distributed parameters and the line voltage and current as Fourier series in the spatial variable z. The modes of propagation (propagation constants) appear as the eigenvalues of an infinite matrix defined in terms of the spatial Fourier series coefficients of the non-uniform line impedance and admittance. [2] Study of the statistics of the line voltage and current (spatial correlations) when the distributed parameters of the line have small random fluctuations. The study is based on perturbation theory applied to matrix eigenvalue problems and is very similar to time independent perturbation theory used in quantum mechanics. [3] Analysis of transmission lines when the distributed parameters are randomly fluctuating functions of both space and time. We focus on estimating the distributed parameter statistical correlations from measurements of the line voltage and current and applying the ergodic hypothesis for estimating the line voltage and current correlations and then matching these correlations with the theoretically derived correlations. [4] Analysis of transmission lines with random loading along the line using infinite dimensional stochastic differential equations. The voltage and current loading along the line are assumed to be expandable in terms of basis functions of the spatial variable with the coefficients being white noise processes in time, ie, derivatives of Brownian motion processes. We then calculate the probability law of the line voltage and current [5] Equivalence of transmission lines and waveguides obtained by expanding the guide electric and magnetic fields in terms of basis functions of (x, y) and the coefficients being functions of z. From the Maxwell equations, we derive an infinite series of first order linear differential equations for the coefficient functions of z and compare these equations with an infinite sequence of coupled transmission lines. The basis functions of (x, y) used in the expansion of the electric and magnetic fields must satisfy the boundary conditions, namely, that Ez and the normal derivative of Hz vanish on the boundary. [6] Study of nonlinear hysteresis and nonlinear capacitive effects on the dynamics of a transmission line. Hysteresis is related to a nonlinear B − H curve having memory and is a consequence of the Landau-Lifshitz theory of magnetism in which the magnetic moment of an atom precesses in an external magnetic field due to the M × H torque on it. The solution to this equation is a Dyson series for the magnetization M in terms of H and truncated upto second degree terms, this leads to a quadratic expression for the hysteresis voltage term as a function of the line current. In other words, we have a second order Volterra relation between the hysteresis voltage and current. This is a consequence of magnetic properties of the material of which the line is made. Nonlinear capacitive effects can be explained from the nonlinear-memory relation between the dipole moment/polarization of an electron relative to its nucleus when an external electric is incident on it. The binding of the electron to the atom has harmonic as well as anharmonic terms which causes the differential equation satisfied by the position of the electron to be nonlinear and hence solving this equation using perturbation theory, we obtain the dipole moment as a Volterra series in the external electric field. When applied to transmission lines, this manifests itself as a Volterra relation between the line charge (which is proportional to the electric displacement vecto D = �0 E + P where P is the polarization/dipole moment per unit volume). The time derivative of this charge is the capacitor current and this component is incorporated into the line current equation and analysis is done using perturbation theory. [7] Quantization of the line equations using the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) formalism. The line equations for a lossless line are distributed parameter analogs of LC circuits. The lossless line equations like an LC circuit can be derived from a Lagrangian and hence from a Hamiltonian that is a quadratic function of the phase variables. The effect of noise on such a system is obtained by adding a GKSL term to the dynamics of states and observables. These GKSL terms can natually be obtained using the Hudson-Parthasarathy quantum stochastic calculus by tracing out over the bath variables. By choosing our GKSL operators L as complex linear functions of the phase variables, we obtain resistive damping terms in the dynamical equations and hence we are able to obtain a quantum mechanical model for a lossy line. A.7. Creativity in the mathematical, physical and the engineering sciences. In this section, we give a brief history of the various intellectual achievements in the mathematical, physical and engineering sciences showing how creativity in these sciences very often comes from a need to understand nature and the working of the world around us. The examples we choose are Newton, Maxwell, Einstein, Planck, Bose, Rutherford, Bohr, Heisenberg, Schrodinger, Dirac, Dyson, Feynman, Schwinger, Tomonaga, Weinberg, Salam, Glashow and Hawking in the physical sciences, Faraday and Edison and the inventor Esaki of the tunnel diode in the engineeering sciences and in the mathematical sciences, Euler, Gauss, Fourier, Fermat, Galois, Abel, Cauchy, Hilbert, Poincare, Kolmogorov,

Quantum Mechanics for Scientists and Engineers 113

116

Ramanujan, Harish-Chandra and more recently, Edward Witten, a pioneer in mathematical string and superstring theory. The creation of quantum electrodynamics: After the creation of the quantum theory of atoms and molecules, there remained several gaps in our understanding of the physical world. For example, it was not clear how to explain various experimental observations like the force of an electron on itself, the electron self energy ie, the movement of an electron produces an em field which acts back on the electron, the phenomenon of vacuum polarization according to which a photon propagates in vacuum to produce an electron-positron pair which propagate and again annihilate each other to produce once again a photon, the anomalous magnetic moment of the electron which gives radiative corrections to the magnetic moment caused once again by the the em field generated by the electron acting back on itself. Compton scattering of an electron/positron by a photon also remained unexplained. In other words, a satisfactory quantum theory describing the interactions of electrons, positrons and photons and how to calculate probabilities of scattering processes of these particles remained to be carried out. Thus because of the need to understand these unexplained physical processes, Feynman, Schwinger and Tomonaga created new mathematical tools which eventually were sharpened by Dyson and the succeeding generation of theoretical physicists like Wienberg, Salam, Glashow, Witten, and more recently by the Indian physicists Sen and Ashtekar. Feynman proposed a path integral formulation for calculating the scattering matrix for particles. This involved identifying the Lagrangian density L0 [φ] of the electron-positron  Dirac field without their interactions and that of the electromagnetic field and then evaluating the action S[φ] = Ld4 x for these fields.  Feynman followed it with evaluation of the Gaussian path integrals exp(iS[φ])Dφ taking into account the Berezin   change for Fermionic path integrals. Then the interaction term Sint [φ] = J μ Aμ d4 x = ψ ∗ γ 0 γ μ ψAμ d4 x (φ = (Aμ , ψ)) is considered and its contribution was evaluated by expanding the exponential exp(iSint [φ]) in as a power series in S[φ] and Feynman associated a diagram with each term in this series explaining how each term gives rise to a different order term in the scattering matrix and how these terms can be calculated easily by a diagrammatic algorithm. On the other hand Schwinger and Tomonaga proposed an operator theoretic approach to calculating the scattering matrix. Their algorithm was based on first quantizing the electromagnetic field using creation and annihilation operators. This had already been observed by Paul Dirac when he wrote down the energy of the electromagnetic field as a quadaratic function of the four vector potential in the spatial frequency domain and deduced that this quadratic structure meant that the em field should be considered as an ensemble of harmonic oscillators with two oscillators associated with each spatial frequency (two degrees of polarization arise from the fact that in the coulomb gauge, in the absence of charges, the electric scalar potential is zero while divA = 0 for the Coulomb gauge implies that in the spatial frequency domain, the magnetic vector potential is orthogonal to the wave vector). The next idea of Schwinger was to substitute this quantum electromagnetic field consisting of a superposition of operators into the atomic Hamiltonian described by position and momentum operators and obtain an interaction term between the position-momentum pair for the atom with the quantum electromagnetic field operators. This interaction Hamiltonian was then used to calculate the scattering matrix elements and deduce corrections to the magnetic moment of the electron. Schwinger and Tomonaga also proposed a Lorentz invariant scheme for writing down the Schrodinger equation (which is not Lorentz covariant) for field theories. The idea basically involved replacing the time t variable by a three dimensional surface variable σ. In other words, just as the t = constt surface is the three dimensional Euclidean space R3 as a subspace of R4 , likewise, σ = constt. could be an arbitrary three dimensional submanifold of R4 . The Schrodinger equation i ∂ψ(t) = H(t)ψ(t) ∂t was replaced by the Lorentz covariant equation δψ(σ) = H(σ)ψ(σ) δσ and this formalism was used with great power by Schwinger and Tomonaga to calculate the scattering matrix element between two three dimensional surfaces. Unification of the Feynman and Schwinger-Tomonaga theory was performed by Freeman Dyson who simply showed why one should obtain the same results using Feynman diagrams and the operator theoretic approach. For a very long time, Dyson’s notes on this was the standard textbook for all courses in quantum field theory all over the world. Dyson’s work led to renormalization theory developed by himself and subsequently othe researchers. This involved getting rid of the infinities in quantum field theory by renormalizing charge, mass and fields, ie, scaling these quantities with numbers depending on an ultraviolet and infrared cutoff while integrating in the four frequency domain. After the great riddle of construcing a cogent quantum theory of electrodynamics was solved almost completely by these four powerful mathematical physicists, the problem of understanding nuclear weak and strong forces remained and also how to unify these with quantum electrodynamics. The problem of unifying quantum electrodynamics with the weak forces, called the Electroweak theory was successfully solved by Weinberg, Salam and Glashow using group theoretic formalism, more precisely the SU (2) × U (1) formalism. Both the weak forces and electromagnetic forces appeared as gauge fields in this theory. Principles of symmetry breaking were used in this unification giving rise to mass of electrons and other nuclear particle. Goldstone had a say in this unification when he proposed the idea of how when a Lagrangian of fields that is invariant under a group G has a vacuum state that is not G-invariant because of degeneracies of the vacuum state, the symmetry of the Lagrangian as viewed from the i

Quantum Mechanics for Scientists and Engineers 114

117

vacuum state is broken to s smaller group H ⊂ G and associated with each degree of broken symmetry is a massless particle called a massless Goldstone boson. The unbroken symmetries correspond to massive particles. Symmetry can also be broken by adding a term to the G-invariant Lagrangian density. This is what happens in the electroweak theory. The electroweak-strong unification was achieved by Gell-Mann and Nee-Mann who based their theory on the group SU (3) × SU (2) × U (1). The idea is to derive all the coupling constants of electrdynamics, the weak and the strong theories from one unified theory. The entire idea of unifying gauge fields is based on the basic principle of Yang and Mills, namely that one can construct a covariant derivative ∇μ = ∂μ + iAμ (x) acting on a vector space (Cn ) valued function of x in such a way that the gauge field Aμ (x) takes values in a Lie algebra g of a subgroup G of the unitary group U (n). The wave function on which ∇μ acts is Cn . Further, this covaraiant derivative satisfies the property that under a local G-transformation by g(x) ∈ G, the gauge field Aμ (x) which takes values in g transforms in such a way to A�μ (x) so that g(x)(∂μ + A�μ (x)) = (∂μ + Aμ (x))g(x) or equivalently as

g(x)∇�μ = ∇μ g(x)

where both sides are regarded as first order differential operators acting on wave functions ψ(x) ∈ Cn . This gives iA�μ (x) = g(x)−1 ∂μ g(x) + ig(x)−1 Aμ (x)g(x) This idea of gauge transformation in which the massive field wave function ψ(x) ∈ Cn transforms to g(x)ψ(x) while the massless gauge field Aμ (x) transforms in the above way leads to the conclusion that given a Lagrangian density of the form L(ψ(x), ∇μ ψ(x)) that is G − invariant, ie, L(gψ(x), g∇μ ψ(x)) = L(ψ(x), ∇μ ψ(x)) for all g ∈ G, it follows that L is invariant also under local G-actions g(x) provided that the Aμ (x) sitting inside the covariant derivative ∇μ is also subject to the above gauge transformation. When the group G is the Abelian group U (1), Aμ (x) is simply a real valued function for each μ and the above gauge transformation reduces to the Lorentz gauge transformation for the electromagnetic potentials. Thus, the non-commutative Yang-Mills theory provides a sweeping generalization of the commutative em field theory. It can also be applied to describe the interaction of the Dirac field ψ(x) with a noncommutative gauge field with the electromagnetic potentials also coming as an extra component of the gauge potential. Although this idea of Yang and Mills is a purely group theoretic construction, it turned out to be one of the most fruitful constructs for unifying almost all the quantum particle fields. What remains now is the development of a quantum theory of gravity which would enable one to associate a particle which we may call a graviton and to describe its interaction with other quantum particles like the photon, electron, positron, and the propagators of the weak and strong forces. It should be borne in mind that the action of a classical gravitational field on any quantum particle described by a relativistic wave equation like the Dirac equation can be achieved using the Idea of Yang and Mills, namely by introducing a gravitational connection Γμ (x) which is a matrix. For example, if the gravitational field is described by a tetrad Vaμ (x) so that the metric is g μν (x) = η ab Vaμ (x)Vbν (x) with η ab being the Minkowski metric of flat space-time, then this tetrad can be understood as a local transformation of curved space-time to an inertial frame. We then construct a covariant derivative ∂μ + Γμ (x) and transform it locally to an inertial frame by using Vaμ (x)(∂μ + Γμ (x)) and setting up the Dirac equation in a gravitational field as [iγ a Vaμ (x)(∂μ + Γmu (x)) − m]ψ(x) = 0

where γ a , a = 0, 1, 2, 3 are the usual Dirac gamma matrices. To qualify as a valid general relativistic wave equation, this must be invariant under local Lorentz transformations Λ(x). Under such a local Lorentz transformation, ψ(x) transforms to D(Λ(x))ψ(x) where D is the Dirac representation of the Lorentz group and its Lie algebra generators are J ab = 14 [γ a , γ b ]. Suppose that under such a local Lorentz transformation Γμ (x) which is a 4 × 4 matrix changes to Γ�μ (x). Then we should have with ψ � (x) = D(Λ(x))ψ(x), the equation [iγ a Λba (x)Vbμ (x)(∂μ + Γ�μ (x) − m]ψ � (x) = 0 where the change of space-time coordinates has been accounted for by the factor matrix elements Λba (x) of the local Lorentz transformation of the inertial frame index a in the tetrad frame Vaμ (x). Using the identity D(Λ)γ b D(Λ)−1 = Λba γ a we get from the above [D(Λ(x))(iγ b Vbμ (x)(D(λ(x))−1 (∂μ + Γ�μ (x)) − m]D(Λ(x))ψ(x) = 0 This is equivalent to iγ b Vbμ (x)((D(Λ(x))−1 ∂μ D(Λ(x))) + ∂μ + D(Λ(x))−1 Γ�μ (x)D(Λ(x))) − m]ψ(x) = 0

Quantum Mechanics for Scientists and Engineers 115

118

It follows that for this to coincide with the untransformed Dirac equation in curved space-time, the connection Γμ (x) of the gravitational field in the Dirac representation should transform to Γ�μ (x) where D(Λ(x))−1 Γ�μ (x)D(Λ(x)) + (D(Λ(x))−1 ∂μ D(Λ(x))) = Γμ (x) or equivalently,

Γ�μ (x) = D(Λ(x))Γμ (x)D(Λ(x))−1 − (∂μ D(Λ(x)))D(Λ(x))−1

Such a connection has been constructed and is given by

Γμ (x) =

1 ab ν J Va Vbν:μ 2

(Reference: Steven Weinberg, ”Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity”, Wiley.) Gravity was unified with classical electromagnetism by Einstein in this beautiful theory the general theory of relativity. This theory said that gravity is not a force, it is simply a curvature of space-time and when matter moves in such a curved space time, follows geodesics which are shortest paths on the curved four dimensional manifold of space-time. These shortest paths are curved because any path on a curved surface is curved. By saying that geodesics are curved, we mean that the relation between the spatial and time coordinates of a moving particle is nonlinear and hence the motion appears to us as being accelerated motion. A.8 Classification and representation theory of semisimple Lie algebras. By Serre’s theorem, a complex semisimple Lie algebra L is generated by 3n generators ei , hi , fi , i = 1, 2, ..., n satisyfying the commutation relations [ei , ej ] = [fi , fj ] = [hi , hj ] == [ei , fj ] = 0, [ei , fj ] = δij hi , [hi , ej ] = aij ej , [hi , fj ] = −aij fj

where aij are integers called the Cartan integers. This result of Serre follows from the Cartan’s theory which says that L has a maximal Abelian subalgebra h (Called a Cartan Algebra) and that any two maximal Abelian subalgebras are mutually conjugate. This result is not true for real semisimple Lie algebras where there can be more than one non-conjugate Cartan algebras. The reprsentation theory for real semisimple Lie algebras was developed almost single handedly by the great Indian mathematician Harish-Chandra who derived generalizations of the character formula of H.Weyl using the theory of distributions and also obtained the complete Plancherel formula for such algebras by introducing in addition to the principal and supplementary series of irreducible representations (introduced by Gelfand for obtaining the Plancherel formula for complex semisimple Lie groups), the discrete series of irreducible representations. Coming back to the theme of complex semisimple Lie algebras, Cartan proved that the elements of h in the adjoint representation, act in a semsimple way on L, ie, each operator ad(H), H ∈ h acting on the vector space L is diagonable. It follows from basic Linear algebra, that the operators ad(H), H ∈ h are simulatneously diagonable and hence we have the following direct sum decomposition of L:  Lα L=h⊗ α∈Φ

where Φ is the set of all roots of L, ie, for each α ∈ Φ and X ∈ Lα , we have [H, X] = α(H)X

We are here defining

Lα = {X ∈ L : [H, X] = α(H)X∀H ∈ h}

α is a non-zero linear functional on h, ie, α ∈ h∗ and all the α� s are distinct linear functionals. Moreover, Cartan’s theory states that there exists a subset Δ ⊂ Φ called a set of simple roots such that any α ∈ Φ is either a purely positive or purely negative integer linear combination of elements of Δ. This means that writing Δ = {α1 , ..., αp }, we have that for any α ∈ Φ, there exist integers m1 , ..., mp such that mj ≥ 0∀j or mj ≤ 0∀j and α=

p 

mj αj

j=1

and further, no α ∈ Δ has such a decomposition. Actually, there exist many such sets Δ. Further, dimLα = 1∀α ∈ Φ. This follows from the following elementary argument, From the Jacobi identity, [Lα , Lβ ] ⊂ Lα+β , α, β ∈ Φ. Thus

Quantum Mechanics for Scientists and Engineers 116

119

[Lα , L−α ] ⊂ h. Choose an eα ∈ Lα , fα ∈ L−α such that B(eα , fα ) = b(α) where B(X, Y ) = T r(ad(X)ad(Y )) and b(α) is a constant to be chosen appropriately. Cartan proved that B defines an non-degenerate symmetric bilinear form on L (only if L is semisimple). Now define tα = a(α)[eα , fα ] where a(α) is a constant to be chosen appropriately. Then, tα ∈ h and [tα , eα ] = α(tα )eα , [tα , fα ] = −α(tα )fα and more generally, for any α, β ∈ Φ,

[tα , eβ ] = β(tα )eβ , [tα , fβ ] = −β(tα )fβ

Consistency is checked as follows using B([X, Y ], Z) = B(X, [Y, Z]):

= b(β)β(tα ) = β(tα )B(eβ , fβ ) = B([tα , eβ ], fβ )

We further have

= −B(eβ , [tα , fβ ]) = β(tα )B(eβ , fβ ) = b(β)β(tα ) B(tα , tβ ) = a(α)B([eα , fα ], tβ ) = a(α)B(eα , [fα , tβ ]) = a(α)α(tβ )B(eα , fα ) = a(α)b(α)α(tβ )

and we denote this by (α, β). The above formula implies that (α, β) = (β, α) = a(α)b(α)α(tβ ) = a(β)b(β)β(tα ) Now define Then,

hα = 2tα /(α, α) [hα , eα ] = 2α(tα )eα /(α, α) = 2eα

provided that we choose the a(α)� s and the b(α)� s so that α(tα ) = (α, α) For such a choice of the constants, we also have [hα , fα ] = −2fα , and

[eα , fα ] = tα /a(α) = (α, α)hα /(2a(α)) = hα

provided that we choose the a(α)� s and the b(α)� s so that (α, α) = 2a(α) = α(tα ) In other words, we have found eα ∈ Lα , fα ∈ L−α , hα ∈ h so that {eα , fα , hα } satisfy the same commutation relations as the standard generators of sl(2, C). We denote the Lie algebra generated by these three elements by slα (2, C). We note that in the adjoint representation, slα (2, C) is (a module for) an irreducible representation of the Lie algebra slα (2, C). We are assuming here that relative to the set of simple roots Δ, α is a positive root, ie it is expressible as a positive integer linear combination of the elements of Δ. We note that if X ∈ Lα , Y ∈ Lβ , then B([H, X], Y ) = −B(X, [H, Y ]) implies

(α(H) + β(H))B(X, Y ) = 0, H ∈ h

and hence B(X, Y ) = 0 unless β = −α. It is therefore clear from the non-degeneracy of B that for each X ∈ Lα , B(X, Y ) �= 0 for some Y ∈ L−α It is also clear that dimLα = 1 This can be seen as follows: Consider the sum Mα = h ⊕

 c

Lcα

Quantum Mechanics for Scientists and Engineers 117

120

the sum being over all the irreducible representations of slα (2, C) in which the weights are multiples of α. (By weight β, here we mean that if X is a weight-vector with weight β where β is a linear functional on h, then ad(H)(X) = β(H)X∀H ∈ h. We note that Mα is a module for slα (2, C). When H = hα , then β(hα ) becomes an eigenvalue of ad(hα ) and hence β can be regarded as a weight for the Lie algebra slα (2, C). Since slα (2, C) as a Lie algebra is isomorphic to sl(2, C), in any irreducible representation of slα (2, C), either a weight zero or a weight one will occur with a unique weight vector. Clearly, ad(hα ) when acting on Mα has weight zero iff the weight vector is in h. Mα contains the module h + slα (2, C) of slα (2, C) and hence it cannot contain any irreducible even module that has zero intersection with h + slα (2, C) appearing in Mα as a direct summand for any even irreducible module for slα (2, C) must necessarily contain a zero weight vector which must be an element of h and hence will intersect the module h + slα (2, C)(Note that h + slα (2, C) is a module for slα (2, C), ie, it is left invariant by the adjoint action of the latter and hence this module can be decomposed into irreducible modlues for slα (2, C)). Therefore, we must have Mα = h + slα (2, C) and in particular,

dimLα = 1, α ∈ Φ

By an even module g of slα (2, C), we mean that ad(hα ) has an even eigenvalue when operating on the root vectors in g in the adjoint representation. For example, if the eigenvalues {−2q, −2q + 2, ..., 0, 2, 4, ..., 2p}} occured in an irreducible submodule of Mα for the Lie algebra slα (2, C) as a direct summand different from slα (2, C) in the adjoint representation, then the zero weight vector in this represenation would be hα which is a contradiction. Further, an odd summand (ie in which a vector having weight one occurs) also cannot occur, for then α/2 would be a root ((α/2)(hα ) = 1) and hence α = 2(α/2) cannot be a root by the above argument(Note that in an irreducible representation of slα (2, C), only the weights from the set 2Z or only weights from 2Z + 1 can occur. We have thus proved that if α is a root and cα is also a root, then c = ±1. Remark: If g is a semisimple Lie algebra and, then we can decompose g=

N 

gi

i=1

as a direct sum of vector spaces gi where each gi is an ideal in g, ie [g, gi ] ⊂ gi ∀i and hence [gi , gj ] ⊂ gi



gj = {0}, i �= j.

A.8.Schrodinger wave equations for quantum general relativity. A manifold specified by space-time coordinates x ˜μ is given. Another coordinate system for this manifold is X μ . The metric tensor relative to the former is g˜μν and the metric tensor for the latter is gμν . Thus, we have μ ν X,σ = g˜ρσ gμν X,ρ μ

a b μ By X,ρ we mean ∂X ∂xρ . We denote the spatial coordinates of the former system by x , x ,etc, where a, b = 1, 2, 3. Thus, the spatial components of the metric in the former system are μ ν X,b g˜ab = gμν X,a

We define

qab = g˜ab

We denote by ((q )) the 3 × 3 matrix that is the inverse of ((qab )). Now, write ab

μ = T μ = N μ + N nμ X,0

where N μ is purely spatial, ie, of the form

μ N μ = N a X,a

and the vectors N μ and nμ are orthogonal with nμ normalized by the factor N . This means that N a is selected so that μ ν , gμν nμ X,a =0 N μ = N a X,a

The first is the condition for N μ to be a spatial vector and the second is the condition for nμ to be orthogonal to all spatial vectors. We can visualize this by saying that the three dimensional spatial manifold Σt defined by x0 = t = constt is embedded in the four dimensional manifold specified by the coordinates X μ . The vector (nμ ) is the unit normal to the

Quantum Mechanics for Scientists and Engineers 118

121

μ 3 manifold Σt and the vectors (X,a )μ=0 , a = 1, 2, 3 are tangential to the manifold Σt . We thus get the following equations for N a : μ μ ν − N a X,a )X,b =0 gμν (X,0

or

g˜0b − N a g˜ab = 0

or equivalently,

qab N b = g˜0a

We now prove the following decomposition:

g μν = q μν + nμ nν

where

q μν nν = 0

In other words, g μν can be decomposed as a sum of a purely spatial part and a purely normal part with regard to the surface Σt . To see this, we write μ ν g μν = g˜αβ X,α X,β = ν g 0a (N μ + N nμ )X,a + g˜00 (N μ + N nμ )(N ν + N nν ) + 2˜ μ ν g˜ab X,a + X,b μ We have to show that the cross term in this expansion is zero, ie, terms involving products of spatial parts X,a and the normal part nμ . The cross part here is ν g 0a N nμ X,a 2N g˜00 N μ nν + 2˜

To prove that this is zero amounts to proving that ν =0 g˜00 N μ nν + g˜0a nμ X,a

To prove this it suffices to show that or equivalently,

g˜00 N a + g˜0a = 0

Proving this is equivalent to proving that which is the same as (since g˜ba g˜

0a

μ =0 g˜00 N μ + g˜0a X,a

+ g˜b0 g˜

00

=

δba

g˜ba g˜0a + g˜00 g˜ab N a = 0 −˜ gb0 g˜00 + g˜00 g˜ab N a = 0

= 0). Thus, we have to show that

qab N a = g˜b0 μ But this has already been established using the orthogonality of the normal vector nμ with the spatial vectors X,a .

A.9. Time travel in the special and general theories of relativity and the revised notions of space-time in quantum general relativity. Gravitational red-shift: Let U (r) be the gravitational potential. Then the approximate (Newtonian) metric of spacetime is given by dτ 2 = (1 + 2U (r)/c2 )dt2 − c−2 (dx2 + dy 2 + dz 2 )

We are assuming that U depends only on the radial coordinate relative to a system. The radial null geodesic (radial propagation of photons) is given by 0 = dτ 2 = (1 + 2U (r)/c2 )dt2 − dr2 or equivalently,

dr/dt = (1 + 2U (r)/c2 )1/2

Thus assuming r1 < r2 , a photon pulse starting from r1 at time t1 arrives at r2 at time t2 given by  r2 t2 − t1 = (1 + 2U (r)/c2 )−1/2 dr r1

119 Quantum Mechanics for Scientists and Engineers

122

In this expression, t1 , t2 are coordinate times, ie, times measured by a clock at a large distance from the gravitiational field, ie, at a point where the gravitatitional field is zero. Now if another pulse starts from r1 at coordinate time t1 + δt1 , then it will arrive at r2 at time t2 + δt2 where  r2 t2 + δt2 − t1 − δt1 = (1 + 2U (r)/c2 )−1/2 dr r1

Note that we are assuming a static gravitational field. It thus follows that δt2 = δt1 The proper time intervals measured by clocks static at r1 and r2 for the pulses are respectively given by dτ1 = (1 + 2U (r1 )/c2 )1/2 dt1 , dτ2 = (1 + 2U (r2 )/c2 )1/2 dt2 It follows therefore that dτ1 /dτ2 = [ where

1 + 2U1 /c2 1/2 ] 1 + 2U2 /c2

U1 = U (r1 ), U2 = U (r2 )

Hence, if U2 < U1 , we get dτ1 > dτ2 or in terms of frequencies,

ν1 = 1/dτ1 < 1/dτ2 = ν2

or more precisely,

1 + 2U2 /c2 1/2 ν1 =[ ] dτ2 , it follows that clocks run slower in a strong gravitational field, ie when the gravitational potential is more negative. A.10[a].Transmission lines with random fluctuations in the parameters and random line loading: v,z (t, z) + (R0 (z) + δR(t, z))i(t, z) + L0 (z)i,t (t, z) + (δL(t, z)i(t, z)),t = wv (t, z) i,z (t, z) + (G0 (z) + δG(t, z))v(t, z)) + C0 (z)v,t (t, z) + (δC(t, z)v(t, z)),t = wi (t, z) In these equations, δR(t, z), δL(t, z), δC(t, z), δG(t, z), wv (t, z), wi (t, z) are random Gaussian space-time Gaussian fields. We wish to solve this system of pde’s approximately using first order perturbation theory and hence calculate the approximate statistical correlations of the fluctuations in the line voltage and line current in terms of the correlations in the parameter flucutations and the voltage and current loading terms wv and wi . A.10[b]. Taking non-linear hysteresis and nonlinear capacitive effects into account, generalize the problem of A.9. A.11. Estimating parameters in statistical image models described by linear and nonlinear partial differential equations: First consider the linear case: The model for the image field φ(x, y), (x, y) ∈ D is given by p 

a,b=1

A(a, b, θ)∂xa ∂yb φ(x, y) = s(x, y) + w(x, y), (x, y) ∈ D

where w(x, y) is zero mean coloured Gaussian noise. θ ∈ Rm is the parameter vector to be estimated from measurements on φ. Here, s(x, y) is a given input non-random signal field. We assume that an initial guess estimate θ0 of θ is known and that the correction δθ to this estimate is to be made. We write φ(x, y) = φ0 (x, y) + δφ(x, y)

Quantum Mechanics for Scientists and Engineers 120

123

where φ0 is the solution with the guess parameter θ0 and zero noise and δφ is the first order correction to φ0 arising from the parameter estimate correction term δθ and the noise w. We regard δφ, δθ, w all as being of the first order of smallness. We define  A(a, b, θ)(jω1 )a (jω2 )b H(ω1 , ω2 , θ) = a,b

Then if two dimensional spatial Fourier transforms are denoted by placing a hat on top of a signal/noise field, we get ˆ 1 , ω2 ) = sˆ(ω1 , ω2 ) + w(ω H(ω1 , ω2 , θ)φ(ω ˆ 1 , ω2 ) Thus to zeroth order, we get

H(ω1 , ω2 , θ0 )φˆ0 (ω1 , ω2 ) = sˆ(ω1 , ω2 ), ˆ 1 , ω2 ) + (Hr (ω1 , ω2 , θ0 )δθr )φˆ0 (ω1 , ω2 ) H(ω1 , ω2 , θ0 )δ φ(ω = w(ω ˆ 1 , ω2 )

where Hr (ω1 , ω2 , θ0 ) = Thus, if

∂H(ω1 , ω2 , θ0 ) ∂θr

g(x, y) = F −1 (H(ω1 , ω2 , θ0 )−1 )

it then follows that

φ0 (x, y) = g(x, y) ∗ s(x, y) = Likewise, if we write



g(x − x� , y − y � )s(x� , y � )dx� dy �

hr (x, y) = F −1 (Hr (ω1 , ω2 , θ0 ))  Ar (a, b, θ0 )δ (a) (x)δ (b) (y) = a,b

where

Ar (a, b, θ0 ) = then we get

∂A(a, b, θ0 ) ∂θr

δφ(x, y) = −g(x, y) ∗ hr (x, y)δθr + g(x, y) ∗ w(x, y) − − − (2)

where summation over the repeated index r is implied. We measure δφ(x, y) as follows: First since we know θ0 and the input signal field s(x, y), we can calculate φ0 (x, y) using (1) and then we measure the actual noise perturbed image field φ(x, y) and calculate δφ(x, y) = φ(x, y) − φ0 (x, y). Now using (2), we calculate the maximum likelihood estimator of δθ as ˆ = δθ  argminδθ ( (δφ(x, y) + gr (x, y)δθr )Q(x, y|x� , y � )(δφ(x� , y � ) + gs (x� , y � )δθs )dxdydx� dy � ) where Q(x, y|x� , y � ) is the inverse Kernel of

ie,

and

We write



R(x, y|x� , y � ) = E[(g(x, y) ∗ w(x, y)).(g(x� , y � ) ∗ w(x� , y � ))] g(x − x1 , y − y1 )g(x� − x�1 , y � − y1� )E(w(x1 , y1 )w(x�1 , y1� ))dx1 dy1 dx�1 dy1� 

Q(x, y|x� y � )R(x� , y � |x�� , y �� )dx� dy � = δ(x − x�� )δ(y − y �� )

gr (x, y) = g(x, y) ∗ hr (x, y) =



Ar (a, b, θ0 )∂xa ∂yb g(x, y)

a,b

g(x, y) = ((gr (x, y)))

121 Quantum Mechanics for Scientists and Engineers

124

Then,

  ˆ = [ Q(x, y|x� , y � )g(x, y)g(x� , y � )T dxdydx� dy � ]−1 [ Q(x, y|x� , y � )g(x, y)δφ(x� , y � )dxdydx� dy � ] δθ

A simple computation gives us the covariance of this parameter vector estimator:  ˆ = [ Q(x, y|x� , y � )g(x, y)g(x� , y � )T dxdydx� dy � ]−1 Cov(δ θ)

Wavelet based image parameter estimation: Let ψn (x, y), n = 1, 2, ... be a wavelet orthonormal basis for L2 (R2 ). Here, the index n consists of the scaling and translational index in both the dimensions, ie, n corresponds to four ordered integer indices. We define  c(n, φ) =< φ, ψn >=

Then,

φ(x, y) =



φ(x, y)ψn (x, y)dxdy

c(n, φ)ψn (x, y)

n

Substituting this into the image pde model gives  c(n, φ)A(a, b, θ)∂xa ∂yb ψn (x, y) = s(x, y) + w(x, y) n,a,b

Taking the inner product on both sides with ψm (x, y) gives  c(n, φ)A(a, b, θ) < ψm , ∂xa ∂yb ψn >= c(m, s) + c(m, w) n,a,b

Define

P (m, n|θ) =



A(a, b, θ) < ψm , ∂xa ∂yb ψn >

a,b

The above equation can then be expressed as  P (m, n|θ)c(n, φ) = c(m, s) + c(m, w) n

Writing

θ = θ0 + δθ, φ(x, y) = φ0 (x, y) + δφ(x, y)

gives us on applying first order perturbation theory,  P (m, n|θ0 )c(n, φ0 ) = c(m, s), n

  (( Pr (m, n|θ0 )δθr )(¸n, φ0 ) + P (m, n|θ0 )δc(n)) = c(m, w) n

where

r

Pr (m, n|θ0 ) = and We write

∂P (m, n|θ0 ) ∂θr

δc(n) = c(n, φ0 + δφ) − c(n, φ0 ) c0 (n) = c(n, φ0 ), Pr (m, n) = Pr (m, n|θ0 ), P0 (m, n) = P (m, n|θ0 )

and thus get  n,r



P0 (m, n)c0 (n) = c(m, s)

n

Pr (m, n)δθr c0 (n) +

 n

P0 (m, n)δc(n) = c(m, w)

Quantum Mechanics for Scientists and Engineers 122

125

A.12 Mackey’s theory on the construction of the basic quantum observables from unitary representations of the Galilean group. (a, v)inV = R3 × R3 . V is the Abelian group of translations and uniform velocity motions acting on the space-time manifold M = {(t, x) : t ∈ R, x ∈ R3 }. This action is given by (a, v)(t, x) = (t + x + vt + a) (τ, g) ∈ R × SO(3). R × SO(3) is the non-Abelian group of time translations and rotations acting on M: (τ, g)(t, x) = (t + τ, gx) The Galiean group G is the semidirect product of V and H: G = V ⊗s H where H acts on V as follows:

(τ, g).(a, v).(τ, g)−1 = (a� , v � )

or equivalently,

(τ, g).(a, v) = (a� , v � ).(τ, g)

Acting both sides on (t, x) ∈ R gives 4

(τ, g)(t, x + vt + a) = (a� , v � )(t + τ, gx) or equivalently,

(t + τ, gx + tgv + ga) = (t + τ, gx + v � t + a� + v � τ )

or equivalently,

v � = gv, a� = g(a − vτ )

Thus, the semimdirect product structure is given by

(τ, g).(a, v).(τ, g)−1 = (g(a − vτ ), gv) ∈ V Any element of the Galilean group G can be expressed in two ways, one as an element (a, v, τ, g) defined by its action on R4 by (a, v, τ, g)(t, x) = (t + τ, gx + vt + a) and in another way as the element (a, v).(τ, g). The action of this on R4 is given by (a, v).(τ, g)(t, x) = (a, v)(t + τ, gx) = (t + τ, gx + vt + vτ + a) It follows that the relationship between these two methods of expressing an element of the Galilean group is given by (a + vτ, v, τ, g) = (a, v).(τ, g) or equivalently by

(a − vτ, v).(τ, g) = (a, v, τ, g)

2j+1 Let now h be a Hilbert space (like for a spin j particle) and H = L2 (R3 , h) the Hilbert space of all measurable  C functions f : R3 → h for which R3 � f (x) �2 d3 x < ∞. The projective unitary representations of G are obtained by using the multiplier m((a, v).(τ, g), (a� , v � ).(τ � , g � )) = exp(iB((a, v), (τ, g)[(a� , v � )])

where B : V × V → R (with V = R3 × R3 ) being a skew symmetric bilinear form invariant under H = {(τ, g) : τ ∈ R, g ∈ SO(3)}. that is invariant under H or equivalently under (τ, g). Note that the action of H on V is defined by (τ, g)[(a, v)] = (τ, g).(a, v).(τ, g)−1 = (g(a − vτ ), gv) We thus find that

m((a, v).(τ, g), (a� , v � ).(τ � , g � )) = exp(iB((a, v), (g(a − vτ ), gv))) = exp(iλ((a, gv) − (v, g(a − vτ ))))

for some λ ∈ R where (u, v) = u v. Let U be a projective unitary representation of G with this multiplier. Then, T

U ((a, v).(τ, g)) = U (a, v)U (τ, g)

Quantum Mechanics for Scientists and Engineers 123

126

We can write

U (a, 0) = V1 (a), U (0, v) = V2 (v), U (τ, g) = W1 (τ )W2 (g)

where V1 , V2 are unitary representations of the Abelian group V = R3 × R3 and W1 and W2 are unitary representations of R and SO(3). By Stone’s theorem on unitary representations of Abelian groups, it follows that there exist Hermitian operators Q = (Q1 , Q2 , Q3 ) and P = (P1 , P2 , P3 ) in L2 (R3 , h) such that V1 (a) = exp(−ia.P ), V2 (v) = exp(−iv.Q) and also a Hermitian operator H in the same space such that W1 (τ ) = exp(−iτ H) We have In particular,

U (a1 , v1 )U (a2 , v2 ) = exp(iλ(aT1 v2 − aT2 v1 ))U (a1 + a2 , v1 + v2 ) V1 (a)V2 (v) = U (a, 0)U (0, v) = exp(iλaT v)U (a, v), V2 (v)V1 (a) = U (0, v)U (a, 0) = exp(−iλaT v)U (a, v)

Thus, we get the Weyl commutation relations V1 (a)V2 (v) = exp(i2λaT v)V2 (v)V1 (a) which can be expressed as exp(−ia.P ).exp(−iv.Q) = exp(i2λaT v)exp(−iv.Q).exp(−ia.P ) and hence by considering infinitesimal a and v in R3 . we get −Pi Qj + Qj Pi = 2iλδij or equivalently,

[Qi , Pj ] = 2iλδij

To get agreement with quantum mechanics that the momentum operators P generate translations and the position operators Q generate uniform velocities, we must take λ = 1/2 and thus, we get [Qi , Pj ] = iδij The Stone-Von-Neumann theorem then implies that the Hilbert space � can be chosen so that the actions of the Qi s and the Pj s in L2 (R3 , h) are such that (Qj f )(x) = xj f (x), (Pj f )(x) = −i

∂f (x) ∂xj

In other words, regarding L2 (R3 , h) as L2 (R3 ) ⊗ h (This is a Hilbert space isomorphism), we have that Qj = xj ⊗ Ih ∂ and Pj = −i ∂x ⊗ Ih . We note compute j U (τ, g)U (a, v) = U ((τ, g).(a, v)) Now,

(τ, g).(a, v)(t, x) = (τ, g)(t, x + vt + a) = (t + τ, gx + tgv + ga) = (ga, gv, τ, g)(t, x)

ie, On the other hand,

(τ, g).(a, v) = (ga, gv, τ, g) (a, v).(τ, g)(t, x) = (a, v).(t + τ, gx) = (t + τ, gx + vt + vτ + a) = (a + vτ, v, τ, g)(t, x)

Quantum Mechanics for Scientists and Engineers 124

Thus,

127

(τ, g).(a, v) = (g(a − vτ ), gv).(τ, g)

So we get

U (τ, g)U (a, v)U (τ, g)−1 = U (g(a − vτ ), gv)

Taking g = I, this gives

W1 (τ )U (a, v)W1 (−τ ) = U (a − vτ, v)

Setting a = 0 in this formula gives

W1 (τ )V2 (v)W1 (−τ ) = U (−vτ, v)

while setting v = 0 gives

W1 (τ )V1 (a)W1 (−τ ) = V1 (a)

The second equation implies

[H, Pj ] = 0, j = 1, 2, 3

We now note that Thus, we get from the first equation

V1 (−vτ )V2 (v) = exp(−iτ |v|2 /2)U (−vτ, v)

W1 (τ )V2 (v)W1 (−τ ) = exp(iτ |v|2 /2)V1 (−vτ )V2 (v) = exp(iτ |v|2 /2)exp(iτ v.P )V2 (v) For infinitesimal τ , this gives or equivalently,

−i[H, exp(−iv.Q)] = (i|v|2 /2 + iv.P )exp(−iv.Q) H − exp(−iv.Q).H.exp(iv.Q) = −|v|2 /2 − v.P − − − (1)

The O(v) term of this equation gives

i[v.Q, H] = −v.P

or equivalently,

i[H, Qj ] = Pj

Combining this with the equation

[H, Pj ] = 0

we may conclude using the commutation relations [Qi , Pj ] = iδij that 3

H = P 2 /2 + E =

1 2 P + E − − − (2) 2 j=1 j

where E is an operator of the form I ⊗ E1 in L2 (R3 ) ⊗ h, ie, for f (x) ∈ h, x ∈ R3 , we have (Ef )(x) = E1 f (x) By considering the O(v 2 ) term in (1), we get [v.Q, [v.Q, H]] = −v 2 or equivalently, This is verified by (2):

[[H, Qi ], Qj ] = −δij /2 [P 2 /2, Qi ] = −iPi , [[P 2 /2, Qi ], Qj ] = −i[Pi , Qj ] = −δij

We now consider the equation U (τ, g)U (a, v)U (τ, g)−1 = U (g(a − vτ ), gv) with τ = 0. We get W2 (g)U (a, v)W2 (g)−1 = U (ga, gv) In particular, we get

W2 (g)V1 (a)W2 (g)−1 = V1 (ga), W2 (g)V2 (v)W2 (g)−1 = V2 (gv)

Quantum Mechanics for Scientists and Engineers 125

128

These equations are the same as W (2(g)Pi W2 (g)−1 =

3 

gji Pj ,

3 

gji Qj

j=1

W2 (g)Qi W2 (g)−1 =

j=1

Here, g ∈ SO(3). Thus, W2 (g) has the effect of rotating the position and momentum operators. N-particle system: We assume that Hi is the Hilbert space for the ith particle and that the projective unitary representation U of the Galilean group G acting in H = ⊗N i=1 Hi has the form U (a, v, 0, g) = ⊗N i=1 Ui (a, v, 0, g) In other words, as regards translation, motion with uniform velocities and rotations, the actions of these on each particle in the system is the same. The above discussion for a single particle implies that each Ui (a, v, 0, g) acts in the same way on the corresponding particle. However, time evolution described by the operator U (0, v, τ, I) acts on the entire system and may not be factorizable into a tensor product of single particle operators. This is because, the3 generator of this group which is the energy/Hamiltonian H is the sum of the individual kinetic energies and an interaction potential energy and the latter depends on some complex combination of all the particle position operators. So for the present, we can let U (0, 0, τ, I) = exp(−iτ H) where H is a Hermitian operator acting in H. We also assume the existence of velocity operators Vi = (Vi1 , Vi2 , Vi3 ), i = 1, 2, ..., N acting in H satisfying the following properies: U (0, v, 0, I)Vij U (0, v, 0, I)−1 = Vij + vj , 1 ≤ j ≤ 3, i = 1, 2, ..., N, Since

Ui (0, v, 0, I) = exp(−iv.Q)

it follows that i[v.

 k

Qk , Vij ] = −vj

where Qk = (Qk1 , Qk2 , Qk3 ) are the position operators acting in Hk and likewise Pk = (Pk1 , Pk2 , Pk3 are the momentum operators acting in Hk . It should be noted that by the theory for one particle  discussed above and the separability of U (a, v, 0, g) we have that U (a, 0, 0, I) = exp(−ia.P ) = ⊗k exp(−iaPk ), P = k Pk and likewise for Q. Thus, we get  Qkl , Vij ] = −δjl i[ k

We also have from the one particle theory and separability of U (a, v, 0, g) that i[Qkl , Pij ] = −δki δlj So, if we postulate that

[Qkl , Vij ] = 0, k �= i

(this is true if we assume that Vij acts in Hi ), then we get

i[Qkl , Vij ] = −δki δij and hence we derive

[Qkl , Pij − Vij ] = 0

which implies that

Pij − Vij = Aij (Q)

where Aij (Q) is a function of Q = (Qij : 1 ≤ i ≤ N, 1 ≤ j ≤ 3) only. We define H0 =

N

N

k=1

k=1

3

1 2 1  2 Vk = Vki 2 2 i=1

Quantum Mechanics for Scientists and Engineers 126

129

From the composition theory of Galilean group representations developed above, we have that U (0, 0, τ, I)U (a, 0, 0, I) = U (a, 0, τ, I) = U (a, 0, 0, I)U (0, 0, τ, I) and hence

[H,



Pki ] = 0

k

Note that

U (a, 0, 0, I) = exp(−ia.



Pk ) = exp(−i



ai Pki )

k,i

This means that the total momentum of the system of N particles is conserved. Now, we consider 3 3   2 [H0 , Qki ] = [ Vkr /2, Qki ] = [ (Pkr − Akr (Q))2 /2, Qki ] = r=1

r=1

−i(Pki − Aki (Q)) = −iVki

We also assume (by definition of the velocity as the time derivative of the position), d U (0, 0, −τ, I).Qki U (0, 0, τ, I)|τ =0 = Vki dt This gives

[H, Qki ] = −iVki

and hence

[H − H0 , Qki ] = 0

and therefore,

H − H0 = V0 (Q) + E

where E is of the form I ⊗ E1 with E1 acting in h and V0 (Q) and arbitrary function of the positions Q = (Qki : 1 ≤ k ≤ N, i = 1, 2, 3). Thus, we finally get the general form of the total system Hamiltonian: H=

1 (Pki − Aki (Q))2 + V0 (Q) + E 2 k,i

(Ref: K.R.Parthasarathy, ”Mathematical Foundations of Quantum Mechanics”, Hindustan Book Agency) A.13. A remark on quantum stochastic calculus. Let ut ∈ H, Ut ∈ U(H), t ≥ 0 Let P be a spectral measure on [0, ∞) with values in P(H) and assume that P commutes with all the Ut s. We can write Ut = exp(iHt ) where Ht is a Hermitian operator in H. Suppose that the Ht s commute with each other and that Ht acts in Pt H where Pt = P [0, t] (For example we can choose a Hermitian operator H in H that commutes with P and then define Ht = Pt H = HPt ). More generally, we shall assume that for s < t, Ht − Hs acts in P [s, t]H. Then we have dUt = iUt dHt and since dHt acts in dPt H = P [t, t + dt]H while Ut acts in Pt H, it follows that for v, w ∈ H, we have with Γ(Ut ) denoting the second quantization of Ut (ie, Γ(Ut ) = W (0, Ut )), d < e(v)|Γ(Ut )|e(w) >=< e(v)|dΓ(Ut )|e(w) >=

and hence Γ(Ut ) satisfies the qsde

i < e(v)|Γ(Ut )|e(w) > .d < v|Ht |w > dΓ(Ut ) = Γ(Ut )dΛ(Ht )

where Λ(X) is the second quantization of X defined by < e(v)|exp(Λ(X))|e(w) >=< e(v)|e(exp(X)w) >= exp(< v|exp(X)|w >)

127 Quantum Mechanics for Scientists and Engineers

130

A.14. Scattering theory with time dependent interactions with the scattering centre. The free particle Hamiltonian is H0 and the Hamiltonian after the particle starts interacting with the scattering centre is H(t) = H0 + δ.V (t). We assume that V (t) = V0 is a constant operator for |t| > T . Let |φ1 > be a free particle state evolving according to H0 in the remote past (the ”in state”) Let |ψ1 > be the corresponding scattered state evolving according to H(t). Likewise, let |φ2 > be a free particle state evolving according to H0 in the future, ie, as t → ∞ (ie, the ”out state”) and |ψ2 > the corresponding scattered state evolving according to H(t). Define for t2 > t1 , U0 (t2 − t1 ) = exp(−i(t2 − t1 )H0 ), U (t2 , t1 ) = T {exp(−i where T {.} denotes the time ordering operator. We must have



t2

H(t)dt)}

t1

limt→∞ (U (t, 0)|ψ2 > −U0 (t)|φ2 >) = 0 and hence

|ψ2 >= limt→∞ U (t, 0)−1 U0 (t)|φ2 >

We write

Ω2 (t) = U (t, 0)−1 U0 (t), t ≥ 0,

Then on an appropriate domain of out states, we have the operator

Ω2 = limt→∞ Ω2 (t) Thus,

|ψ2 >= Ω2 |φ2 >

Likewise,

limt→−∞ (U (0, t)−1 |ψ1 > −U0 (−t)|φ1 >) = 0

or equivalently,

|ψ1 >= limt→−∞ U (0, t)U0 (−t)|φ1 > = limt→∞ U (0, −t)U0 (t)|φ1 > = Ω1 |φ1 >

where

Ω1 = limt→∞ Ω1 (t)|φ1 >

where The scattering matrix is given by

Ω1 (t) = U (0, −t)U0 (t) S = Ω∗2 Ω1 = limt→∞ Ω2 (t)∗ Ω1 (t) = limt→∞ U0 (−t)U (t, 0)U (0, −t)U0 (t)

= limt→∞ U0 (−t)U (t, −t)U0 (t)  t = limt→∞ exp(itH0 ).T {exp(−i H(s)ds)}.exp(−itH0 ) −t

Note: In discussing scattering theory with noise, we assume that {V (t)} is an operator valued random process and then compute the average value of the scattering matrix with respect to the probability distribution of {V (t)}. By the Dyson series expansion,  t

T {exp(−i

exp(−2itH0 ) + exp(−itH0 )(

∞ 

n=1

where

(−i)n



H(s)ds)} =

−t

V˜ (s1 )...V˜ (sn )ds1 ...dsn )exp(−itH0 )

−t= k,m

where

 k,m

|I[k, m]|2 = 1

This means that given that we measure pixel number m, the probability of getting the grey scale amplitude and phase level k is given by PI (k|m) = I[k, m]|2 More generally, the image can be in a mixed state ρI defined by  ρI = I[k, m, r, s]|k, m >< r, s| 1≤k,r≤p,1≤m,s≤N

where the condition implies that

T r(ρI ) = 1 

I[k, m, k, m] = 1

k,m

Then if the image is in the mixed state ρI , the probability of getting the grey scale level amplitude specified by the index k given that the mth pixel is measured is given by < k, m|ρI |k, m > PI (k|m) =  r < r, m|ρI |r, m >

Quantum Mechanics for Scientists and Engineers 129

132

I[k, m, k, m] = r I[r, m, r, m]

We shall now express ρI in the frequency domain of the grey scale amplitudes: The quantum Fourier transform of the grey scale state |k > is given by p  exp(−i2πkr/p)|r > |k˜ >= p−1/2 r=1

The inverse quantum Fourier transform is thus given by

p 

|k >= p−1/2 Then, ρI =

where



exp(i2πkr/p)|˜ r>

r=1

 I[k, m, r, s]|k, m >< r, s| = p−1 I[k, m, r, s]exp(i2π(kk � − rr� )/p)|k˜� >< r˜� | ⊗ |m >< s|  ˆ � , m, s, r� ]|k˜� >< r˜� | ⊗ |m >< s| = I[k ˆ � , m, s, r� ] = p−1 I[k

 k,r

We may express this as

I[k, m, r, s]exp(i2π(kk � − rr� )/p)



ˆ m, r, s]|k˜ >< r˜| ⊗ |m >< s| I[k,  ˜ m >< r˜, s| ˆ m, r, s] = |k, = I[k,

ρI =

The average image energy at pixel number m and frequency k˜ is thus given by

˜ m|ρI |k, ˜ m> ˆ m, k, m] < k, I[k,  = ˆ ˜, m|ρI |˜ r, m > r < s| m,s

where Tms is a p × p matrix for each m, s ∈ {1, 2, ..., N }. We assume that T is a unitary matrix, ie, T ∗ T = IpN This condition is equivalent to requiring that  ∗ Tms Tm s ⊗ |s >< m|m� >< s� | = IpN or equivalently,



m,s,s

or equivalently,

∗ Tms Tms ⊗ |s >< s� | = IpN



∗ Tms Tms = δss Ip

m

Applying the unitary operator T to ρI gives a transformed image field specified by the density matrix  ρ�I = T ρI T ∗ = ρI = I[k, m, r, s]T (|k >< r| ⊗ |m >< s|)T ∗ =

Now,



ˆ m, r, s]T (|k˜ >< r˜| ⊗ |m >< s|)T ∗ I[k, T (|k >< r| ⊗ |m >< s|) =

Quantum Mechanics for Scientists and Engineers 130

(

 jl

133

Tjl ⊗ |j >< l|)(|k >< r| ⊗ |m >< s|)(

=



 j  l

Tj∗ l ⊗ |l� >< j � |)

Tjl |k >< r|Tj∗ l ⊗ |j >< l|m >< s|l� > |m >< j � |  Tjm |k >< r|Tj∗ s ⊗ |j >< j � | = jj 

A.16. Quantization of the em fields inside a rectangular waveguide. (∇2⊥ + h2 )Ez = 0, (∇2⊥ + h2 )Hz = 0 jωμ γ ∇⊥ Ez − 2 ∇⊥ Hz × zˆ, h2 h γ jω� H⊥ = − 2 ∇⊥ Hz + 2 ∇⊥ Ez × zˆ h h We wish to select potentials Φ, A such that when the em fields satisfy the above, then E⊥ = −

E = −nablaΦ − jωA, μH = ∇ × A or equivalently,

Ez = γΦ − jωAz , E⊥ = −∇⊥ Φ − jωA⊥ , μHz zˆ = ∇⊥ × A⊥ ,

μH⊥ = ∇⊥ Az × zˆ − γ zˆ × A⊥

In addition, we wish that the potentials Φ, A satisfy the Lorentz gauge condition divA = −jω�μΦ, or equivalently,

∇⊥ .A⊥ − γAz = −jω�μΦ

It is easily seen that the most general potentials satisfying the above requirements are given by A⊥ = −(∇⊥ Φ + E⊥ )/jω, Az = (γΦ − Ez )/jω,

where Φ is any scalar field that satisfies the Helmholtz equation

(∇2⊥ + h2 )Φ = 0 In particular, we can take Φ = 0 and then

A = −E/jω

The general solution for the electromagnetic fields in the guide with the boundary conditions that Ez and the normal components of H vanish on the side boundaries is given by   Ez = C(n, m)exp(−γnm z)unm (x, y), Hz = D(n, m)exp(−γnm z)vnm (x, y) where

These functions are normalized:

γnm = (h2nm − ω 2 μ�)1/2 , √ unm (x, y) = (2/ ab)sin(nπx/a)sin(mπy/b), √ vnm (x, y) = (2/ ab)cos(nπx/a)cos(mπy/b) 

0

a



0

b

u2nm dxdy =



0

a



0

b

2 vnm dxdy = 1

Quantum Mechanics for Scientists and Engineers 131

134

and further they are orthogonal





We thus find that zˆ





C(n, m)unm (x, y)exp(−γnm z)−

We note that

unm un m dxdy = 0, (n, m) �= (n� , m� ) vnm vn m dxdy = 0, (n, m) �= (n� , m� ) E=

[(γnm /h2nm )C(n, m)∇⊥ unm (x, y)+(jωμ/hnm2 )D(n, m)∇⊥ vnm (x, y)׈ z ]exp(−γnm z)  =

=



(∇⊥ unm , ∇⊥ vn m × zˆ)dxdy



(∇⊥ unm × ∇⊥ vn m , zˆ)dxdy

(unm,x vn m ,y − unm,y vn m ,x )dxdy = 0

on integration by parts (we are left with only boundary terms which vanish). We note that  E⊥ = − [C(n, m)γnm /h2nm )∇⊥ unm (x, y) + D(n, m)(jωμ/hnm2 )∇⊥ vnm (x, y) × zˆ]exp(−γnm z)

and hence

= where



a

0





b

|E|2 dxdy

0

(|C(n, m)|2 (1 + |γnm |2 /h2nm ) + |D(n, m)|2 (μω)2 /h2nm )exp(−2αnm z) γnm (ω) = αnm (ω) + jβnm (ω)

Thus, � 



a 0



b

0



0

d

|E|2 dxdydz =

(λ(n, m)|C(n, m)|2 + μ(n, m)|D(n, m)|2 )

n,m

where

λ(n, m) = (1 + |γnm |2 /h2nm )(1 − exp(−2αnm d))/2αnm μ(n, m) = ((μω)2 /h2nm )(1 − exp(−2αnm d))/2αnm

Note that the energy in the magnetic field is given by  |∇ × A|2 dxdydx/2μ = and

(∇ × A, B) = ∇.(A × B) − (A, ∇ × B)

The first term on the rhs is a perfect divergence and by Gauss’ theorem, its volume integral over the guide volume is zero if assuming that the fields vanish on the surface. Further, ∇ × B = ∇ × (∇ × A) = ∇(divA) − nabla2 A = −∇2 A since divE = 0 implies divA = 0. Further, ∇2 A = −∇2 (E/jω) = ω 2 �μE/jω = −jω�μE and hence,  (2μ)−1 |∇ × A|2 dxdydz = (�/2)



|E|2 dxdydz

Quantum Mechanics for Scientists and Engineers 132

135

In other words, the energy in the magnetic field is same as the energy in the electric field. Thus, the total field energy is given by  � |E|2 dxdydz 

=

(λ(n, m)|C(n, m, ω)|2 + μ(n, m)|D(n, m, ω)|2 )

n,m

and this energy can be quantized using creation and annihilation operators in place of C(n, m, ω), D(n, mω) and their conjugates. A.17. Image processing for non-Gaussian noise models √ based on the Edgeworth expansion. The Edgeworth expansion: Let φ(x) = exp(−x2 /2)/ 2π, the standard normal density. Define the Hermite polynomials by Hn (x) = (−1)n exp(x2 /2)Dn exp(−x2 /2), D = d/dx  The Edgeworth expansion of a density f (x) for which all the moments R |x|k f (x)dx, k = 1, 2, ..., are finite is given by f (x) = φ(x)(1 +



c[n]Hn (x)) = φ(x) + (2π)−1/2

n≥1



c[n]Dn exp(−x2 /2)

n≥1

Generating function and orthogonality of the Hermite polynomials:   tn Hn (x)/n! = exp(x2 /2)( (−t)n Dn /n!)exp(−x2 /2) = exp(x2 /2)exp(−tD)exp(−x2 /2) n≥0

Thus,

n≥0

= exp(x2 /2)exp(−(x − t)2 /2) = exp(tx − t2 /2) 

n,m≥0

tn sm



R

Hn (x)Hm (x)φ(x)dx =



φ(x)exp((t + s)x − t2 /2 − s2 /2)dx

= exp(ts)

and hence



Hn (x)Hm (x)φ(x)dx = n!δ[n − m], n, m ≥ 0

√ Thus {Hn (x)/ n! : n ≥ 0} forms an orthnormal basis for the Hilbert space L2 (R, φ(x)dx). It therefore follows that the coefficients c[n], n ≥ 0 for the Edgeworth expansion of f (x) are given by  f (x)Hn (x)dx/n!, n ≥ 0 c[n] = R

Note that H0 (x) = 1. Now consider a multivariate Edgeworth pdf defined by ψ(x) = |A|ΠM i=1 f ((Ax)i ) where A is an M × M matrix and x ∈ RM . We have ψ(x) = (2π)−M/2 |A|exp(−xT AT Ax/2)ΠM i=1 (1 +



c[n]Hn ((Ax)i )

n≥1

We shall calculate is moment generating function: Let X ∈ RM have ψ as its pdf. Then  ˆ = Eexp(< t, X >) = exp(< t, x >)ψ(x)dx ψ(t) = (2π)−M/2



RM

RM

exp(< t, A−1 y >)exp(−y T y/2)ΠM i=1 (1 +

= (2π)−M/2 ΠM i=1





c[n]Hn (yi ))dy

n≥1

exp((A−T t)i ξ)exp(−ξ 2 /2)(1 +



n≥1

c[n]Hn (ξ))dξ

Quantum Mechanics for Scientists and Engineers 133

136

To calculate this integral, we first observe that (2π)−1/2



R

exp(tξ)exp(−ξ 2 /2)Hn (ξ)dξ

= (2π)−1/2 (−1)n = tn





exp(tξ).Dξn exp(−ξ 2 /2)dξ

φ(ξ)exp(tξ)dξ = tn exp(t2 /2)

where integration by parts has been used. Thus for the above multivariate case, we get  ˆ = exp(tT A−1 A−T t/2)ΠM (1 + c[n]((A−T t)i )n ) ψ(t) i=1 n≥1

= exp(tT (AT A)−1 t/2)ΠM i=1 (1 +



c[n](A−T t)i )n )

n≥1

The approximate maximum likelihood estimator for an Edgeworth distribution: Suppose y = Ax + w where w has a multivariate Edgeworth expansion and x also has a multivariate Edgeworth expansion. We wish to estimate x based on y by maximizing p(x|y), ie, the MAP estimate. We have p(x|y)py (y) = p(y|x)px (x) = pw (y − Ax)px (x) Discrete time non-linear filtering theory applied to real time image parameter estimation. The parameter vector v[n] at time n satisfies the stochastic difference equation v[n + 1] = f (n, v[n]) + �v [n + 1] where �v [n] is an iid sequence. Thus, v[n] is a discrete time Markov process with transition density p(v[n + 1]|v[n]) = p�v (v[n + 1] − f (n, v[n])) The image vector x[n] is partitioned into patches Pi x[n] with each patch given by Pi x[n] = vi [n] + �i [n], i = 1, 2, ..., L or equivalently,

P x[n] = v[n] + �[n]

where P is a non-singular square matrix and �[n] is an iid sequence independent of the sequence �v [n], n ≥ 1. Finally, the measurement model for the image vector is given by y[n] = x[n] + w[n] where w[n] is again an iid sequence independent of both the sequences �v and �. We assume that all the three random sequences �v , �, w have multivariate Edgeworth probability densities with possibly different linear combination coefficients. The aim is to dynamically estimate v[n] based on Yn = {y[k] : k ≤ n} We have p(v[n + 1]|Yn+1 ) = 

We now observe that where



p(y[n + 1]|v[n + 1])p(v[n + 1]|v[n])p(v[n]|Yn )dv[n] p(y[n + 1]|v[n + 1])p(v[n + 1]|v[n])p(v[n]|Yn )dv[n]dv[n + 1]

y[n] = P −1 (v[n] + �[n]) + w[n] = P −1 v[n] + d[n] d[n] = P −1 �[n] + w[n]

Quantum Mechanics for Scientists and Engineers 134

137

is again an iid vector valued noise. Thus, the MAP estimate of v[n + 1] given Yn+1 is given by  vˆ[n + 1] = argmaxv pd (y[n + 1] − Av � )p�v (v � − f (n, v))p(n, v|Yn )dv where A = P −1 . We assume that vˆ[n + 1] = f (n, vˆ[n]) + δv � = vˆ0 [n + 1] + δv � where vˆ0 [n + 1] = f (n, vˆ[n]), expand the above integral upto O((δv � )2 ) and then maximize this w.r.t δv � to get the extra correction. We have v0 [n + 1] + δv � )) = pd (y[n + 1] − A(ˆ 1 v0 [n + 1]) − p�d (y[n + 1] − Aˆ v0 [n + 1])T Aδv � + δv �T AT p��d (y[n + 1] − Aˆ v0 [n + 1])Aδv � pd (y[n + 1] − Aˆ 2 with neglect of O(|δv � |3 ) terms. Likewise, v0 [n + 1] + δv � − f (n, v)) = p�v (ˆ v0 [n + 1] − f (n, v)) + p��v (ˆ v0 [n + 1] − f (n, v))T δv � p�v (ˆ 1 v0 [n + 1] − f (n, v))δv � + δv �T p���v (ˆ 2 with neglect of O(|δv � |3 ). We thus obtain upto O(|δv � |2 ),  v0 [n + 1]) − p�d (y[n + 1] − Aˆ v0 [n + 1])T Aδv � vˆ[n + 1] = vˆ0 [n + 1] + argmaxδv (pd (y[n + 1] − Aˆ 1 + δv �T AT p��d (y[n+1]−Aˆ v0 [n+1])Aδv � )(p�v (ˆ v0 [n+1]−f (n, v))+p��v (ˆ v0 [n+1] 2 T � 1 v0 [n+1]−f (n, v))δv � )p(n, v|Y −f (n, v)) δv + δv �T p���v (ˆ 2  = vˆ0 [n+1]+argmaxδv [( 

+δv �T (

(pd (y[n+1]−Aˆ v0 [n+1])p��v (ˆ v0 [n+1]−f (n, v))−p�v (ˆ v0 [n+1]

−f (n, v))AT p�d (y[n+1]−Aˆ v0 [n+1]))p(n, v|Yn 1 (AT p�d (y[n+1]−Aˆ v0 [n+1])p��v (ˆ v0 [n+1]−f (n, v))T + (p���v (ˆ v0 [n+1] 2

This equation is of the form

v0 [n+1])A)p(n, v|Yn )dv)δv −f (n, v))+AT p��d (y[n+1]−Aˆ

1 vˆ[n + 1] = vˆ0 [n + 1] + argmaxδv [f (n, Yn+1 , vˆ0 [n + 1])T δv � + δv �T F (n, Yn+1 , vˆ0 [n + 1])δv � 2 = vˆ0 [n + 1] − F (n, Yn+1 , vˆ0 [n + 1])−1 (f (n, Yn+1 , vˆ0 [n + 1]))

and provides the desired recursion. Remark: The following approximation provides an alternate technique for improving the speed of the recursion:  ψ(n, y[n + 1], v)p(n, v|Yn )dv ≈

1 v [n])Cov(v[n]|Yn )) ψ(n, y[n + 1], vˆ[n]) + T r(ψv�� (n, y[n + 1]ˆ 2 To apply this formula, we note that the vector and matrices f (n, Yn+1 , vˆ0 [n + 1]), F (n, Yn+1 , vˆ0 [n + 1]) can be expressed as integrals of the form f (n, Yn+1 , vˆ0 [n + 1]) = intψ1 (n, y[n + 1], vˆ0 [n + 1], v)p(n, v|Yn )dv  F (n, Yn+1 , vˆ0 [n + 1]) = ψ2 (n, y[n + 1], vˆ0 [n + 1], v)p(n, v|Yn )

A.18.Cartan’s classification of the simple Lie algebras and the Weyl character formula for the irreducible representations of Compact semisimple Lie groups. A scheme S is a finite set linearly independent vectors (elements) α1 , ..., αn in a real vector space with an inner product (., .) such that a(α, β) = 2(α, β)/(α, α)

Quantum Mechanics for Scientists and Engineers 135

138

is a non-positive integer for all α �= β, α, β ∈ S. The Cauchy Schwarz inequality then implies that 0 ≤ a(α, β)a(β, α) ≤ 3, α, β ∈ S, α �= β ie, the product a(α, β)a(β, α) assumes only the values 0, 1, 2, 3. It is known from the general theory of semisimple Lie algebras, that a set of simple positive roots of a semisimple Lie algebra forms a scheme. The numbers a(α, β) are called the Cartan integers. Obviously a(α, α) = 2. To pictorially display a scheme S having n elments, we arrange these elements as vertices with the weight of each vertex α marked by a number proportional to (α, α) = |α|2 . Theorem 1: A connected scheme with n elements cannot have more than n − 1 links. For suppose that the elments of the scheme are αk , k = 1, 2, ..., n. Then consider n n   0 for all k �= m forms a dense subset of H. Hence, there exists a vector u such that < u, uk >, k = 1, 2, ..., p are all distinct. Hence the Vand-der-Monde matrix ((< u, uk >n ))1≤k,m≤p is non-singular implying that c(k) = 0, k = 1, 2, ..., p. Now, since  √ e(u) = 1 ⊕ tn u⊗n / n! �

n≥1

we get

√ dn e(tu)|t=0 = n!u⊗n dtn and since any symmetric tensor can be expressed as a linear combination of tensors of the form u⊗n , it follows that the exponential vectors e(u), u ∈ H span a dense subspace of Γs (H). An adapted process Xt , t ≥ 0 is a family of ˜ t |e(ut] )) ⊗ |e(u(t ) > for all t ≥ 0 where we have used the isomorphism that operators in Γs (H) such that Xt |e(u) >= (X identifies |e(u ⊕ v) > with |e(u) > ⊗|e(v) >. Here ut] = u ⊗ χ[0,t] and u(t = u ⊗ χ (t,∞) . Ideally speaking, if T denotes the isomorphism that identifies the vector e( i ui ) in Γs ( i Hi ) with ⊗i e(ui ) in i Γs (Hi ), then we should write the definition of an adapted process as ˜ t |e(ut] ) >) ⊗ e(u(t ) T (Xt |e(u) >) = (X Let P : 0 = t0 < t1 < ... < tn = T be a partition of [0, T ]. Its size is defined as |P | = max0≤k≤n−1 (tk+1 − tk ) and we define the partial sum I(X, A, P ) =

n−1  k=0

Xtk (Atk+1 (m) − Atk (m))

Note that X(t) is adapted so it acts in the Fock space Γs (Ht] ) while Atk+1 − Atk acts in the Fock space Γs (H(tk ,tk+1 ] since  (Atk+1 (m) − Atk (m))|e(u) >= (

tk+1

¯ dt)|e(u) >

tk

Note that time unfolds in quantum stochastic calculus as a continuous tensor product of Hilbert spaces. This can be visualized also in the classical probabilisitc setting by noting that if Ft , t ≥ 0 is a filtration on a probability space (Ω, F, P ) generated by a stochastic process X(t), t ≥ 0, then for any t1 < t2 < t3 , if F(t1 ,t2 ] denotes the σ field σ(X(t) : t1 < t ≤ t2 ), we can write L2 (F(t1 ,t3 ] ) = L2 (F(t1 ,t2 ] ) ⊗ L2 (F(t2 ,t3 ] ) in the sense that any measurable functional of X(t), t1 < t ≤ t3 can be expressed as a sum (poissbily infinite) of products of functions of {X(t) : t1 < t ≤ t2 } and of {X(t) : t2 < t ≤ t3 }. More generally, we can write 2 L2 (F(0,∞) = ⊗∞ i=1 L (F(ti ,ti+1 ] )

where We now have

0 = t0 < t1 < ...., tn → ∞ I(X, A, P )|e(u) >=

n−1  k=0

X(tk )|e(u) >> ((tk , tk+1 ])

Quantum Mechanics for Scientists and Engineers 138

141

where > ((s, t]) =



t

s

< m(t� ), u(t� ) > dt� , s ≤ t

Note that > can be extended to a complex measure on (R+ , B(R+ )). If Q is a partition finer that P , then it is clear that for each k = 0, 1, ..., n − 1, there exist integers a(k) < b(k) such that b(k)

(tk , tk+1 ] =



(sl , sl+1 ]

l=a(k)

and the points sl , l = 0, 1, ...m − 1 all form the partition Q. Thus, we have X(tk ) > ((tk , tk+1 ]) −

=

b(k) 

l=a(k)

So

b(k) 

X(sl ) > ((sl , sl+1 ])

l=a(k)

(X(tk ) − X(sl )) > ((sl , sl+1 ])

� (I(X, A, P ) − I(X, A, Q))|e(u) >�≤

max|t−s|≤|P |,s,t∈[0,T ] � (X(t) − X(s))|e(u) >� | > ([0, T ])|

Assume that X(t) is strongly uniformly continuous on [0, T ]. Then

limδ→0 max|t−s|≤δ,s,t∈[0,T ] � (X(t) − X(s))|e(u) >�= 0 and hence it follow that if Pn , n = 1, 2, ... is an increasing sequence of partitions, ie, Pn+1 > Pn ∀n and |Pn | → 0 as n → ∞, then |(I(X, A, Pn+m ) − I(X, A, Pn ))|e(u) >�→ 0, n → ∞, m = 1, 2, ...

which implies that I(X, A, Pn )|e(u) >, n = 1, 2, ... is a Cauchy sequence in the Boson Fock space Γs (H ⊗ L2 (R+ )). and hence converges to an element of this space. Further, the limit is independent of the sequence of partitions Pn for if Qn , n = 1, 2, ... is another increasing sequence of partitions such that |Qn | → 0, then � (I(X, A, Pn ) − I(X, A, Qn ))|e(u) >�≤� (I(X, A, Pn ) − I(X, A, Pn ∪ Qn ))|e(u) >� + � (I(X, A, Pn ∪ Qn ), I(X, A, Qn )|e(u) >�

and by the above logic, both of the terms on the rhs converge to zero, proving that the lhs also converges to zero and hence the strong limits of I(X, A, Pn ) and of I(X, A, Qn ) are the same. A.20. Hartree-Fock equations: ψa (x), x ∈ R3 are Fermionic operator fields and they satisfy the standard anticommutation relations {ψa (x), ψb∗ (x� )} = δab δ 3 (x − x� ), {ψa (x), ψb (x� )} = 0, {ψa (x)∗ , ψb (x)∗ } = 0

Let

T (x) = −∇2x /2m

and let V (x, x� ) = V (x� , x) be a scalar potential field. The Hartree Fock seconed quantized Hamiltonian is defined by H = H0 + H1 , H0 =



ψa (x)∗ T (x)ψa (x)d3 x

with summation over the repeated index a being implied,  H1 = V (x, x� )ψa (x)∗ ψa (x)ψb (x� )∗ ψb (x� )d3 xd3 x� Note that

H0∗ = H0

Quantum Mechanics for Scientists and Engineers 139

142

follows by integration by parts and

H1∗ = H1

so that

H∗ = H

and hence H is a valid Hamiltonian. The Fermionic fields at time t are given by the rules of Heisenberg’s matrix mechanics: ψa (t, x) = exp(itH)ψa (x)exp(−itH), and its adjoint

ψa (t, x)∗ = exp(itH)ψa (x)∗ exp(−itH)

Now,

∂ψa (t, x) = iexp(itH)[H, ψa (x)]exp(−itH) ∂t [H, ψa (x)] = [H0 , ψa (x)] + [H1 , ψa (x)]

We have [H0 , ψa (y)] =

=



[H1 , ψa (y)] = Now,

=−





[ψb (x)∗ T (x)ψb (x), ψa (y)]d3 x

{ψb (x)∗ , ψa (y)}T (x)ψb (x)d3 x

δab δ 3 (x − y)T (x)ψb (x)d3 x = −T (x)ψa (x)



V (x, x� )[ψb (x)∗ ψb (x)ψc (x� )∗ ψc (x� ), ψa (y)]d3 xd3 x� ψa (y)ψb (x)∗ ψb (x)ψc (x� )∗ ψc (x� ) =

(δab δ 3 (y − x) − ψb (x)∗ ψa (y))ψb (x)ψc (x� )∗ ψc (x� ) =

δ (y − x)ψa (y)ψc (x� )∗ ψc (x� ) + ψb (x)∗ ψb (x)ψa (y)ψc (x� )∗ ψc (x� ) 3

So

[ψb (x)∗ ψb (x)ψc (x� )∗ ψc (x� ), ψa (y)] = −ψb (x)∗ ψb (x){ψc (x� )∗ , ψa (y)}ψc (x� ) − δ 3 (y − x)ψa (y)ψc (x� )∗ ψc (x� ) = −δac δ 3 (y − x� )ψb (x)∗ ψb (x)ψc (x� ) − δ 3 (y − x)ψa (y)ψc (x� )∗ ψc (x� ) = −δ 3 (y − x� )ψb (x)∗ ψb (x)ψa (y) − δ 3 (y − x)ψa (y)ψc (x� )∗ ψc (x� )

= −δ 3 (y − x� )ψb (x)∗ ψb (x)ψa (y) − δ 3 (y − x)(δac δ 3 (y − x� ) − ψc (x� )∗ ψa (y))ψc (x� )

Thus, we get

= −δ 3 (y − x� )ψb (x)∗ ψb (x)ψa (y) − δ 3 (y − x)ψc (x� )∗ ψc (x� )ψa (y) − δ 3 (y − x)δ 3 (y − x� )ψa (y)

= −iT (t, y)ψa (t, y) − 2i We write this equation as i



∂ψa (t, y) = ∂t V (y, x)ψb (t, x)∗ ψb (t, x)d3 x − iV (y, y)ψa (t, y)

∂ψa (t, y) ˜ y)ψa (t, y) = H(t, ∂t

˜ y) is the effective Hamiltonian operator defined by where H(t,  ˜ y) = T (t, y) + 2 V (y, x)ψb (t, x)∗ ψb (t, x)d3 x + V (y, y) H(t, and This is the Hartree-Fock equation.

T (t, y) = exp(itH)T (y)exp(−itH)

Quantum Mechanics for Scientists and Engineers 140

143

A.21. Quantum scattering theory. Explicit determination of the scattering operator. H0 = P 2 /2m, H = H0 + V . exp(−itH0 )ψ(Q) = ψt (Q) say. Then,

idψt (Q)/dt = H0 ψt (Q) = −∇2Q ψt (Q)/2m ψt (Q) = exp(it∇2Q /2m)ψ(Q)  / exp(it∇Q 2m) = (2πσ 2 )−3/2 exp(−|x|2 /2σ 2 )exp((x, ∇Q))d3 x

Then,

exp(σ 2 ∇2Q /2) = exp(it∇2Q /2m)

Hence,

σ 2 = it/m, σ =

So ψt (Q) = (2πit/m)−3/2



= (2πit/m)−3/2 = (2πit/m)−3/2 Define the Kernel function



exp(−m|x|2 /2it)exp((x, ∇Q ))ψ(Q)dx exp(−m|x|2 /2it)ψ(Q + x)d3 x exp(−m|Q − x|2 /2it)ψ(x)d3 x

Kt (Q) = (2πit/m)−3/2 exp(−m|Q|2 /2it)

Let Then,



 it/m

W (t) = exp(itH)exp(−tH0 ) W � (t) = iexp(itH).V (Q).exp(−itH0 )

(V is assumed to be a function of Q only. Thus, W � (t) = iW (t)exp(itH0 )V (Q)exp(−itH0 ) Define Then, and

Z(t) = exp(itH0 ).V (Q).exp(−itH0 ) Z � (t) = iexp(itH0 )[H0 , V (Q)]exp(−itH0 ) [H0 , V (Q)] = [P 2 , V (Q)]/2m = (2m)−1 ([Pa , V (Q)]Pa + Pa [Pa , V (Q)]) = −i(2m)−1 ((∇Q V (Q), P ) + (P, ∇Q V (Q))) (−i/m)(V � (Q), P ) − (1/2m)∇2Q V (Q)

Another way to evaluate this is to note that

Z(t) = V (exp(itH0 )Q.exp(−itH0 )) Now,

exp(itH0 )Q.exp(−itH0 ) = exp(itad(H0 ))(Q) = Q + it[H0 , Q] + (it)2 [H0 , [H0 , Q]] + ... = Q + tP/m

Thus, Thus,

Z(t) = V (Q + tP/m) W � (t) = iW (t)V (Q + P t/m)

Quantum Mechanics for Scientists and Engineers 141

144

Now suppose |f >, |u >∈ Hac (H0 ). Then, by definition of the absolutely continuous spectrum of an operator, the Radon-Nikodym derivatives d < u|E0 (λ)|u > /dλ and d < f |E0 (λ)|f > /dλ exist and are finite. We have for V = |u >< u| with < u|u >= 1, W (t)|f >= exp(itH)exp(−itH0 )|f >= (i



t

exp(itH)V.exp(−itH0 )dt)|f >

0

and for Ω+ |f >= limt→∞ W (t)|f > to exist, it is sufficient that  ∞ X= � V.exp(−itH0 )|f >� dt < ∞ 0

Now,

V.exp(−itH0 )|f >= |u >< u|exp(−itH0 )|f >

so

� V.exp(−itH0 )|f >�= | < u|exp(−itH0 )|f > |  = | (d < u|E0 (λ)|f > /dλ)exp(−iλt)dλ| R

This is the magnitude of the Fourier transform of the function λ → d < u|E(λ)|f > /dλ. We note that | < u|dE(λ)|f > | ≤< u|dE) (λ)|u >1/2 < f |dE0 (λ)|f >1/2 Hence,

|d < u|E0 (λ)|f > /dλ| ≤ (d � E0 (λ)|u >�2 /dλ)1/2 (d � E0 (λ)|f >�2 /dλ)1/2

A necessary condition for the magnitude of the Fourier transform of a function to be integrable is that the Fourier transform be finite. Thus, a necessary condition for X < ∞ is satisfied since the Radon-Nikodym derivatives d � E(λ)|u >�2 /dλ and d � E0 (λ)|f >�2 /dλ are finite because both |f > and |u > belong to Hac (H0 ). Suppose H0 = P (in one dimension) andV = V (Q). H = H0 + V = P + V (Q). Then, V.exp(itP )f (x) = V (x)f (x + t). So, Ω+ |f > will exist if the function t → ( R V (x)2 |f (x + t)|2 dx)1/2 is integrable on R+ . Consider now two Hamiltonians H0 , H = H0 + V and let φ : R → R. Consider now the Hamiltonians φ(H0 ), φ(H). Let Wφ (t) = exp(itφ(H))exp(−itφ(H0 )) Then, Wφ (t) = I + i



t

0

So Wφ (∞)|f > will exist if

We note that



∞ 0

exp(isφ(H))(φ(H) − φ(H0 ))exp(−isφ(H0 ))ds

� (φ(H) − φ(H0 ))exp(−isφ(H0 ))|f >� ds < ∞

< u|(φ(H) − φ(H0 ))exp(−isφ(H0 ))|f >= −



R



R

exp(−isφ(λ))d < u|φ(H)E0 (λ)|f >

exp(−isφ(λ))φ(λ)d < u|E0 (λ)|f >

In particular, if φ is an invertible function we can write



< u|(φ(H) − φ(H0 ))exp(−isφ(H0 ))|f >= (exp(−isλ)d < u|φ(H)E0 (φ−1 (λ))|f > /dλ)dλ

−



exp(−isλ)λ(d < u|E0 (φ−1 (λ))|f > /dλ)dλ

provided we assume that the concerned Radon-Nikodym derivatives exist. These will exist provided that all |u > , φ(H)|u > and |f > belong to the absolutely continuous parts of the spectral measure E0 oφ−1 ie they belong to Hac (φ(H0 )).

Quantum Mechanics for Scientists and Engineers 142

145

A.22.Hartree Fock approximation to the two electron problem of the Helium atom. The Hamiltonian is H = H1 + H2 + V12 where

H1 = −∇21 /2m − 2e2 /r1 , H2 = −∇22 /2m − 2e2 /r2 , V12 = e2 /r12

Let us try the wave function (antisymmetric because the two electrons form a Fermionic pair) √ ψ = (ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 )/ 2 with the constraints

< ψ1 |ψ1 >=< ψ2 |ψ2 >= 1, < ψ1 |ψ2 >= 0

As in all eigenvalue problems we extremize

S =< ψ|H|ψ > −2E1 (< ψ1 |ψ1 > −1) − 2E2 (< ψ2 |ψ2 >) − 2λ1 Re(< ψ1 |ψ2 >) − 2λ2 Im(< ψ2 |ψ1 >)) We first observe that taking into account the constraints, S =< ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 |(H1 + H2 + V12 )|ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 > −2E1 (< ψ1 |ψ1 > −1) − 2E2 (< ψ2 |ψ2 >) − 2λ1 Re(< ψ1 |ψ2 >) − 2λ2 Im(< ψ2 |ψ1 >))

=< ψ1 |H1 |ψ1 > + < ψ2 |H1 |ψ2 > + < ψ1 |H2 |ψ1 > + < ψ2 |H2 |ψ2 > + < ψ1 ⊗ ψ2 |V12 |ψ1 ⊗ ψ2 > < ψ2 ⊗ ψ1 |V12 |ψ2 ⊗ ψ1 > − < ψ1 ⊗ ψ2 |V12 |ψ2 ⊗ ψ1 >

− < ψ2 ⊗ ψ1 |V12 |ψ1 ⊗ ψ2 > −2E1 (< ψ1 |ψ1 > −1) − 2E2 (< ψ2 |ψ2 >) − 2λ1 Re(< ψ1 |ψ2 >) − 2λ2 Im(< ψ2 |ψ1 >))

Now,

δS/δ ψ¯1 = 0

gives

2H1 ψ1 (r1 ) + 2 < I ⊗ ψ2 |V12 |ψ1 ⊗ ψ2 > −2 < I ⊗ ψ2 |V12 |ψ2 ⊗ ψ1 >

−2E1 ψ1 − λ1 |ψ1 > −iλ2 |ψ2 >= 0

and likewise another equation for δS/δ ψ¯2 = 0. Expanding the above, we get  2(−∇21 /2m − 2e2 /r1 − E1 − λ1 )ψ1 (r1 ) + 2 ψ¯2 (r2 )(e2 /r12 )(ψ1 (r1 )ψ2 (r2 ) − ψ2 (r1 )ψ1 (r2 ))d3 r2 −iλ2 ψ2 (r1 ) = 0

We note that the second term can be expressed as  ψ¯2 (r2 )(e2 /r12 )(ψ1 (r1 )ψ2 (r2 ) − ψ2 (r1 )ψ1 (r2 ))d3 r2 =[





(|ψ2 (r2 )|2 (e2 /r12 )d3 r2 ]ψ1 (r1 ) − (

(e2 /r12 )ψ¯2 (r2 )ψ1 (r2 )d3 r2 )ψ2 (r1 )

The first term in this expression represents the potential energy produced by a smeared second electron charge on the first charge, the charge density of this smeared distribution being given by e|ψ2 (r2 )|2 . The second term in the above expression represents the effect of the spin interaction between the two electrons, the interaction caused by the fact that both the electrons cannot occupy the same state. Remark: The constraint < ψ1 |ψ2 >= 0 need not be introduced. Neither do the constraints < ψ1 |ψ1 >=< ψ2 |ψ2 >= 1 need to be imposed. We only need to introduce the constraint that the overall wave function be normalized, ie, 1 =< ψ|ψ > which is equivalent to

2 =< ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 |ψ1 ⊗ ψ2 − ψ2 ⊗ ψ1 >= 2 < ψ1 |ψ1 >< ψ2 |ψ2 > −2| < ψ1 |ψ2 > |2

Quantum Mechanics for Scientists and Engineers 143

146

or equivalently,

1 =< ψ1 |ψ1 >< ψ2 |ψ2 > −| < ψ1 |ψ2 > |2

This results in the following version of the Hartree Fock equation  2(−∇21 /2m − 2e2 /r1 )ψ1 (r1 ) + ψ¯2 (r2 )(e2 /r12 )(ψ1 (r1 )ψ2 (r2 ) − ψ2 (r1 )ψ1 (r2 ))d3 r2 +



ψ¯2 (r1 )(e2 /r12 )(ψ2 (r1 )ψ1 (r2 ) − ψ1 (r1 )ψ2 (r2 ))d3 r2

−E(< ψ2 |ψ2 > ψ1 (r1 )− < ψ2 |ψ1 > ψ2 (r1 )) = 0

with another equation of the same type, ie, with the same value of E for ψ2 . We can generalize this to a system of N interacting Fermions. The Hamiltonian of such a system is given by H=

N 



Hi +

i=1

Vij

1≤i= N !det((< ψa |ψb >)) = N !

 σ

sgn(σ)ΠN k=1 < ψk |ψσk >= Λ[ψ]

say. We have S1 [ψ] =< ψ|

N 

k=1

=



Hk |ψ >=

σ,τ,k

=



σ,τ,k



σ,τ ∈SN

sgn(στ ) < ψσ1 ⊗ ...ψσN |Hk |ψτ 1 ⊗ ...ψτ N >

sgn(στ )(ΠN j=1,j�=k < ψσj , ψτ j >) < ψσk |Hk |ψτ k >

sgn(στ )(ΠN j=1,j�=k < ψj , ψτ σ −1 j >) < ψk , |Hσ −1 k |ψτ σ −1 k >

=



ρ,σ,k

and

sgn(ρ)(ΠN j=1,j�=k < ψj , ψρj >) < ψk |Hσk |ψρk > S2 [ψ] =< ψ|

 i) < ψσi ⊗ ψσj |Vij |ψρi ⊗ ψρj >



σ,ρ,i,j:σ −1 i=

sgn(ρ)(ΠN k=1,k�=i,j < ψk |ψρk >) < ψi ⊗ ψj |Vσ −1 i,σ −1 j |ψρi ⊗ ψρj > S[ψ] = S1 [ψ] + S2 [ψ]

The variational principle

δψ¯k (S[ψ] − E(Λ[ψ] − 1)) = 0

thus gives

 ρ,σ

+



ρ,σ,j

sgn(ρ)(ΠN j=1�=k < ψj |ψρj >)Hσk |ψρk >

sgn(ρ)|ψρk > (ΠN l=1,l�=k,j < ψl |ψρl >) < ψj |Hσj |ψρj >

N

i=1

Hi .

Quantum Mechanics for Scientists and Engineers 144

+



σ,ρ,j:σ −1 k

sgn(ρ)(ΠN m=1,m�=i,k < ψm |ψρm >)|ψρk >< ψi ⊗ I|Vσ −1 i,σ −1 k |ψρi ⊗ I >

sgn(ρ)(ΠN m=1,m�=i,j,k < ψm |ψρm >)|ψρk >< ψi ⊗ ψj |Vσ −1 i,σ −1 j |ψρi ⊗ ψρj >

= N !E

 σ

sgn(σ)|ψσk > (ΠN j=1,j�=k < ψj |ψσj >), k = 1, 2, ..., N

A.23.Plasmonic waveguides. A rectangular waveguide is filled with a plasma. The charge per particle of the plasma is q and the particle distribution function is f (t, r, v). This distribution function is assumed to vanish on the boundaries of the guide and hence its Fourier transform w.r.t time can be expanded as  f f (ω, r, v) = fnm (ω, v)unm (x, y)exp(−γnm (ω)z) n,m≥1

where

√ unm (x, y) = (2/ ab)sin(nπx/a)sin(mπy/b), 0 ≤ x ≤ a, 0 ≤ y ≤ b

For a fixed n, m, we write f (ω, v) for fnm and γ for γnm . The Maxwell equations curlE = −jωμH, curlH = J + jω�E where J(ω, x, y) =



qvδf (ω, x, y, v)d3 v

where the f (ω, x, y, v)exp(−γz) is a component of the particle density corresponding to propagation constant γ. In component form, we have Ez,y (x, y) + γEy (x, y) = −jωμHx (x, y), γEx + Ez,x = jωμHy , Ey,x − Ex,y = −jωμHz ,

Hz,y + γHy = Jx (x, y) + jω�Ex , γHx + Hz,x = −Jy (x, y) − jω�Ey ,

Hy,x − Hx,y = Jz (x, y) + jω�Ez (x, y)

where the frequency argument ω has been omitted. Here,    Jx (x, y) = qvx δf (x, y, v)d3 v, Jy (x, y) = qvy δf (x, y, v)d3 v, Jz (x, y) = qvz δf (x, y, v)d3 v − − − (a) These equations can be solved for Ex , Ey , Hx , Hy in terms of J(x, y) and the partial derivatives of Ez (x, y), Hz (x, y). Having done so, we substitute these into the Boltzmann kinetic transport equation with the unperturbed Maxwell distribution function f0 (v) being taken in place of f where multiplication with the em fields is concerned. Thus, we get the approximate equation jωδf (x, y, v) + vx δf,x (x, y, v) + vy δf,y (x, y, v) − γvz δf (x, y, v) + q(E + μv × H, ∇v )f0 (v) = −δf (x, y, v))/τ (v) − − − (b) where the relaxation time approximation for the collision term has been used. Here, δf (x, y, v) = f (x, y, v) − f0 (v). To proceed further, we first solve the above equations for Ex , Ey , Hx , Hy :   γ −jωμ (Ex , Hy )T = (−Ez,x , Hz,y − Jx (x, y))T , jω� −γ   γ jωμ (Ey , Hx )T = (−Ez,y , −Hz,x − Jy (x, y))T , jω� γ

Quantum Mechanics for Scientists and Engineers 145

148

Solving these gives

Ex = −(γ/h2 )Ez,x − (jωμ/h2 )(Hz,y − Jx ) − − − (1) Ey = −(γ/h2 )Ez,y + (jωμ/h2 )(Hz,x + Jy ) − − − (2) Hx = (jω�/h2 )Ez,y − (γ/h2 )(Hz,x + Jy ) − − − (3)

Hy = −(jω�/h2 )Ez,x − (γ/h2 )(Hz,y − Jx ) − − − (4)

where

h2 = γ 2 + ω 2 �μ

Substituting these into the third components of the Maxwell curl equations gives (Hz,xx + Jy,x ) + (Hz,yy − Jx,y ) + h2 Hz = 0 (Ez,xx + Ez,yy ) − (γ/jω�)(Jx,x + Jy,y ) + h2 Ez = 0

or equivalently,

(∇2⊥ + h2 )Ez = (γ/jω�)(Jx,x + Jy,y ), (∇2⊥ + h2 )Hz = Jx,y − Jy,x

In accordance with the boundary conditions on the tangential components of E and the normal components of H, we have the expansions   Enm unm (x, y)exp(−γnm z), Hz (x, y, z) = Hnm wnm (x, y)exp(−γnm z) Ez (x, y, z) = n,m

n,m

where

√ wnm (x, y) = (2/ ab)cos(nπx/a)cos(mπy/b)

Formally, we can write

Ez = (∇2⊥ + h2 )−1 (γ/jω�)(∇⊥ .J⊥ ) − − − (5) Hz = (∇2⊥ + h2 )−1 (Jx,y − Jy,x ) − − − (6)

and express Ex , Ey , Hx , Hy in terms of J using eqns. (1)-(6). Finally, these expressions are substituted into the linearized Boltzmann eqn. (b) by making use of (a). The result is a linear integro-partial differential equation for the Boltzmann particle distribution function f (ω, x, y, v) with γ as a parameter. The solutions of this equation will generally lead to discrete values of the propagation constant γ. A.24. Winding number of planar Brownian motion. Let Zt = Xt + iYt be a complex Brownian motion, ie, X, Y are independent real standard Brownian motion processes. We write  ρt = Xt2 + Yt2 = |Zt |, θt = T an−1 (Yt /Xt ) = Arg(Zt ) Then,

log(ρt ) = RelogZt , θt = Imlog(Zt )

Away from the origin, z → logz is an analytic function of a complex variable and hence its Laplacian vanishes. Thus, from Ito’s formula, dlog(Zt ) = dZt /Zt and hence log(Zt ) is a Martingale. Writing log(z) = u(x, y) + iv(x, y), z = x + iy it follows that u(Xt , Yt ) and v(Xt , Yt ) are both real Martingales. We have u(Xt , Yt ) = log(ρt ), v(Xt , Yt ) = θt Further, the quadratic variation of the Martingales u(Xt , Yt ) and v(Xt , Yt ) are the same processes. They are equal to 

0

t

|∇u(Xs , Ys )|2 ds =



0

t

|∇v(Xs , Ys )|2 ds =



0

t

ds/|Zs |2

Quantum Mechanics for Scientists and Engineers 146

149

where we make use of the Cauchy-Riemann equations, u,x = v,y , u,y = −v,x and of course

u,x + iu,y = u,x − iv,x , v,x + iv,y = −u,y + iv,y

while on the other hand, writing f (z) = logz = u + iv, we have

f � (z) = u,x + iv,x , if � (z) = u,y + iv,y Thus, Thus, writing

|f � (z)|2 = |∇u|2 = |∇v|2 = 1/|z|2 = 1/(x2 + y 2 ) = 1/ρ2 C(t) =



t



ds/|Zs |2 =

0

t

0

ds/ρ2s

it follows that there exists a planar Brownian motion β(t) + iγ(t) such that log(ρt ) + iθt = logZt = β(C(t)) + iγ(C(t)) so that

β(C(t)) = log(ρt ), γ(C(t)) = θt

In fact, if τ () is the inverse function of C, then we have that u(Xτ (t) , Yτ (t) ) and v(Xτ (t) , Yτ (t) ) are independent Brownian motion processes which we denote by β(t) and γ(t) respectively. We have f (Zτ (t) ) = β(t) + iγ(t) and

df (Zτ (t) ) = f � (Zτ (t) )dZτ (t)

(since (dZ) = 0). Thus, 2

(df (Zτ (t) ))2 = f � (Zτ (t) )2 (dZτ (t) )2 = 0

which proves independence of the Brownian motions β and γ. Note that we can write β(t) = log(ρτ (t) ), γ(t) = θτ (t) Remark: The real parts β(t) and γ(t) of f (Zτ (t) ) are continuous martingales with quadratic variation  and imaginary  dt 0 matrix equal to = dt.I2 which proves by Levy’s theorem for vector valued Martingales that these two 0 dt processes are independent standard Brownian motion processes. Now, the angle turned by the planar Brownian motion process Zt around the origin respectively inside and outside a circle of radius r are  t  t θr− (t) = Im χ|Zs |r dlog(Zs ) 0

0

and these can be expressed as θr− (t) = = Im θr+ (t) =





t

χρs r dZs /Zs

C(t)

χβ(s)log(r) dγ(s)

Quantum Mechanics for Scientists and Engineers 147

150

Now define

Ta = min(t ≥ 0 : β(t) = a)

and

σr = min(t ≥ 0 : ρt = r)

Then, Thus,

σr = min(t : log(ρt ) = log(r)) = min(t : β(C(t)) = log(r)) C(σr ) = min(C(t) : β(C(t)) = log(r)) = min(t : beta(t) = log(r)) = Tlog(r)

Now for a set E in C, consider



I(t) =

t

σ√ t

We write

Zt = a + Zˆt , a �= 0

We assume that Z0 = a so that Zˆ0 = 0. We have I(t) =



t

χZˆs ∈E−1 dZˆs /(a + Zˆs )

σ√t

Now, the process Zˆts , s ≥ 0 has the same law as ˜ = I(t)



√

tZˆs , s ≥ 0 and hence I(t) has the same distribution as

1

� t−1 σ√

χZ(s)∈E dZs /Zs

√ √ χZˆs ∈(E−1)/√t tdZˆs /(a + tZˆs )

t

√ where σr� is the same as σr but with the process a + tZˆs/t , s ≥ 0 used in place of Zˆs , s ≥ 0 (Note that these two processes have the same law). Now √ √ � t−1 σ√ = min(s/t : |a + tZˆs/t | = t) t √ √ = min(s : |a + tZˆs | = t) √ = min(s : |a/ t + Zˆs | = 1)

� converges as t → ∞ to the random variable min(s : |Zˆs | = 1) = σ say. It follows on This gives the result that t−1 σ√ t taking lim t → ∞ that the limit in law of I(t) as t → ∞ is given by the random variable



1

σ

where

χZˆs ∈E0 dZˆs /Zˆs

√ E0 = limt→∞ (E − 1)/ t

which is a √ finite random variable since dZˆs /Zˆs = dlog(Zˆs ) and σ equals the limit as t → ∞ of the random variable min(s : |1/ t + Zˆs | = 1). In particular, we get that limt→∞ |θr− (t) − θr− (σ√t )|/logt = 0 and likewise for θr+ . Thus, it follows that if limt→∞ θr− (σ√t )/log(t) exists in law, then this limit coincides in law with the limit in law of θr− (t)/log(t) and likewise for θr+ (t). Now, θr− (σ√t ) = =





C(σ√t )

χβ(s)

where the expected value is taken in the ground state of HF , ie vacuum state |0 > in which each an has eigenvalue zero. T is the time ordering operator. We can write G(t, r, t� , r� ) = θ(t − t� ) < ψ(t, r)ψ(t� , r� )∗ > +θ(t� − t) < ψ(t� , r� )∗ ψ(t, r) >= Now we note that < ψ(t, r)ψ(t� , r� )∗ >=

 n,m

cn c¯m exp(−i(En t − Em t� ))un (r)¯ um (r� ) < an a∗m >

and since

< an a∗m >= δn,m

(since an a∗m + a∗m an = δn,m and an |0 >= 0), it follows that  exp(−iEn (t − t� )un (r)¯ um (r� ) < ψ(t, r)ψ(t� , r� )∗ >= n

We note that

< ψ(t� , r� )∗ ψ(t, r) >= 0

since ψ(t, r)|0 >= 0. Thus, G(t, r, t� , r� ) = θ(t − t� ) where We then get

 n

exp(−iEn (t − t� ))un (r)¯ un (r� ) = G0 (t − t� , r, r� )

G0 (t, r, r� ) = G(t, r, 0, r� ) 

R

G0 (t, r, r� )exp(i(ω + i�)t)dt =

 un (r)¯ un (r� ) i(ω − E n + i�) n

Quantum Mechanics for Scientists and Engineers 156

We have

159

G,t (t, r, t� , r� ) = δ(t − t� ) < ψ(t, r)ψ(t, r� )∗ > +θ(t − t� ) < ψ,t (t, r)ψ(t� , r� )∗ > = δ(t − t� )δ 3 (r − r� ) − iθ(t − t� ) < H(r)ψ(t, r)ψ(t� , r� )∗ > = δ(t − t� )δ 3 (r − r� ) − iH(r)G(t, r, t� , r� )

A.28.Design of quantum gates using scattering theory. Let H0 , H be two self-adjoint operators in the same Hilbert space H. We define W (t) = exp(itH)exp(−itH0 ), t ∈ R The limits

Ω+ = slimt→∞ W (t), Ω− = slimt→−∞ W (t)

may exist on different domains. Let D+ be the domain of Ω+ and D− that of Ω− . Then Ω∗+ : H → D+ is defined and hence S = Ω∗+ Ω− : D− → D+ is defined. It is clear that W (t) is a unitary operator on H and hence

< f |Ω∗+ Ω+ |g >=< Ω+ f |Ω+ g >= limt→∞ < W (t)f |W (t)g >=< f |g >, f, g ∈ D+ Thus Ω+ is unitary when restricted to D+ . Thus, Ω+ Ω∗+ : H → H is an orthogonal projection of H onto D+ . We write V = H − H0 and then find that  t  t W (t) − I = W � (s)ds == i exp(isH)V.exp(−isH0 )ds 0

0

In particular,



Ω+ = I + i

∞

exp(isH)V.exp(−isH0 )ds

0

Let E0 be the spectral measure of H0 and E that of H. We have  ∞  d/dt(exp(itH0 ).exp(−itH))dt = I − i Ω∗+ = I + i 0

exp(itH0 )V.exp(−itH)dt

0

Thus, I = Ω∗+ Ω+ = Ω+ − i Now,

∞



∞

exp(itH0 )V.exp(−itH)Ω+ dt

0

exp(−itH)Ω+ = lims→∞ exp(i(s − t)H)exp(−isH0 ) = lims→∞ exp(isH).exp(−i(s + t)H0 ) = Ω+ exp(−itH0 )

Thus, I = Ω+ − i or equivalently, Ω+ = I + i





∞

[0,∞)×R

=I−

exp(itH0 )V Ω+ exp(−itH0 )dt

0



R

exp(it(H0 − λ + i�))V Ω+ dtE0 (dΛ)

(H0 − λ + i�)−1 V Ω+ E0 (dλ)

The right side integral is to be interpreted as the limit of � → 0+. This is the rigorous statement of the first LippmanSchwinger equation in scattering theory. Roughly speaking, if |φ+ > is an output free particle state corresponding to an ”eigenfunction” of H0 with ”eigenvalue” λ and |ψ+ >= Ω+ |φ+ > is the corresponding output scattered state, which is an ”eigenfunction” of H with the same eigenvalue λ (energy is conserved during the scattering process), then we have |ψ+ >= |φ+ > −(H0 − λ + i�)−1 V |ψ+ >

Quantum Mechanics for Scientists and Engineers 157

160

In a similar way, we get that if |φ− > is an input free particle state corresponding to an energy eigenfunction of H0 with eigenvalue λ and |ψ− >= Ω− |φ− > the corresponding input scattered state corresponding to an eigenfunction of H with eigenvalue λ, then we get the second Lippman-Schwinger equation |ψ− >= |φ− > −(H0 − λ − i�)−1 V |ψ− > This is seen as follows: Ω∗− = limt→−inf ty exp(itH0 )exp(−itH) = I + i =I +i



∞



0

exp(itH0 )V.exp(−itH)dt

−∞

exp(−itH0 )V.exp(itH)dt

0

Thus,



I = Ω∗− Ω− = Ω− + i 

= Ω− + i or equivalently, Ω− = I − i

∞

0



∞

0



=I−

∞

0

exp(−itH0 )V.exp(itH)Ω− dt

exp(−itH0 )V Ω− .exp(itH0 )dt

exp(−it(H0 − λ − i�))V Ω− dtE0 (dλ) (H0 − λ − i�)−1 V Ω− E0 (dλ)

R

which gives the second Lippman-Schwinger equation. Now, we derive an explicit form of the scattering matrix. We define R0 (λ) = (H0 − λ)−1 , R(λ) = (H − λ)−1 respectively for λ belonging to the resolvent set of H0 and of H. We have

(I − i



S = Ω∗+ Ω− = ∞

exp(itH0 )V.exp(−itH)dt)Ω−

0

= Ω− − i



= Ω− +

∞

0



R

(H0 − λ + i�)−1 V.Ω− E0 (dλ)

= Ω− + Now,

=I −i =I−



R



Ω− = I − i





exp(itH0 )V.Ω− exp(−itH0 )dt

R0 (λ − i�)V Ω− E0 (dλ)

0

exp(itH).V.exp(−itH0 )dt

−∞

∞

exp(−itH)V exp(itH0 )dt

0

×R

(H − λ − iδ)−1 V E0 (dλ)

Substituting this into the previous expression gives    R0 (λ − i�)V E0 (dλ) − R0 (λ − i�)V R(λ + iδ)V E0 (dλ) S − I = − R(λ + iδ)V E0 (dλ) + R

R

A.29. Perturbed Einstein field equations with electromagnetic interactions. The unperturbed metric is g00 = 1, grs = −S(t)2 δrs , g0r = 0,

Quantum Mechanics for Scientists and Engineers 158

161

A small change in the coordinate system can be made such that the first order perturbations in the metric δgμν satisfies δg0μ = 0. The unperturbed energy-momentum tensor of the matter field is Tμν = (ρ(t) + p(t))Vμ Vν − p(t)gμν where Vr = 0, V0 = 1 (This is possible because the above unperturbed metric satisfies the comoving condition, ie, particles with Vr = 0 satisfy the geodesic equation). Thus, T00 = ρ(t), Trs = p(t)S 2 (t)δrs , g0r = 0 We also define the tensor We have so that since δg00 = 0,

Sμν = Tμν − T gμν /2, T = g μν Tμν = ρ + p − 4p = ρ − 3p δSμν = δTμν − T δgμν /2 − gμν δT /2 δS00 = δT00 − δT /2 = δρ − δρ/2 + 3δp/2 = (3δp + δρ)/2

Note that δρ, δp, δvμ , δgμν are in general functions of space and time, ie, of x = (t, r). δSrs = δTrs − T δgrs /2 − grs δT /2 = −δpgrs − (ρ − 3p)δgrs /2 − grs (δρ − 3δp)/2 = (3p − ρ)δgrs /2 + S 2 (δρ − δp)δrs /2

Note that we have the sequence of implications

vμ v μ = 1, Vμ δv μ = 0, δv 0 = 0 More precisely, gives which implies

δ(gμν v μ v ν ) = 0 V μ V ν δgμν + gμν (V μ δv ν + V ν δv μ ) = 0 δg00 + 2δv 0 = 0

and since by the coordinate condition, δg00 = 0, it follows that δv 0 = 0. Hence, δv0 = (δg0μ v μ ) = δg0μ V μ + g0μ δv μ = 0 which gives

δv0 = 0

since by the coordinate condition, δg0μ = 0 and g0r = 0. So δT00 = (p + ρ)(2V0 δv0 ) + V02 (δp + δρ) − pδg00 − g00 δp = δρ Further,

δSr0 = δS0r = (ρ + p)δvr

since δg0r = δgr0 = 0. Finally, we need to compute δRμν and equate this to K.δSμν . We have α δRμν = (δΓα μα ):ν − (δΓμν ):α

(δΓα μα ):ν = α (δΓα μν ):β = (δΓμν ),β ρ ρ α +Γα ρβ δΓμν − Γμβ δΓρν

Thus,

−Γρνβ δΓα ρμ (δΓα μν ):α

Quantum Mechanics for Scientists and Engineers 159

162

= (δΓα μν ),α ρ +Γα ρα δΓμν

− 2Γρμα δΓα ρν

Thus final form of the first order perturbation to the Ricci tensor is given by δRμν = ρ α (δΓα μα ),ν − Γμν δΓρα

−(δΓα μν ),α

ρ ρ α −Γα ρα δΓμν + 2Γμα δΓρν

The perturbed Einstein field equations have the general form

A1 (μν, rs, t)δgrs (t, r) + A2 (μν, rsl, t)δgrs,l (t, r) + A3 (μν, rslp, t)δgrs,lp (t, r) +A4 (μν, rs, t)δgrs,0 (t, r) + A5 (μν, rsl, t)δgrs,l0 (t, r) + A6 (μν, rs, t)δgrs,00 (t, r) +A7 (μν, t)δρ(t, r) + A8 (μν, r, t)δvr (t, r) = KδSμν (t, r) where Sμν (t, r) is the energy-momentum tensor of the electromagnetic field. We have, Sμν = (−1/4)Fαβ F αβ gμν + Fμα Fνα √ Fμν = Aν,μ − Aμ,ν , (F μν −g),ν = 0

F0r = Ar,0 − A0,r = −(S 2 (t)Ar ),0 − A0,r Frs = As,r − Ar,s = S 2 (t)(Ar,s − As,r )

The gauge condition is

√ (Aμ −g),μ = 0

which reads

(A0 S 3 (t)),0 + S 3 (t)Ar,r = 0

The unperturbed Maxwell equations are F,r0r = 0, (F r0 S 3 ),0 + S 3 F,srs = 0 F 0r = g 00 g rr F0r = (1/S 2 )((S 2 Ar ),0 − A0,r )

and so the first Maxwell equation gives

(S 2 Ar ),0r − ∇2 A0 = 0

or which reads on using the gauge condition,

(S 2 )� Ar,r + S 2 Ar,0r − ∇2 A0 = 0

−(S 2 )� S −3 (A0 S 3 ),0 − S 2 (S −3 (A0 S 3 ),0 ),0 − ∇2 A0 = 0 and the second Maxwell equation gives −S(As,r − Ar,s ),s − ((A0,r + Ar,0 )S),0 = 0 which simplifies to

S∇2 Ar − SAs,rs − (S(A0,r + Ar,0 )),0 = 0

and this becomes on using the gauge condition,

S∇2 Ar + S −2 (A0 S 3 ),0r − (SA0,r ),0 − (SAr,0 ),0 = 0 or

(∇2 Ar − Ar,00 ) + S −3 (A0 S 3 ),0r − S −1 (SA0,r ),0 − S −1 S � Ar,0 = 0

Quantum Mechanics for Scientists and Engineers 160

163

Since the radiation field is homogeneous and isotropic, we assume that the vector potential has mean zero and statistical correlations of the form < Ar (t, r)As (t� , r� ) >= P (t, t� , r − r� )δrs We find that since

(A0 S 3 (t)),0 + S 3 (t)Ar,r = 0

we have

A0,0 (t, r)S 3 (t) + 3S 2 (t)S � (t)A0 (t, r) + S 3 (t)Ar,r (t, r) = 0

and hence we can assume that A0 = 0 provided that we admit the Coulomb gauge constraint Ar,r (t, r) = 0 which gives

∂P (t, t� , r) =0 ∂xs � and so we can assume that P (t, t , r) is independent of the spatial vector r = (xs ). Thus, we can write < Ar (t, r)As (t� , r� ) >= P (t, t� )δrs

Now, the Maxwell equation is automatically satisfied since A = 0 and 0

Ar,r

(S 2 Ar ),0r − ∇2 A0 = 0

= 0. The second Maxwell equation

(∇2 Ar − Ar,00 ) + S −3 (A0 S 3 ),0r − S −1 (SA0,r ),0 − S −1 S � Ar,0 = 0 gives using A0 = 0,

∇2 Ar − Ar,00 − S −1 S � Ar,0 = 0

and hence taking correlations with As (t� , r� ), we get

∇2 P (t, t� ) − P,tt (t, t� ) − S −1 (t)S � (t)P,t (t, t� ) = 0 and likewise, with t and t� interchanged. We assume that we have solved this equation to obtain P (t, t� ). Now, the average energy-momentum tensor of the radiation field before the metric has been perturbed by δgμν (t, r) is given by Sμν = (−1/4) < Fαβ F αβ > gμν + < Fμα Fνα > Now,

Fαβ F αβ = 2F0r F 0r + Frs F rs = −2S −2 (Ar,0 − A0,r )2 + S −4 (As,r − Ar,s )2

= −2S −2 ((−S 2 Ar ),0 )2 − A0,r )2 + (As,r − Ar,s )2 = 2S −2 ((S 2 Ar ),0 )2 + (As,r − Ar,s )2 and its average value is

= 2S 2 (Ar,0 )2 + 2S −2 ((S 2 )� )2 (Ar )2 + (As,r − Ar,s )2 < Fαβ F αβ >= 6S P,tt (t, t� ) + 2S −2 ((S 2 )� )2 P (t, t� ) 2

since P does not depend on the spatial coordinates. Note that by P,tt (t, t), we mean that P,tt (t, t) = limt →t P,tt (t, t� ) Thus,

< Fαβ F αβ >= 6S 2 (t)P,tt (t, t) + 8(S � )2 P (t, t� )

We next compute < Fμα Fνα >. For μ = ν = 0 this is < F0r F0r >= −S −2 < (Ar,0 )2 >= −S 2 < (Ar,0 )2 >= −3S 2 (t)P,tt (t, t)

Quantum Mechanics for Scientists and Engineers 161

164

For μ = r, ν = 0, we get < Frα F0α >=
= −S

−2

< (As,r − Ar,s )As,0 >= −S 2 < (As,r − Ar,s )As,0 >= 0

since P is independent of the spatial coordinates. Thus,



< S00 >= (−3/2)S 2 (t)P,tt (t, t) − 2(S � )2 P (t, t� ) − 3S 2 P,tt (t, t) = (−9/2)S 2 (t)P,tt (t, t) − 2S 2 P (t, t� ) < Sr0 >=< S0r >= 0 Finally,

< Srs >= (3/2)S 4 (t)P,tt (t, t)δrs + < Fr0 Fs0 + Frm Fsm > < Fr0 Fs0 >=< Ar,0 As,0 >= S 4 (t) < Ar,0 As,0 >= S 4 (t)P,tt (t, t)δrs < Frm Fsm >= (−1/S 2 ) < Frm Fsm >= (−1/S 2 ) < (Am,r − Ar,m )(Am,s − As,m ) >= 0

since P is independent of the spatial coordinates. Thus,

< Srs >= (−1/2)S 4 (t)P,tt (t, t)δrs The perturbed Maxwell equations and the perturbed energy-momentum tensor of the electromagnetic field: The perturbed Maxwell equations are √ (δ(F μν −g),ν = 0

The perturbed gauge condition is

√ (δ(Aμ −g)),μ = 0

which gives

√ (S 3 (t)δAμ + Aμ δ −g),μ = 0

If we assume δA0 = 0, then this gauge condition gives

√ S 3 (δAr ),r + (Ar δ −g),r = 0 Now,

δg = g(−S −2 )δgrr

(Note that our coordinate system is chosen so that δg0μ = 0. Thus, we get √ √ δ −g = −δg/2 −g = −Sδgrr /2 Thus the gauge condition with δA0 = 0 gives S 2 (δAr ),r − Ar δgss,r /2 = 0 where we have used Ar,r = 0 ,ie, the Coulomb gauge for the unperturbed em field. This gives us the following gauge condition for the perturbed em field (δAr ),r = S −2 Ar δgss,r /2 We now write down the first perturbed Maxwell equation using δA0 = 0 as S 2 δF,r0r − (F 0r δgss /2),r = 0 or which is since Ar,r = 0, the same as

S 2 δAr,0r − (Ar,0 δgss /2),r = 0 S 2 δAr,0r − Ar,0 δgss,r /2 = 0

Substituting for δAr,r from the perturbed gauge condition, we get

S 2 (S −2 Ar δgss,r ),0 − Ar,0 δgss,r = 0 or equivalently

Ar (S −2 δgss,r ),0 = 0

Quantum Mechanics for Scientists and Engineers 162

165

and this condition is impossible to fulfill in general. So we cannot assume the Coulomb condition δA0 = 0 for the perturbed situation. Remark: We have assumed that A0 = 0 and hence that the gauge condition reads Ar,r = 0. We have also assumed that < Ar (t, r)As (t , s) >= P (t, t , r − r )δrs and have hence derived the condition that P (t, t , r) is independent of r. This is rather unrealistic since P being independent of r implies that the vector potentials between all the spatial points have the same correlation. A more realistic assumption would be to take < Ar (t, r)As (t , s) >= P (t, t )f r (r)f s (r ) and hence derive from the gauge condition that

r divf = f,r =0

A more general approach: Assume that the unperturbed metric is gμν (x) and the unperturbed vector potential is Aμ (x). Their perturbed versions are gμν (x) + δgμν (x) and Aμ (x) + δAμ (x) respectively. The unperturbed velocity field of matter is v μ (x) and it gets perturbed by δv μ (x). Likewise the unperturbed density and pressure are ρ(x) and p(x) and they get perturbed by δρ(x) and δp(x) respectively. The unperturbed four vector potential correlations are Lμν (x, x ) =< Aμ (x)Aν (x ) > The Einstein field equations are

Rμν = K(SM μν + SEμν ))

where K = −8πG, where so that

SM μν = TM μν − TM gμν /2 TM μν = (ρ + p)vμ vν − pgμν , TM = g μν TM μν = ρ − 3p SM μν = (ρ + p)vμ vν − (p + ρ)gμν /2

and

SEμν = (−1/4)Fαβ F αβ gμν + Fμα Fνβ g αβ < SEμν >= gμν g αρ g βσ < Fαβ Fρσ > +g αβ < Fμα Fνβ >

Now,

< Fμα Fνβ >=< (Aα,μ − Aμ,α )(Aβ,nu − Aν,β ) >

=< (gαρ Aρ ),μ − (gμρ Aρ ),α )((gβσ Aσ ),ν − (gνσ Aσ ),β ) > = (gαρ,μ − gμρ,α )(gβσ,ν − gνσ,β )Lρσ (x, x)

+(gαρ gβσ < Aρ,μ Aσ,ν > +gαρ gβσ,ν < Aρ,μ Aσ > +(gαρ,μ gβσ < Aρ Aσ,ν > −gαρ gνσ,β < Aρ,μ Aσ > −gαρ,μ gνσ < Aρ Aσ,β > −gαρ gνσ < Aρ,μ Aσ,β > −gμρ gβσ < Aρ,α Aσ,ν >

−gμρ,α gβσ < Aρ Aσ,ν > −gμρ gβσ,ν < Aρ,α Aσ >

+gμρ gνσ < Aρ,α Aσ,β > +gμρ,α gνσ < Aρ Aσ,β > +gμρ gνσ,β < Aρ,α Aσ > Using this formula, we can express < SEμν (x) >= C1 (μναβ, x) < Aα (x)Aβ (x) > +C2 (μναβρ, x) < Aα (x)Aβ,ρ (x) > β +C3 (μναβρσ, x) < Aα ,ρ (x)A,σ (x) >

where the functions Ck are constructed using the unperturbed metric gμν (x) and its first order partial derivatives gμν,α (x). The perturbation δ < SEμν (x) > of the average Maxwell energy momentum tensor is to be evaluated in terms of the metric perturbations. For evaluating this, we have to first express using the Maxwell equations, the perturbation δAμ (x) of the electromagnetic four vector potential in terms of the unperturbed potentials Aμ (x) and the metric perturbations δgμν (x). The Maxwell equations are √ (F μν −g),ν = 0

Quantum Mechanics for Scientists and Engineers 163

166

and its perturbation is We have

√ √ ( −gδF μν + F μν δ −g),ν = 0 √ √ √ √ δ −g = −δg/2 −g = −gg μν δgμν /2 −g = ( −g/2)g μν δgμν δF μν = δ(g μα g νβ Fαβ )

= δ(g μα g νβ )Fαβ + g μα g νβ δFαβ Now,

δ(g μα g νβ ) = −g μα g νρ g βσ δgρσ −g νβ g μρ g ασ δgρσ

δFαβ = δ(Aβ,α − Aα,β ) = (δAβ ),α − (δAα ),β =

δAβ = δ(gβmu Aμ ) = Aμ δgβμ + gβμ δAμ The perturbation of the gauge condition is given by Now

√ (Aμ −g),μ = 0 √ √ ( −gδAμ ),μ + (Aμ δ −g),μ = 0

√ √ √ √ δ −g = −δg/2 −g = −gg μν δgμν /2 −g = −gg μν δgμν /2

Combining all these equations, it follows that δAμ satisfies an equation of the form

D1 (μνρσ, x)δAν,ρσ (x) + D2 (μνρ, x)δAν,ρ (x) + D3 (μνραβ, x)Anu ,ρ (x)δgαβ (x) +D4 (μναβ, x)Aν (x)δgαβ (x)+ D5 (μνραβσ, x)Aν,ρ (x)δgαβ,σ (x) +D6 (μναβσ, x)Aν (x)δgαβ,σ (x) = 0 This equation can formally be solved to give  δAμ (x) = M (μν, αβ, x, x� , x�� )Aν (x� )δgαβ (x�� )d4 x� d4 x�� The perturbation to δ < SEμν (x) >=< δSEμν (x) > to the average energy momentum tensor of the electromagnetic field can be expressed as follows: δSEμν (x) = (−1/4)δ(Fαβ F αβ gμν ) + δ(Fμα Fνβ g αβ ) = (E1 (μνρσαβ)(x)Aρ,σ (x) + E2 (μνραβ, x)Aρ (x))δAα ,β (x) +(E3 (μνρσα)(x)Aρ,σ (x) + E4 (μνρα, x)Aρ (x))δAα (x) +(E5 (μνρδαβσ, x)Aδ (x)Aρ (x) + E6 (μνργδαβσ, x)Aδ (x)Aρ,γ (x))δgαβ,σ (x) +(E7 (μνρδχαβ, x)Aδ,χ Aρ (x) + E8 (μνρδχγαβ, x)Aδ,χ (x)Aρ,γ (x))δgαβ (x)

A.30. BCS theory of superconductivity. The second quantized Hamiltonian is given by   K = ψa (x)∗ (−∇2 /2m + μ)ψa (x)d3 x + V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y with the Fermionic fields ψa satisfying the anticommutation relations {ψa (x), ψb (y)∗ } = δab δ 3 (x − y), {ψa (x), ψb (y)} = 0

Quantum Mechanics for Scientists and Engineers 164

We define

167

ψa (tx) = exp(itK).ψa (x).exp(−itK)

Then,

ψa,t (tx) = iexp(itK)[K, ψa (x)]exp(−itK)

The Green’s function is defined as Gab (t, x|s, y) =< T (ψa (tx)ψb (s, y)∗ ) >= T r(ρG T (ψa (tx)ψb (sy)∗ )) where

ρG = exp(−βK)/T r(exp(−βK))

We observe that for t > s, Gab (tx|sy) = T r(exp(−βK).exp(itK)ψa (x)exp(i(s − t)K)ψb (y)∗ .exp(−isK)) = T r(exp((i(t − s) − β)K)ψa (x)exp(i(s − t)K))ψb (y)∗ )

We make the approximation of replacing  by



V (x, y)ψa (x)∗ ψa (x)ψb (y)∗ ψb (y)d3 xd3 y

V (x, y)(2 < ψa (y)∗ ψa (y) > ψb (x)∗ ψb (x)d3 xd3 y + +



V (x, x)ψa (x)∗ ψa (x)d3 x +





V (x, y) < ψa (x)ψb (y)∗ > ψa (x)∗ ψb (y)d3 y

V (x, y) < ψa (x)ψb (y) > ψa (x)∗ ψb (y)∗ d3 xd3 y

plus other quadratic terms. We can write down the general form of the above approximation to the potential energy integral as  F1 (x, y) < ψb (y)∗ ψb (y) > ψa (x)∗ ψa (x)d3 xd3 y + + +





F4 (x, y) < ψb (y)∗ ψa (x)∗ > ψa (x)ψb (y)d3 xd3 y = V1 + V2 + V3 + V4 

[V1 , ψc (x� )] = 

F3 (x, y) < ψb (y)ψa (x) > ψa (x)∗ ψb (y)∗ d3 xd3 y



say. We have

=−

F2 (x, y) < ψb (x)∗ ψa (y) > ψa (y)∗ ψb (x)d3 xd3 y

F1 (x, y) < ψb (y)∗ ψb (y) > [ψa (x)∗ ψa (x), ψc (x� )]d3 xd3 y �

F1 (x, y) < ψb (y)∗ ψb (y) > δac δ 3 (x − x� )ψa (x)d3 xd3 y = −(

[V2 , ψc (x� )] = =−







F2 (x, y) < ψb (x)∗ ψa (y) > [ψa (y)∗ ψb (x), ψc (x� )]d3 xd3 y

F2 (x, y) < ψb (x)∗ ψa (y) > δac δ 3 (y − x� )ψb (x)d3 xd3 y 

= −( 

F1 (x� , y) < ψb (y)∗ ψb (y) > d3 y)ψc (x� )

F2 (x, x� ) < ψb (x)∗ ψa (x� ) > ψb (x)d3 x [V3 , ψc (x� )] =

F3 (x, y) < ψb (y)ψa (x) > [ψa (x)∗ ψb (y)∗ , ψc (x� )]d3 xd3 y

Quantum Mechanics for Scientists and Engineers 165

168

Now,

ψa (x)∗ ψb (y)∗ ψc (x� ) = ψa (x)∗ (δbc δ 3 (y − x� ) − ψc (x� )ψb (y)∗ )

= δbc δ 3 (y − x� )ψa (x)∗ − (δac δ 3 (x − x� ) − ψc (x� )ψa (x)∗ )ψb (y)∗

= δbc δ 3 (y − x� )ψa (x)∗ − δac δ 3 (x − x� )ψb (y)∗ + ψc (x� )ψa (x)∗ ψb (y)∗

Thus,

[V3 , ψc (x� )] = = Finally,





F3 (x, y) < ψb (y)ψa (x) > (δbc δ 3 (y − x� )ψa (x)∗ − δac δ 3 (x − x� )ψb (y)∗ )d3 xd3 y

F3 (x, x� ) < ψc (x� )ψa (x) > ψa (x)∗ d3 x − [V4 , ψc (x� )] =





F3 (x� , y) < ψb (y)ψc (x� ) > ψb (y)∗ d3 y

F4 (x, y) < ψb (y)∗ ψa (x)∗ > [ψa (x)ψb (y), ψc (x� )]d3 xd3 y = 0

A.31. Maxwell’s equations in the closed isotropic model of the universe. The metric is dτ 2 = dt2 − f 2 (r)S 2 (t)dr2 − r2 S 2 (t)(dθ2 + sin2 (θ)dφ2 ) where

f (r) = (1 − kr2 )−1/2

Thus,

g00 = 1, g11 = −f 2 (r)S 2 (t), g22 = −r2 S 2 (t), g33 = −r2 S 2 (t)sin2 (θ) √ −g = r2 f (r)S 3 (t)sin(θ)

Assume A0 = 0. The gauge condition then gives

√ (Aμ −g),μ = 0

sin(θ)(A1 r2 f ),1 + r2 f (A2 sin(θ)),2 + r2 f sin(θ)A3,3 = 0

or equivalently,

r−2 (r2 f A1 ),1 + (sin(θ))−1 (A2 sin(θ)),2 + A3,3 = 0

The Maxwell equation under the assumption A0 = 0 gives or equivalently,

√ (F 0k −g),k = 0 √ (g kk Ak,0 −g),k = 0 √ (g kk −g(gkk Ak ),0 ),k = 0

or equivalently,

(r2 f sin(θ)(S 2 Ak ),0 ),k = 0

The above two equations are not compatible. So we assume that A0 �= 0 and then the above two equations, ie, the gauge condition and Gauss’ law respectively become sin(θ)r2 f (S 3 A0 ),0 + sin(θ)(A1 r2 f ),1 + r2 f (A2 sin(θ)),2 + r2 f sin(θ)A3,3 = 0 and

This further simplifies to

√ 0 = (F 0k −g),k √ = (g kk F0k −g),k = √ (g kk (Ak,0 − A0,k ) −g),k = √ √ (g kk (gkk Ak ),0 −g),k − (g kk A0,k −g),k ((S 2 Ak ),0 r2 f sin(θ)),k − S 2 (g kk A0,k r2 f sin(θ)),k = 0

Quantum Mechanics for Scientists and Engineers 166

The Maxwell equation gives taking r = 1,

169

√ √ (F rs −g),s + (F r0 −g),0 = 0 √ √ √ (F 12 −g),2 + (F 13 −g),3 + (F 10 −g),0 = 0

and likewise for r = 2, 3. This equation can be expressed as √ √ √ 0 = (F 12 −g),2 + (F 13 −g),3 + (F 10 −g),0 √ √ = (g 11 g 22 (A2,1 − A1,2 ) −g),2 + (g 11 g 33 (A3,1 − A1,3 ) −g),3 √ +(g 11 (A0,1 − A1,0 ) −g),0 = (g 11 g 22 ((g22 A2 ),1 − g 11 A1,2 ),2 + √ (g 11 g 33 (g33 A3 ),1 − (g11 A1 ),3 ) −g),3 √ +(g 11 (A0,1 − (g11 A1 ),0 ) −g),0

A.32. Intuitive proof of Cramer’s theorem for large deviations for iid random variables. Let Xi , i = 1, 2, ... be iid random variables with Eexp(λX1 ) = exp(Λ(λ)) Define Also define

I(x) = supλ∈R (λx − Λ(λ)) Sn =

n 

Xi

i=1

Then, we have for any Borel set E and any λ ∈ R, the following: Let x ∈ E be arbitrary. Then, 1 = E[exp(λSn − nΛ(λ))] ≥ E[exp(λSn − nΛ(λ))χSn /n∈E ] ≥ E[infx∈E exp(nλx − nΛ(λ))χSn /n∈E ]

and therefore, or equivalently,

= infx∈E exp(n(λx − Λ(λ)))P (Sn /n ∈ E) n−1 .log(P (Sn /n ∈ E) + infx∈E (λx − Λ(λ)) ≤ 0 n−1 .log(P (Sn /n ∈ E) ≤ −infx∈E (λx − Λ(λ))

If we assume that for each x ∈ E the function λ → (λx − Λ(λ)) attains it supremum at some λ(x) ∈ R, then we can conclude that n−1 .log(P (Sn /n ∈ E) ≤ −infx∈E I(x) where

I(x) = supλ∈R (λx − Λ(λ)) = λ(x)x − Λ(λ(x))

This is called the Large deviations upper bound. To get the lower bound, we define a new probability measure P˜n by the prescription, P˜n = exp(λSn − nΛ(λ)).Pn where Pn is the distribution of (X1 , ..., Xn ), then we have  1 = P˜n (Rn ) = exp(λξ − nΛ(λ))dPn (ξ)

and we obtain on differentiating both sides of this equation w.r.t. λ,  0 = (ξ − nΛ� (λ))dP˜n (ξ)

Quantum Mechanics for Scientists and Engineers 167

170

and hence if η denotes the mean of X1 under P˜1 , then we have 0 = η − Λ� (λ) ie,

η = Λ� (λ)

˜ 1 (x) = exp(xλ − Λ(λ))dP1 (x). Thus, we get Note that X1 , ..., Xn under P˜n are iid random variables with distribution dP for λ > 0 and η > 0,  Pn (E) = exp(−λξ + nΛ(λ))dP˜n (ξ) = E

EP˜ [exp(−n(λSn /n − Λ(λ)))χSn /n∈E ]

≥ EP˜ [exp(−n(λSn /n − Λ(λ)))χSn /n∈E χ|Sn /n−η|≤� ]

We are assuming that λ is a function of η determined by the condition Λ� (λ) = η. It follows that  Pn (E) ≥ supη+�∈E exp(−n(λ(η + �) − Λ(λ)))P˜n (|{Sn /n ∈ E} {|Sn /n − η| ≤ �})

Now assume that E = (−∞, a]. By the law of large numbers for iid random variables, we have limn→∞ P˜n (|Sn /n − η| ≤ �) = 1

∀� > 0. Note that P˜ is the probability measure on RZ+ having marginals P˜n , n = 1, 2, .... If we further assume that η < a, the it follows by the law of large numbers that P˜n ({Sn /n ∈ E} ∩ {|Sn /n − η| < �) → 1 and hence We can write the above as

liminfn→∞ n−1 .log(Pn (E)) ≥ supη∈E (−λη + Λ(λ))

liminfn→∞ n−1 .log(Pn (E)) ≥ supη∈E (−λη + Λ(λ)) = −infη∈E I(η) A similar argument works for η > a. A.33. Current in the BCS theory of superconductivity: ψα (x) are the Fermionic field operators. They satisfy the anticommutation rules {ψα (x), ψβ (y)∗ } = δαβ δ 3 (x − y), {ψα (x), ψβ (y)} = 0, {ψα (x)∗ , ψβ (y)∗ } = 0

The BCS Hamiltonian has the form H = (−1/2m) + + + +











ψα (x)∗ (∇ + ieA(x))2 ψα (x)d3 x + μ



ψα (x)∗ ψα (x)dx

V1 (x, y) < ψα (x)ψβ (y) > ψα (x)∗ ψβ (y)∗ dxdy V¯1 (x, y) < ψβ (y)∗ ψα (x)∗ > ψβ (y)ψα (x)dxdy V2 (x, y) < ψα (x)ψβ (x)∗ > ψα (y)∗ ψβ (y)dxdy V3 (x, y) < ψα (x)ψβ (y)∗ > ψα (x)∗ ψβ (y)dxdy

where for Hermitianity of H, we require that V¯2 (x, y) = V2 (x, y), V¯3 (x, y) = V3 (y, x)

Quantum Mechanics for Scientists and Engineers 168

171

The Gibbs density operator for this Hamiltonian is ρG = exp(−βH)/T r(exp(−βH)) and averages are defined w.r.t this density, ie for any operator X constructed out of the quantum fields ψα (x), ψα (x)∗ , α = 1, 2, x ∈ R3 , we define < X >= T r(ρG X) We write T = (−1/2m) Then,

ψα (x)∗ (∇ + ieA(x))2 ψα (x)dx

[T, ψα (x)] = (1/2m)(∇ + ieA(x))2 ψα (x)

Also [

 + + + =





 



psiβ (y)∗ psiβ (y)d3 y, ψα (x)] = −ψα (x)

[V, ψγ (z)] = V1 (x, y) < ψα (x)ψβ (y) > [ψα (x)∗ ψβ (y)∗ , ψγ (z)]dxdy V¯1 (x, y) < ψβ (y)∗ ψα (x)∗ > [ψβ (y)ψα (x), ψγ (z)]dxdy V2 (x, y) < ψα (x)ψβ (x)∗ > [ψα (y)∗ ψβ (y), ψγ (z)]dxdy V3 (x, y) < ψα (x)ψβ (y)∗ > [ψα (x)∗ ψβ (y), ψγ (z)]dxdy

δ1αγ (x, z)V1 (x, z)ψα (x)∗ dx − +

ie,







Δ1γα (z, x)V1 (z, x)ψα (x)∗ dx

V2 (x, z)Δ2γα (x, x)ψα (z)dx

A.34. Some aspects of nonlinear filtering theory. Let Xt be a Markov process with transition generator kernel Kt (x, y),  E(f (Xt+dt )|Xt = x) − f (x) = dt Kt (x, y)f (y)dy + o(dt) = (Kt f )(x)dt + o(dt)

Assume that the measurement process is zt defined by

dzt = ht (Xt )dt + dvt where vt is a Levy process, ie, an independent increment process with moment generating function given by Eexp(svt ) = exp(tψ(s)) Let Zt = σ(zs , s ≤ t), ie, the measurement process upto time t and for any function φ(x) defined on the state space of the Markov process Xt , we define πt (φ) = E(φ(Xt )|Zt ) We can write a stochastic differential equation for the process πt (φ) as dπt (φ) = Ft (φ)dt +

∞ 

Gkt (φ)(dzt )k

k=1

where Ft (φ), Gkt (π) are Zt measurable functions. The processes Ft (φ) and Gkt (φ) need to be calculated. We define a process yt via the sde  dyt = fk (t)(dzt )k yt , y0 = 1 k≥1

Quantum Mechanics for Scientists and Engineers 169

172

where the fk� s are arbitrary non-random functions. Then yt is a process adapted to Zt and we have the relation E[(φ(Xt ) − πt (φ))yt ] = 0 by the definition of conditional expectation. We write φt = πt (φ). It follows that E[(dφt − dπt (φ))yt ] + E[(φt − pit (φ))dyt ] + E[(dφt − dπt (φ))dyt ] = 0 by Ito’s formula. This equation can be expressed as   E[(dφt − dπt (φ))yt ] + fk (t)E[(φt − πt (φ))(dzt )k yt ] + E[(dφt − dπt (φ))(dzt )k yt ] = 0 k≥1

k

and from the arbitrariness of the functions fk , we get E[(dφt − dπt (φ))|Zt ] = 0 − − − (1) We note that

E[(φt − πt (φ))(dzt )k |Zt ] + E[(dφt − dπt (φ))(dzt )k |Zt ] = 0, k ≥ 1 − − − (2) (dzt )k = (dvt )k , k ≥ 2

and hence writing

E[(dvt )k ] = μk dt

we get from (1) and (2), assuming μ1 = 0, Ft (φ) +



k≥1

μk Gkt (φ) = πt (Kt φ) − − − (3),

(πt (ht φ) − πt (φ)πt (ht )) + πt ((Kt φ)ht ) − Ft (φ)πt (ht ) − μk πt (Kt (φ)) −

 r≥1



μr+1 Grt (φ)πt (ht ) = 0,

r≥1

μr+k Grt (φ) = 0, k ≥ 2

A.35. Some aspects of supersymmetry. Assume that xμ , μ = 0, 1, 2, 3 are the Bosonic spatial variables and θμ , μ = 0, 1, 2, 3 are the Fermionic anticommuting variables. A super field S(x, θ) can be expanded as S(x, θ) = S0 (x) + S1μ (x)θμ + S2μν (x)θμ θν + S3μνρ (x)θμ θν θρ +S4 (x)θ1 θ2 θ3 θ4 Note that a product of more than four θ s is zero. The sum in each term is over the repeated variables and we may assume that the summation is over μ < ν < ρ < σ. We now need to introduce an infinitesimal supersymmetry transformation as a super vector field and describe the transformation laws of the component fields S0 , S1μ , S2μν , S3μνρ , S4 under such a supersymmetry transformation. The infinitesimal super-symmetry transformation may be derived by assuming the following group conservation law for Bosonic and Fermionic variables (x, θ), x = (xμ ), θ = (θμ ): �







(xμ , θα ).(x� μ, θ α ) = (xμ + x μ + θT γ μ θ� , θα + θ alpha ) We express this equation as (x, θ).(x� , θ� ) = m((x, θ), (x� , θ� )) = (m1 ((x, θ), (x� , θ� )), m2 ((x, θ), (x� , θ� ))) where



m1 ((x, θ), (x� , θ� )) = (xμ + x μ + θT γ μ θ) 

m2 ((x, θ), (x� , θ� )) = (θα + θ α ) Then to get left and right-invariant vector fields on the supermanifold described by Bosonic and Fermionic coordinates (x, θ), we define ∂mν1 ∂ ∂mν ∂ Dxμ = +  μ2 ν μ ν ∂x ∂x ∂x ∂θ

Quantum Mechanics for Scientists and Engineers 170

173

∂ = ∂xμ ∂xμ ν ∂m1 ∂ ∂mν ∂ = +  α2 ν α ν ∂θ ∂x ∂θ ∂θ =

Dθα

= (θT γ ν )α ∂xμ + ∂θα = (θT γ μ )α

∂ ∂ + α ∂xμ ∂θ

A.36. Comparison between the classical and quantum motions of a simple pendulum perturbed by white Gaussian noise. Classical case: θ�� (t) = −a.sin(θ(t)) + σw(t) w(t) = B � (t)

B(t) is standard Brownian motion. We solve this by perturbation theory: Let θ(t) = θ0 (t) + σθ1 (t) + σ 2 θ2 (t) + ... Then substituting this expression into the equation of motion and equating coefficients of σ m , m = 0, 1, 2 successively gives us θ0�� (t) = −a.sin(θ0 (t)), θ1�� (t) = −a.cos(θ0 (t))θ1 (t) + B � (t), Define the matrix

θ2�� (t) = −a.cos(θ0 (t))θ2 (t) + (a/2)sin(θ0 (t))θ1 (t)2 A(t) =

and the 2 × 2 state transition matrix Φ(t, τ ) by



0 −a.cos(θ0 (t))

1 0



,

∂Φ(t, τ ) = A(t)Φ(t, τ ), t ≥ τ, Φ(τ, τ ) = I2 ∂t Then, θ1 (t) = θ2 (t) = (a/2)





t

Φ12 (t, τ )dB(τ ),

0

Φ12 (t, τ )sin(θ0 (τ ))θ1 (τ )2 dτ

Using these formulas, the statistics of θ(t) can be computed upto O(σ 2 ). We leave the problem of calculating the mean and autorcorrelation of θ(t) upto O(σ 2 ) as an exercise to the reader. The quantum case: Unitarity of the Schrodinger evolution operator U (t) is guaranteed if we take into account an Ito correction term in the Hamiltonian. Thus, the Schrodinger evolution is given by dU (t) = [−(iH0 + σ 2 V 2 /2)dt − iσV dB(t)]U (t) where

H0 = −∂θ2 /2ml2 − mgl.cos(θ)

We note that in this formalism, U (t) should be regarded as an integral kernel U (t, θ, θ� ) so that  (U (t)ψ)(θ) = U (t, θ, θ� )ψ(θ� )dθ� Heisenberg matrix mechanics: Let X be an observable and define X(t) = U (t)∗ XU (t) Then, assuming that V = V (θ) is a multiplication operator, dX(t) = dU (t)∗ XU (t) + U (t)∗ XdU (t) + dU (t)∗ XdU (t)

Quantum Mechanics for Scientists and Engineers 171

174

= U (t)∗ (i[H0 , X]dt − (σ 2 /2)(V 2 X + XV 2 − V XV )dt + iσ[V, X]dB(t))U (t)

For example, taking X = θ, pθ respectively gives

dθ(t) = U (t)∗ (pθ /ml2 − (σ 2 /2)(V 2 θ + θV 2 − V θV ) + iσ[V, θ]dB(t))U (t) = pθ (t)/ml2 , dpθ (t) = U (t) (i[−mglcos(θ), pθ ] − (σ 2 /2)(V 2 pθ + pθ V 2 − V pθ V ) + iσ[V, pθ ]dB(t))U (t) ∗

= U (t)∗ (−mglsin(θ) − (σ 2 /2)(V [V, pθ ] + [pθ , V ]V ) + iσ[V, pθ ]dB(t))U (t) = −mglsin(θ(t)) − σV � (θ(t))dB(t)

Thus, we get the quantum analogue of the classical sde. However, to be more precise and obtain further generalizations, we must take our noise processes as the creation, annihilation and conservation processes of the Hudson-Parthasarathy quantum stochastic calculus. HP calculus generalizations: Consider the qsde dU (t) = (−(iH0 + P )dt + L1 dA(t) − L2 dA(t)∗ + SdΛ(t))U (t) where L1 , L2 , P, S are system operators chosen to make U (t) unitary for all t. Here, H0 = −∇2 /2m + U (r) acts in the system Hilbert space h = L2 (R3 ). Let X be a system observable. At time t it evolves under HP noisy Heisenberg dynamics to X(t) = U (t)∗ XU (t) X(t) is defined on h ⊗ Γs (L2 (R+ )) and A(t), A∗ (t), Λ(t) are the HP noise operator processes satisfying the quantum Ito formula dA(t)dA(t)∗ = dt, dA∗ dA = 0, (dA)2 = 0, (dA∗ )2 = 0, (dΛ)2 = dΛ, dA.dΛ = dA, dΛ.dA∗ = dA∗ , dΛ.dA = 0, dA∗ dΛ = 0 Remark: Formally, we can write

dΛ = dA∗ .dA/dt

and hence using dA.dA∗ = dt, we get We get

dΛ.dA∗ = dA∗ , dA.dΛ = dA

dX(t) = dU (t)∗ XU (t) + U (t)∗ XdU (t) + dU (t)∗ XdU (t) =

U (t)∗ (i[H0 , X]dt−(P X +XP )dt+(L∗1 X −XL2 +S ∗ XL∗1 )dA∗ (−L∗2 X +XL∗1 −L∗2 XS)dA+(S ∗ X +XS ∗ +S ∗ XS)dΛ)U (t)

Exercise: Evaluate the Heisenberg equations of motion taking X = qk , X = pk , k = 1, 2, 3 where r = (q1 , q2 , q3 ), p = (p1 , p2 , p3 ). Assume that the system operators L1 , L2 , S, P are arbitrary functions of q, p subject to the constraint that makes U (t) unitary for all t. Specialize to the case when L1 , L2 , S, P are functions of q alone. A.37. The magnetic field produced by a transmission line current when hysteresis effects are taken into account. The line equations are  V � (ω, z) + Z(ω, z)I(ω, z) + δ H1 (ω1 , ω − ω1 , z)I(ω1 , z)I(ω − ω1 , z)dω1 = 0 I � (ω, z) + Y (ω, z)V (ω, z) + δ Here,



H2 (ω1 , ω − ω1 , z)V (ω1 , z)V (ω − ω1 , z)dω1 = 0

Z(ω, z) = R(z) + jωL(z), Y (ω, z) = G(z) + jωC(z)

We can expand in a Fourier series Z(ω, z) =



Zn (ω)exp(jnβz), β = 2π/d

n∈Z

Y (ω, z) =

 n

Yn (ω)exp(jnβz)

Quantum Mechanics for Scientists and Engineers

175

V (0) (ω, z) =



Vn(0) (ω)exp((γ + jnβ)z),

n

I (0) (ω, z) =



In(0) (ω)exp((γ + jnβ)z)

n

V (ω, z) = V (0) (ω, z) + δV (1) (ω, z) + O(δ 2 ) I(ω, z) = I (0) (ω, z) + δI (1) (ω, z) + O(δ 2 ) Then, substituting these expressions into the above line differential equations and equating coefficients of δ 0 and δ 1 respectively gives us  (0) (γ + jnβ)Vn(0) (ω) + Zn−m (ω)Im (ω) = 0, m

(γ +

jnβ)In(0) (ω)

+



Yn−m (ω)Vm(0) (ω) = 0

m

∂V (1) (ω, z) + Z(ω, z)I (1) (ω, z) + H1 .(I (0 ⊗ I (0) )(ω, z) = 0, ∂z

Let Φ(ω, z, z � ) ∈ C2×2

∂I (1) (ω, z) + Y (ω, z)I (1) (ω, z) + H2 (V (0) ⊗ V (0) )(ω, z) = 0 ∂z denote the state transition matrix corresponding to the forcing matrix   0 −Z(ω, z) A(ω, z) = −Y (ω, z) 0

Then, the solutions to the first order line voltage and current perturbations are given by  z V (1) (ω, z) = − Φ11 (ω, z, z � )H1 .(I (0 ⊗ I (0) )(ω, z � )dz � 0

−



z

0

Φ12 (ω, z, z � )H2 1.(V (0 ⊗ V (0) )(ω, z � )dz �

I (1) (ω, z) = − −



0

z



0

z

Φ21 (ω, z, z � )H1 .(I (0) ⊗ I (0) )(ω, z � )dz �

Φ22 (ω, z, z � )H2 .(V (0) ⊗ V (0) )(ω, z � )dz �

Denoting the infinite dimensional matrix ((Zn−m (ω))) by Z(ω), ((Yn−m (ω))) by Y(ω), the diagonal matrix diag[n : n ∈ (0) (0) Z] by D, the unperturbed voltage and current Fourier coefficient vectors ((Vn (ω))) and ((In (ω))) by V(0) (ω) and (0) I (ω) respectively, the above unperturbed eigenvalue equations can be expressed as T(ω)u(ω) = −γ(ω)u(ω) − − − (1) where



jβD Z(ω) Y(ω) jβD   (0) V (ω) u(ω) = (0) I (ω)

T(ω) =



,

Let −γn (ω), un (ω), n = 1, 2, ... be the complete set of eigenvalues and corresponding eigenvectors of T(ω) as in (1). Then we can write  cn (ω)un (ω) u(ω) = n∈Z

and we have on defining the infinite dimensional column vector

en (ω, z) = ((exp((γn (ω) + jkβ)z)))k∈Z , the following expression for the unperturbed line voltage and current:  V (0) (ω, z) = cn (omega)en (ω, z)T vn (ω), n

Quantum Mechanics for Scientists and Engineers 173

176

I (0) (ω, z) =



cn (ω)en (ω, z)T wn (ω)

n

where

un (ω) =



vn (ω) wn (ω)



The far field magnetic vector potential produced by this unperturbed line current is given by Az (ω, r) = (μ.exp(−jKr)/r)



d

I (0) (ω, ξ)exp(jKξ.cos(θ))dξ

0

= (μ.exp(−jKr)/r)P (ω, θ) where K = ω/c and P (ω, θ) =



d

I (0) (ω, ξ)exp(jkξ.cos(θ))dξ

0

A.38. Problems in quantum mechanics with solutions: [1] Obtain the supersymmetry generators in the form of super vector fields for four Boson and four Fermionic variables (xμ ), (θμ ). hint: Consider the generators of the form Dα = (Aμ θ)α ∂/partialxμ + Bαβ ∂/∂θβ and

¯ α = (C μ θ)α ∂/partialxμ + F β ∂/∂θα D α

The matrices A , C and the complex numbers Bαβ and Fαbeta are to be chosen so that the supersymmetric commutation relations hold in the form: ¯ β ] = K μ ∂/∂xμ [Dα , D αβ μ

μ

[2] Consider a periodic potential V (r), r ∈ R3 with three linearly independent period vectors, a1 , a2 , a3 ∈ R3 so that V (r + n1 a1 + n2 a2 + n3 a3 ) = V (r), n1 , n2 , n3 ∈ Z Expand V as a 3-D Fourier series using the reciprocal lattice vectors and hence formulate the stationary state Schrodinger equation −∇2 ψ(r)/2m + V (r)ψ(r) = Eψ(r), r ∈ R3 so that in view of the periodicity of V , we have

ψ(r + n1 a1 + n2 a2 + n3 a3 ) = C1n1 C2n2 C3n3 ψ(r), n1 , n2 , n3 ∈ Z where |Ck | = 1. If there are Nk atoms along ak k − 1, 2, 3, then we may apply the periodic boundary conditions on ψ without loss of generality in the form ψ(r + Nk ak ) = ψ(r), k = 1, 2, 3 This leads to so that

CkNk = 1, k = −1, 2, 3 Ck = exp(2πilk /Nk ), k = 1, 2, 3

for some lk ∈ {0, 1, ..., Nk − 1}. Now define the Bloch wave functions ul1 l2 l3 (r) by ψ(r) = exp(2πi(l1 b1 /N1 + l2 b2 /N2 + l3 b3 /N3 , r))ul1 l2 l3 (r) where {b1 , b2 , b3 } are the reciprocal lattice vectors corresponding to {a1 , a2 , a3 }. In other words, (bi , aj ) = δij Then, we have

ul1 l2 l3 (r + m1 a1 + m2 a2 + m3 a3 ) = ul1 l2 l3 (r), m1 m2 , m3 ∈ Z

Quantum Mechanics for Scientists and Engineers 174

177

ie, ul1 l2 l3 is also periodic with periods a1 , a2 , a3 like V (r) and hence it can also be developed into a Fourier series:  Ul1 l2 l3 [n1 n2 1n3 ]exp(2πi(n1 b1 + n2 b2 + n3 b3 , r)) ul1 l2 l3 (r) = n1 n2 n3 ∈Z

V (r) =



V [n1 n2 n3 ]exp(2πi(n1 b1 + n2 b2 + n3 b3 , r))

n1 n2 n3 ∈Z

The problem is to subsitute these two Fourier series expansions for the wave function and the potential into the stationary state Schrodinger wave equation and derive a difference equation for Ul1 l2 l3 [n1 n2 n3 ]. [3] Quantum control of the Hudson-Parthasarathy equation based on the Belavkin filter observer. HP equn: dU (t) = (−(iH + P )dt + L1 dA + L2 dA∗ + SdΛ)U (t) For unitarity of U (t), we require

0 = d(U ∗ U ) = dU ∗ .U + U ∗ .dU + dU ∗ .dU

This gives using the quantum Ito formula dAdA∗ = dt, dΛ.dA∗ = dA∗ , dA.dΛ = dA, (dΛ)2 = dΛ, (all the other products of differentials are zero),

P = L∗2 L2 /2, L∗2 + L1 + L∗2 S = 0, L∗1 + L2 + S ∗ L2 = 0, S + S∗ + S∗S = 0

L1 , L2 , S, H, P are system operators with H, P self-adjoint. We have (I + S)∗ (I + S) = I + S ∗ + S + S ∗ S = I so we can write

I + S = exp(iλ(t)Z)

where Z is selfadjoint and λ(t) is a real valued controllable function of time. Then, we have L1 = −L∗2 (I + S) = −L∗2 exp(iλ(t)Z) We write L2 = L =

p 

ck (t)Nk = L({ck (t)})

k=1

Then



L1 = −(

c¯k (t)Nk∗ ))exp(iλ(t)Z) = L1 ({ck (t)}, λ(t)),

k

S = exp(iλ(t)Z) − 1 = S(λ(t))

ck (t) are complex valued controllable functions of time. The real time control algorithms for controlling the ck (t)� s and λk (t) are based on minimizing the expected value of the Belavkin observer error E(t) = Xd (t) − πt (X) where Xd (t) is the desired system state at time t. This is a system observable and πt (X) is a measurable function of the noise algebra σ(Y )t upto time t. The noise is taken as Y (t) = U (t)∗ Yi (t)U (t), Yi (t) = a1 B(t) + a2 Λ(t), B(t) = A(t) + A(t)∗ , a1 , a2 ∈ R It is easily verified that Y statisfies the non-demolition abelian property, ie, [Y (t), Y (s)] = 0∀t, s ≥ 0

Quantum Mechanics for Scientists and Engineers 175

178

and

[Y (s), jt (X)] = 0, t ≥ s, jt (X) = U (t)∗ XU (t) = U (t)∗ (X ⊗ I)U (t)

We have

Y (t) = jt (Yi (t)) = U (t)∗ Yi (t)U (t), dY (t) = dYi (t) + dU (t)∗ dYi (t).U (t) + U (t)∗ dYi (t)dU (t) = dYi (t) + U (t)∗ (L∗1 dA∗ + L∗2 dA + S ∗ dΛ)dYi + dYi (L1 dA + L2 dA∗ + SdΛ))U (t)

Now,

dYi (L1 dA + L2 dA∗ + SdΛ) = (a1 (dA + dA∗ ) + a2 dΛ)(L1 dA + L2 dA∗ + SdΛ) = a1 L2 dt + a2 SdΛ + a2 L2 dA∗ + a1 SdA

We thus get dY (t) = dYi (t) + jt (a1 (L2 + L∗2 ))dt + jt (a2 (S + S ∗ ))dΛ + jt (a2 L2 + a1 S ∗ )dA∗ + jt (a2 L∗2 + a1 S)dA write

πt (X) = E(jt (X)|σ(Y )t )

where the expectation is taken in the state |f φ(u) > with f ∈ h, the system Hilbert space and |φ(u) >= exp(− � u �2 /2)|e(u) > the normalized exponential vector in the Boson Fock space Γs (L2 (R+ )). We assume that the optimal filter is described by the following qsde: dπt (X) = Ft (X)dt + Gt (X)dY (t) where Ft (X), Gt (X) ∈ σ(Y )t . Then applying the orthogonality principle E[(jt (X) − πt (X))C(t)] = 0 for C(t) defined via the qsde

dC(t) = f (t)C(t)dY (t), C(0) = 1

and using the arbitrariness of f (t) gives us E[(djt (X) − dπt (X))|σ(Y )t ] = 0, E[(djt (X) − dπt (X))dY (t)|σ(Y )t ] + E[(jt (X) − πt (X))dY (t)|σ(Y )t ] = 0

These two equations give us two equations for the observables Ft (X) and Gt (X), solving which the Belavkin filter is obtained. To derive these two equations, we compute using quantum Ito’s formula, djt (X) = dU ∗ XU + U ∗ XdU + dU ∗ XdU = = U ∗ (i[H, X]−P X −XP +L∗2 XL2 )dt+(L∗2 X +XL1 +L∗2 XS)dA+(L∗1 X +XL2 +S ∗ XL2 )dA∗ +(S ∗ X +XS +S ∗ XS)dΛ)U = jt (θ0 (X))dt + jt (θ1 (X))dAt + jt (θ2 (X))dA∗t + jt (θ3 (X))dΛt

where

θ0 (X) = i[H, X] − L∗ LX/2 − XL∗ L/2 + L∗ XL∗ θ1 (X) = L∗ X − XL∗ (I + S) + L∗ XS,

θ2 (X) = −(1 + S)∗ LX + XL − S ∗ XL, θ3 (X) = S ∗ X + XS + S ∗ XS

Note that and The equation

L2 = L = L({ck (t)}), S = S(λ(t)) L1 = −L∗ (1 + S) = L1 ({ck (t), λ(t)) E[(djt (X) − dπt (X))|σ(Y )t ] = 0,

Quantum Mechanics for Scientists and Engineers 176

thus gives

179

u(t) + πt (θ3 (X))|u(t)|2 πt (θ0 (x)) + πt (θ1 (X))u(t) + πt (θ2 (x))¯

−Gt (X)[a1 (u(t)+¯ u(t))+a2 |u(t)|2 +πt (a1 (L2 +L∗2 ))+πt (a2 (S+S ∗ ))|u(t)|2 +pit (a2 L2 +a1 S ∗ )¯ u(t)+πt (a2 L∗2 +a1 S)u(t)] = 0 or equivalently,

Ft (X) = u(t) + πt (θ3 (X))|u(t)|2 πt (θ0 (x)) + πt (θ1 (X))u(t) + πt (θ2 (x))¯

−Gt (X)[a1 (u(t) + u ¯(t)) + πt (a1 (L2 + L∗2 )) + πt (a2 (S + S ∗ ))|u(t)|2 + pit (a2 L2 + a1 S ∗ )¯ u(t) + πt (a2 L∗2 + a1 S)u(t)]

The equation gives

E[(djt (X) − dπt (X))dY (t)|σ(Y )t ] + E[(jt (X) − πt (X))dY (t)|σ(Y )t ] = 0 u(t) + πt (θ3 (X))|u(t)|2 πt (θ0 (X)) + πt (θ1 (X))u(t) + πt (θ2 (X))¯

−Ft (X) − Gt (X)E[(dY (t))2 |σ(Y )t ] + πt (X)(a1 (u(t) + u ¯(t)) + a2 |u(t)|2 + πt (a1 (L2 + L∗2 )) + πt (a2 (S + S ∗ ))|u(t)|2 +πt (a2 L2 + a1 S ∗ )¯ u(t) + πt (a2 L2 + a1 S)u(t) = 0

Now, from the equation dY (t) = dYi (t) + jt (a1 (L2 + L∗2 ))dt + jt (a2 (S + S ∗ ))dΛ + jt (a2 L2 + a1 S ∗ )dA∗ + jt (a2 L∗2 + a1 S)dA = jt (a1 (L2 + L∗2 ))dt + jt (a2 (S + S ∗ + 1))dΛ + jt (a2 L2 + a1 (S ∗ + 1))dA∗ + jt (a2 L∗2 + a1 (S + 1))dA we get using quantum Ito’s formula and the homomorphism property of jt , E[(dY (t))2 |σ(Y )t ] = u(t) πt (a22 (S + S ∗ + 1)2 )|u(t)|2 + πt ((a2 L∗2 + a1 (S + 1))(a2 L2 + a1 (S ∗ + 1))) + πt ((a2 (S + S ∗ + 1)(a2 L2 + a1 (S ∗ + 1)))¯ +πt (a2 (a2 L∗2 + a1 (S + 1))(S + S ∗ + 1))u(t)

[4](Reference: S.Wienberg, The quantum theory of fields, vol.II, Cambridge University Press)Evaluate approximately the path integeral   Z(J) = exp(iI(φ) − i J(x)φ(x)d4 x)Dφ

where J(x) is an external current source and  I(φ) = (∂μ φ)(∂ μ φ)/2 − m2 φ2 /2 − �V (φ)d4 x

ie, the Klein-Gordon action functional with a small perturbative correction. Calculate using this expression the propagators  exp(iI(φ)φ(x1 )...φ(xn )Dφ/Z(0) by using the formula

= in



∂ n bZ(J) |J=0 ∂J(x1 )...∂J(xn ) exp(iI(φ))φ(x1 )...φ(xn )dφ

Now derive the equation for J at which log(Z(J)) − this stationary solution by J0 (x), show that



J(x)φ0 (x)d4 x becomes stationary for a fixed field φ0 . Denoting

−φ0 (x) + δlogZ(J0 )/δJ(x) = 0 or equivalently, φ0 (x) + iZ(J0 )−1



exp(i



(I(φ) − J0 (x)φ(x))d4 x)φ(x).Dφ = 0

Quantum Mechanics for Scientists and Engineers 177

180

We write

J0 (x) = J0 (φ0 )(x)

or inverting this,

φ0 (x) = φ0 (J0 )(x)

It follows that

(δφ0 (x)/δJ0 (y) = δ 2 log(Z(J0 ))/δJ(x)δJ(y)

We define Γ(φ0 ) = log(Z(J0 )) − Thus, δΓ(φ0 )/δφ0 (y) =





J0 (x)φ0 (x)d4 x

φ0 (x)(δJ0 (x)/δφ0 (y))d4 x − = −J0 (y)



(δJ0 (x)/δφ0 (y))φ0 (x)d4 x − J0 (y)

Γ(φ0 ) is called the quantum effective action and the above equation is called the equation of motion for the quantum effective ation. [5] Calculation of path integrals for gauge invariant theories: Let φ(x) be the set of fields and f [φ] a gauge fixing functional. We consider a path integral of the form  X = G[φ]B[f [φ]]F [φ]Dφ where F [φ] is the Jacobian determinant:

F [φ] = det(df [φΛ ]/dΛ)|Λ=id

where φ → φΛ is the gauge transformed field. Λ is the gauge transorming function. We wish to show that in a certain sense, X does not depend on the gauge fixing functional f . We assume that the combined action functional G[φ] along with the measure Dφ, ie, G[φ]Dφ is invariant under the gauge transformation, ie, for all gauge transformations Λ, we have G[φΛ ]DφΛ = G[φ]Dφ Then, it follows that X= =





G[φΛ ]B[f [φΛ ]]F [φΛ ]DφΛ G[φ]B[f [φΛ ]]F [φΛ ]Dφ

Now, if Λ and λ are two gauge transformations, then det(df [φΛoλ ]/dλ)|λ=id = F [φΛ ]/ρ(Λ) where

ρ(Λ)−1 = det(d(Λoλ)/dλ)λ=id

So, X=



G[φ]B[f [φΛ ]]ρ(Λ)det(df [φΛ ]/dΛ)Dφ

Integrating this equation w.r.t Λ over the gauge group gives us after appropriately normalizing the Haar measure,   X = G[φ].B[f ]]df.Dφ = C G[φ]Dφ where C is a constant defined by

C=



B[f ]Df =



B[f [φΛ ]]det(df [φΛ ]/dΛ)ρ(Λ)dΛ

where we use the fact that ρ(Λ) as defined above is the left invariant Haar density on the gauge group and also the assumption that the gauge group acts transitively on the matter fields φ.

Quantum Mechanics for Scientists and Engineers 178

181

[6] Quantum teleportation: The simplest version of this idea involves transmitting one qubit of information by transmitting just two classical bits of information. Alice and Bob share an entangled state |00 > +|11 > apart from normalization factor of 2−1/2 . Alice prepares another state |ψ >= c1 |1 > +c2 |0 > where c1 , c2 ∈ C and |c1 |2 + |c2 |2 = 1 which she wishes to transmit to Bob by making use of the entangled state that she shares with Bob. The overall state of Alice and Bob is thus |ψ > (|00 > +|11 >) = (c1 |1 > +c2 |0 >)(|11 > +|00 >) = c1 |111 > +c1 |100 > +c2 |011 > +c2 |000 >

The first two qubits of this state can be controlled only by Alice and the last only by Bob. Alice applies a phase gate to her two qubits thus obtaining the overall state as |φ >= c1 |111 > +c1 |100 > +c2 |011 > −c2 |000 > We express this state in terms of the orthogonal one qubit state |+ >= |1 > +|0 >, |− >= |1 > −|0 > or equivalently,

|1 >= |+ > +|− >, |0 >= |+ > −|− >

Thus the overall state of Alice and Bob is given by

|φ >= c1 (|+ > +|− >)(|+ > +|− >)|1 > +c1 (|+ > +|− >)(|+ > −|− >)|0 > +c2 (|+ > −|− >)(|+ > +|− >)|1 > −c2 (|+ > −|− >)(|+ > −|− >)|0 >

Alice now performs a measurement on her two qubits in this shared state using the orthonormal basis | + + >, | + − > , | − + >, | − − >. If she measures | + + >, then clearly from the above expression for |φ >, Bob’s state collapses to c1 |1 > +c1 |0 > +c2 |1 > −c2 |0 >= c1 (|1 > +|0 >) + c2 (|1 > −|0 >) If Alice measures | + − >, then Bob’s state collapses to c1 |1 > −c1 |0 > +c2 |1 > +c2 |0 >= c1 (|1 > −|0 >) + c2 (|1 > +|0 >) If Alice measures | − + >, then Bob’s state collapses to c1 |1 > +c1 |0 > −c2 |1 > +c2 |0 >= c1 (|1 > +|0 >) + c2 (|0 > −|1 >) Finally, if Alice measures | − − >, then Bob’s state collapses to c1 |1 > −c1 |0 > −c2 |1 > −c2 |0 >= c1 (|1 > −|0 >) − c2 (|1 > +|0 >) Using two classical bits, Alice reports to Bob the outcome of her measurements. If she reports | + + >, then Bob applies the unitarty gate U1 to his state defined by √ √ U1 (|1 > +|0 >) 2 = |1 >, U1 (|1 > −|0 >)/ 2 = |0 >, to recover |ψ >. If Alice reports | + − >, then Bob applies the unitary gate U2 to his state defined by √ √ U2 (|1 > −|0 >)/ 2 = |1 >, U2 (|1 > +|0 >)/ 2 = |0 > to recover |ψ >. If Alice reports | − + >, then Bob applies the unitary gate U3 to his state defined by √ √ U3 (|1 > +|0 >)/ 2 = |1 >, U3 (|1 > −|0 >)/ 2 = −|0 > to recover |ψ >. Finally, if Alice reports | − − >, then Bob applies the unitary gate U4 to his state defined by √ √ U4 (|1 > −|0 >)/ 2 = |1 >, U4 (|1 > +|0 >)/ 2 = −|0 >

Quantum Mechanics for Scientists and Engineers 179

182

to recover the state |ψ >. [7] Quantum Boltzmann equation. N identical particles. ρ(t) is the joint density matrix of the particles. The k th particle acts in the Hilbert space Hk . The Hilbert space of the whole system is H=

N  k=1

Hk

Each Hk is an identical copy of a fixed Hilbert space H0 . ρ satisfies the Schrodinger-Von-Neumann-Liouville equation iρ� (t) = [H, ρ(t)] where H=

N 

k=1

Hk +



Vkj

1≤k)W (z)W (u) where W (z) = exp(

N  j=1

(¯ zj aj − zj a∗j )), z = [z1 , ..., zN ]T

Let ρ be a state in the Hilbert space L2 (RN ) = Γs (C). Then its quantum Fourier transform is given by ρˆ(z) = T r(ρW (z)) We have

ρˆ(za − zb ) = T r(ρW (za − zb )) =

T r(ρW (za )W (zb )∗ )exp(iIm(< za , zb >)) and hence the matrix ((ˆ ρ(za − zb )exp(−iIm(< za , zb >)))1≤a,b≤p is positive definite for any p complex vectors za , a = 1, 2, ..., p in CN . Let ρ be a Gaussian state, ie, its quantum Fourier transform has the form ρˆ(x + iy) = exp(lT x + mT y − [xT , y T ]S[xT , y T ]T ) where l, m are complex vectors and S is a symmetric complex matrix. We write R(z) = [xT , y T ]T and hence, we can write ρˆ(z) = exp(pT R(z) − R(z)T SR(z)/2) We have for z, u ∈ CN ,

Im(< z, u >) = R(z)T JR(u)

where J=



0N −IN

IN 0



∈ R2N ×2N

Quantum Mechanics for Scientists and Engineers 184

187

Thus, the requirement of the the above positive definitivity amounts to requiring that 2S − iJ ≥ 0 When this condition is satisfied, we call ρ a quantum Gaussian state. [13] Problem: Consider Dirac’s equation in a noisy electromagnetic field described by the qsde β α β dU (t) = [(α, pdt + e(Lα β dΛα (t))) + βmdt) + (V (q)dt + eMβ dΛα (t))U (t)

where the system Hilbert space is h = L2 (R3 ) and the noise Boson-Fock space is Γs (L2 (R+ )⊗Cd ). The system operators α Lα β andMβ are assumed to be functions of q, p only. The electron charge e is a perturbation parameter. Formally, the noisy magnetic vector potential is β At = Lα β dΛα (t)/dt If we assume that Lα β are functions of the position vector observable q only, then At becomes a a multiplication operator when restricted to the system Hilbert space h only. The noisy electric scalar potential is Φt = Mβα dΛβα (t)/dt and if we assume that Mβα are functions of q only, then the electric scalar potential becomes a multiplication operator when restricted to the system Hilbert space only. To get an approximate solution for U (t), we write U (t) = U0 (t) + eU1 (t) + e2 U2 (t) + O(e3 ) Problem: Calculate the matrix elements of U (t) between the states |f φ(u) > and |gφ(v) > upto O(e2 ) where |f >, |g >∈ h and u, v ∈ L2 (R+ ) ⊗ Cd , |φ(u) >= exp(− � u �2 /2)|e(u) > and likewise for |φ(v) >. Interpret this result in terms of transition probabilities for the system between two energy levels by taking an average over the noise space assuming that the noise space remains in the coherent state |φ(u) > before and after the transition. [14] Alternative description of the induced representation of a semidirect product: G = N ⊗s H, hN h−1 = N, ∀h ∈ H, N ˆ . Let is Abelian. Choose χ0 ∈ N H0 = {h ∈ H : h.χ0 = χ0 } where

ˆ h.χ(n) = χ(h−1 nh), h ∈ H, n ∈ N, χ ∈ N

Let Oχ0 denote the orbit of χ0 under H, ie,

ˆ Oχ0 = {hχ0 : h ∈ H} ⊂ N The manifolds H/H0 and Oχ0 are isomorphic under the H-group action. This follows by defining the map φ : H/H0 →   Oχ0 by φ(hH0 ) = hχ0 , h ∈ H. The map is well defined since hH0 = h� H0 implies h −1 h ∈ H0 implies h −1 hχ0 = χ0  � � � implies hχ0 = h χ0 . The map φ is a bijection since φ(hH0 ) = φ(h H0 ) implies hχ0 = h χ0 implies h −1 hχ0 = χ0  implies h −1 h ∈ H0 implies hH0 = h� H0 . This proves the injectivity of φ. Surjectivity is obvious. Let L be a representation of H0 in a vector space V . Now Let Y denote the vector space of all functions f : H → V such that L L : H → Y by U L (h1 )f (h) = f (h−1 is a f (hh−1 0 ) = L(h0 )f (h), h0 ∈ H0 , h ∈ H. Define U 1 h), h1 , h ∈ H. Then, U representation of H in Y and is called the representation of H induced by the representation L of H0 . Now we give an alternate description of the induced representation. Let Y˜ denote the vector space of all functions f : Oχ0 → V . Define T : Y˜ → Y by the formula T f (h) = A(h)f (hχ0 ), h ∈ H where A(h) is an appropriate matrix, ie A : H → GL(V ). In order that T f ∈ Y , we require that T f (hh−1 0 ) = L(h0 )T f (h), h0 ∈ H0 , h ∈ H, ie, −1 A(hh−1 0 )f (hh0 χ0 ) = L(h0 )A(h)f (hχ0 ), h ∈ H, h0 ∈ H0

or equivalently, since h−1 0 χ0 = χ0 ,

A(hh−1 0 ) = L(h0 )A(h), h0 ∈ H0 , h ∈ H

This can be satisfied for example by choosing

A(h) = L(γ(hχ0 )−1 h)−1 = L(h−1 γ(hχ0 ))

Quantum Mechanics for Scientists and Engineers 185

188

We then check that

−1 −1 = L(h0 h−1 γ(hh−1 h) = L(h0 ) A(hh−1 0 )A(h) 0 χ0 )L(γ(hχ0 )

Thus, We have for h, h1 ∈ H,

(T f )(h) = L(h−1 γ(hχ0 ))f (hχ0 ), h ∈ H −1 −1 (U L (h1 )T f )(h) = T f (h−1 h1 γ(h−1 1 h) = L(h 1 hχ0 ))f (h1 hχ0 )

If we equate this to T V L (h1 )f (h) = L(h−1 γ(hχ0 ))(V L (h1 )f )(hχ0 ), then we must have (V L (h1 )f )(hχ0 ) = L(γ(hχ0 )−1 hh−1 h1 γ(h−1 1 hχ0 )) = L(γ(hχ0 )−1 h1 γ(h−1 1 hχ0 )) or equivalently, writing χ = h.χ0 , we get V L (h1 )f (χ) = L(γ(χ)−1 h1 γ(h−1 1 χ)) V L therefore defines a representation of H in Y˜ that is equivalent to the induced representation U L of H in Y . as

[15] EM wave propagation in an inhomogeneous waveguide: The permittivity and permeability fields are represented  �[n, x, y]exp(−nβz), �(x, y, z) = n

μ(x, y, z) =



μ[n, x, y]exp(−nβz)

n

For example, if the length of the guide is d, we can take β = j2π/d, ie, develop � and μ as Fourier series in the z variable. We also expand the electric and magnetic fields as Fourier series in the z variable with an additional complex propagation constant γ:  E(x, y, z) = E[n, x, y]exp(−(γ + nβ)z), n

H(x, y, z) =



H[n, x, y]exp(−(γ + nβ)z)

n

We also expand the current density J(x, y, z) within the guide as  J[n, x, y]exp(−jnβz) J(x, y, z) = n

Substituting these into the Maxwell curl equations curlE(x, y, z) = −jωμ(x, y, z)H(x, y, z), curlH(x, y, z) = J(x, y, z) + jω�(x, y, )E(x, y, z) and equating coefficients of exp(−jnβz) for each integer n gives us z × E⊥ [n, x, y] = −jω ∇⊥ Ez [n, x, y] × zˆ − (γ + nβ)ˆ z × H⊥ [n, x, y] = jω ∇⊥ Hz [n, x, y] × zˆ − (γ + nβ)ˆ +J⊥ [n, x, y]

 m

 m

for the x − y components. For the z component, we have (ˆ z , ∇⊥ × H⊥ ) = Jz + jω�Ez , (ˆ z , ∇⊥ × E⊥ ) = −jωμHz

μ[m, x, y]H⊥ [n − m, x, y], �[m, x, y]E⊥ [n − m, x, y]

Quantum Mechanics for Scientists and Engineers 186

189

or in the transform domain, (ˆ z , ∇⊥ × E⊥ [n, x, y]) = −jω

 m

μ[m, x, y]Hz [n − m, x, y],

(ˆ z , ∇⊥ × H⊥ [n, x, y]) = Jz [n, x, y] + jω

 m

�[m, x, y]Ez [n − m, x, y]

To cast these equations in the format of perturbation theory, we assume that �[n, x, y] and μ[n, x, y] for n �= 0 are small compared to �[0, x, y] and μ[0, x, y] and hence attach a small pertrubation parameter δ to the former terms. We then get  z × E⊥ [n, x, y] + jωμ[0, x, y]H⊥ [n, x, y] = −jδω ∇⊥ Ez [n, x, y] × zˆ − (γ + nβ)ˆ μ[m, x, y]H⊥ [n − m, x, y], m�=0

z × H⊥ [n, x, y] − jω�[0, x, y]E⊥ [n, x, y] − J⊥ [n, x, y] = jδω ∇⊥ Hz [n, x, y] × zˆ − (γ + nβ)ˆ or in the transform domain, (ˆ z , ∇⊥ × E⊥ [n, x, y]) + jωμ[0, x, y]Hz [n, x, y] = −jδω



m�=0



m�=0

�[m, x, y]E⊥ [n − m, x, y]

μ[m, x, y]Hz [n − m, x, y],

(ˆ z , ∇⊥ × H⊥ [n, x, y]) − Jz [n, x, y] − jω�[0, x, y]Ez [n, x, y] = jωδ



m�=0

�[m, x, y]Ez [n − m, x, y]

[16] Symmetry breaking: Let ψ(x) ∈ CN be the matter field. It transforms under gauge group G ⊂ U (N ). Under a local gauge transformation, this is given by ψ(x) → g(x)ψ(x), where g(x) ∈ G. Now, assume that we can write ˜ ˜ ψ(x) = γ(x)ψ(x) where γ(x) is a representative element of a coset in G/H with H a subgroup of G. Thus, ψ(x) ˜ transforms according to the ”broken subgroup” H and g(x) ∈ G transforms ψ(x) to g(x)ψ(x) = g(x)γ(x)ψ(x) and / H. Assume that Aμ (x) are we can write g(x)γ(x) = γ � (x)h(x) where h(x) ∈ H and either γ � (x) = e or else γ � (x) ∈ the gauge fields taking values in the Lie algebra of G, ie in g. ψ(x) transforms according to G and the corresponding transformation law of Aμ (x) under G is obtained by using the fact that g(x)(∂μ + ieAμ (x))ψ(x) = (∂μ + A�μ (x))g(x)ψ(x) for this ensures that if the Lagrangian density L(ψ(x), ∇μ ψ(x)) is globally G-invariant, ie L(gψ(x), g∇μ ψ(x)) = L(ψ(x), ∇μ ψ(x)) for all g ∈ G, where

∇μ = ∂μ + ieAμ (x)

is the gauge covariant derivative, then it would follow that L is also locally G-invariant, ie, L(ψ � (x), ∇�μ ψ � (x)) = L(ψ(x), ∇μ ψ(x)) where

∇�μ = ∂μ + ieA�μ (x)

It follows that the transformation law of the matter field ψ(x) under the gauge transformation is ψ � (x) = g(x)ψ(x) while the corresponding transformation law of the gauge field Aμ (x) under the gauge transformation is given by (∂μ + ieA�μ (x)) = g(x)(∂μ + ieAμ (x))g(x)−1 (as an operator equation), or equivalently, ieA�μ (x) = ieg(x)Aμ (x)g(x)−1 + g(x)∂μ (g(x)−1 ) or

A�μ (x) = g(x)Aμ (x)g(x)−1 + (i/e)(∂μ g(x))g(x)−1

Quantum Mechanics for Scientists and Engineers 187

190

This equation is the non-Abelian generalization of the Abelian U(1) gauge transformation of the Dirac field in an electromagnetic field where the wave function ψ(x) ∈ C4 transforms to ψ � (x) = exp(iφ(x))ψ(x) where φ(x) ∈ R and correspondingly the electromagnetic four potential Aμ (x) changes to A�μ (x) so that (∂μ + ieA�μ )exp(iφ) = exp(iφ)(∂μ + ieAμ ) or equivalently,

ieA�μ = ieAμ − i∂μ φ

or equivalently,

A�μ (x) = Aμ (x) − e−1 ∂μ φ(x)

which is the standard Lorentz gauge transformation. Coming back to the general non-Abelian case, we assume that if {ta } are the generators of the Lie algebra of G, then  Dba (g)tb gta g −1 = b

In other words, D(g) is the adjoint representation of G acting in g. Then,  Aaμ ta )g −1 gAμ (x)g −1 = g( a

=



Aaμ gta g −1 =

a

This means that

A�μ (x) =





Aaμ Dba (g)tb

a

Dba (g)Aaμ (x)tb + (i/e)(∂μ g(x))g(x)−1

b

Now suppose that the matter Lagrangian density in the absence of the gauge field has the form L=

1 (∂μ ψ(x))∗ (∂ μ ψ(x)) − m2 ψ(x)∗ ψ(x) 2

This Lagrangian density is invariant under global G-transformations since g ∈ U (N ). Then since ˜ ψ(x) = γ(x)ψ(x) we get so that

∂μ ψ(x) = (∂μ γ)ψ˜ + γ∂μ ψ˜ ˜ μ ψ)∗ + ψ˜∗ (∂μ γ)∗ (∂ μ γ)ψ˜ L = (∂μ ψ)(∂ ˜ − m2 ψ(x) ˜ ∗ ψ(x) ˜ +2Re(ψ˜∗ ∂μ γ)∗ γ∂ μ ψ)

It is clear that This Lagrangian density of the matter part ie those terms invoving only ψ˜ and not γ is invariant under ˜ ˜ ψ(x) → hψ(x), h ∈ H and further that the above Lagrangian does not contain terms that are quadratic in γ(x) not ˜ involving their derivatives. This means that ψ(x) has mass while the other part γ(x) does not have mass. γ(x) is called the Goldstone-massless part of the field and ψ˜ is called the massive part. Thus, broken symmetries are associated with ˜ the productions of massless Goldstone Bosons. Just as ψ was separated into a massive part ψ(x) and a massless part γ(x), we can also separate the gauge field Aμ (x) into two parts: ˜ ψ(x) = γ(x)−1 ψ(x), A˜μ (x) = γ(x)−1 Aμ (x)γ(x) or equivalently, Then we find that

A˜aμ (x) =



Dab (γ(x)−1 )Abμ (x)

˜ ψ � (x) = g(x)ψ(x) = g(x)γ(x)ψ(x), g(x)γ(x) = γ � (x)h(x),

Quantum Mechanics for Scientists and Engineers 188

191

(∂μ g)γ + g∂μ γ = (∂μ γ � )h + γ � ∂μ h and hence

Thus, implies

γ −1 g −1 (∂μ g)γ + γ −1 ∂μ γ = h−1 γ � − 1(∂μ γ � )h + h−1 partialμ h A�μ (x) = g(x)Aμ (x)g(x)−1 + (i/e)(∂μ g(x))g(x)−1 − − − (1)  A˜�μ = γ −1 A�μ γ � =

γ



−1

g(x)Aμ (x)g(x)−1 γ � + (i/e)γ

Noting that γ



−1

it follows that this equals



−1

(∂μ g(x))g(x)−1 γ �

g = hγ −1

A˜�μ = hγ −1 Aμ γh−1 + (i/e)γ



−1

((−g(∂μ γ)γ −1 g −1 + (∂μ γ � )γ

+γ � (∂μ h)h−1 γ



−1

−1

)γ �

= hA˜μ h−1 + (i/e)(−hγ −1 (∂μ γ)h−1 + γ or equivalently,





−1

∂μ γ � + (∂μ h)h−1 )

 A˜�μ − (i/e)γ −1 ∂μ γ � =

h(A˜μ (−i/e)γ −1 ∂μ γ)h1 + (i/e)(∂μ h)h−1 − − − − − (2)

where we have used the fact that the equation

γ −1 g −1 (∂μ g)γ + γ −1 ∂μ γ =

implies which again implies

h−1 γ � − 1(∂μ γ � )h + h−1 partialμ h (∂μ g)γ + g∂μ γ = (∂μ γ � )h + γ � (∂μ h) (∂μ g)g −1 + g(∂μ γ)γ −1 g −1 = (∂μ γ � )hγ −1 g −1 +γ � (∂μ h)γ −1 g −1

which implies

(∂μ g)g −1 + g(∂μ γ)γ −1 g −1 = (∂μ γ � )γ +γ � (∂μ h)h−1 γ





−1

−1

Equation (2) should be compared to equation (1). (1) tells us how the gauge fields Aμ transform under a local gauge transformation g(x) coming from the big group G. (2) tells us how the massless components of the gauge fields A˜μ transform locally under the broken subgroup H. It tells us that when symmetry breaking takes place, then the massless gauge field should be taken as A˜μ − (i/e)γ −1 ∂μ γ rather than as A˜μ . In other words, the massless Goldstone components γ of the matter field ψ also becomes a part of the massless component of the gauge field A˜μ and the sum total of these two determines the effective gauge field which transforms under the broken subgroup H just as the original gauge field Aμ transformed under the larger group G. It is possible to express all these formulae in terms of functions of the space-time coordinates by first choosing a set of generators ti , i = 1, 2, ..., p of the Lie algebra of H and then extending this to a basis of the Lie algebra of G, ie, {t1 , ..., tp , x1 , ..., xq }, p + q = n = dimG. Then, we can write γ(x) = exp(

q 

(ξi (x)xi )

i=1

ξi (x) can be taken as the Goldstone Boson fields and quantities like (∂μ γ(x))γ(x)−1 can be expressed as nonlinear functions of the ξi (x). In this way, the Lagrangian density can be expressed in terms of the massless part A˜μ of the Gauge fields, the massive part ψ˜ of the matter field ψ and the massless Goldstone Boson fields ξi (x).

Quantum Mechanics for Scientists and Engineers 189

192

[17] Supersymmetric field theories: Let θa , a = 0, 1, 2, 3 be anticommuting Majorana Fermions: Writing θ = (θa ), we have   0 e θ θ∗ = −e 0 where

e= with





0 iσ 2

iσ 2 0

σ μ , μ = 0, 1, 2, 3

begin the Pauli spin matrices:

 0 1 , 1 0     0 −i 1 0 , σ3 = σ2 = i 0 0 −1 σ 0 = I2 , σ 1 =

We write



σμ = ημν σ ν

so that The Dirac gamma matrices are defined as

σ0 = σ 0 , σr = −σ r , r = 1, 2, 3 γμ =

They satisfy



0 sigmaμ

σμ 0



{γ μ , γ ν } = 2η μν I4

Define the left Chiral and right Chiral fermionic variables

θL = ((1 + γ5 )/2)θ, θR = ((1 − γ5 )/2)θ where γ5 = Clearly,



I2 0

0 −I2



γ 0 γ 1 γ 2 γ 3 = iγ5

and hence from the anticommutativity of the γ , we deduce that μ

{γ μ , γ5 } = 0 We also have

θ = θ L + θR , ∂/∂θ = ((1 + γ5 )/2)(∂/∂θL ) + ((1 − γ5 )/2)(∂/∂θR )

Define the supersymmetry generators as where Also define where

L = γ5 �∂/∂θ + γ μ θ∂μ ∂μ = ∂/∂xμ ¯ = −γ5 �L L �=

Clearly,



e 0

0 e



e2 = −I2 , �2 = −I4 , eT = −e, �T = −�, γ5T = γ5 , [γ5 , �] = 0

Quantum Mechanics for Scientists and Engineers 190

and hence

193

(γ5 �)T = −γ5 �, (γ5 �)2 = −I4

so we can also write

¯ T = LT γ5 � L

We get

¯= L ∂/∂θ − γ5 �γ μ θ∂μ

Then using

{∂/∂θa , θb } = δba

we get

¯ b } = {(γ μ θ)a ∂μ , ∂/∂θb } {La , L

+{(γ5 �)ac ∂/∂θc , −(γ5 �γ μ θ)b ∂μ }

= (γ μ )ab ∂μ − (γ5 �)ac (γ5 �γ μ )bd δcd ∂μ = (γ μ )ab ∂μ − (γ5 �)ac (γ5 �γ μ )bc ∂μ

= (γ μ )ab ∂μ − [(γ5 �)γ μT (γ5 �)T ]ab ∂μ μ = (γab + γ5 �γμT γ5 �)∂μ μ = 2γab ∂μ

since γ5 �γ μT γ5 � =



e 0

0 −e

  (

0 σ

μT

σμT 0

Formally, we can write this anticommutation relation as



e 0

0 −e



= γμ

¯ = 2γ μ ∂μ {L, L} It is easy to see that the six matrices �, γ5 �, �γ μ , μ = 0, 1, 2, 3 form a basis for the space of all skew-symmetric 4 × 4 matrices. We can thus expand any such matrix X as X = a0 � + a1 γ5 � + bμ �γ μ where a0 , a1 , bμ are complex constants. To evaluate these constants, we note that T r(�2 ) = −4, T r(�γ5 �) = 0, T r(γμ �2 ) = 0, T r((γ5 �)(γ5 �)) = −4, T r((γ5 �)γμ �) = 0,

It follows that

T r(γν )�2 γ μ ) = −4δνμ a0 = −T r(�X)/4, a1 = −T r(γ5 �X)/4, bμ = −T r(γμ �X)/4

In particular, we find that since θθT is a 4 × 4 skew-symmetric matrix,

θθT = (−1/4)((θT �θ)� + (θT γ5 �θ)γ5 � + (θT γμ �θ)�γ μ ) Now let S(x, θ) be any superfield. It can be expanded in powers of θ upto fourth degree with the coefficients being functions of x only. It follows that we can expand it as S(x, θ) = C(x) + θT �ω(x) + θT �θM (x) + θT γ5 �θN (x) + θT �γ μ θVμ (x)+ (θT �θ)θT γ5 �β(x) + (θT �θ)2 K(x) This is a consequence of the fact that any first degree polynomial in θ can be expressed as θT �ω where ω has four components, any second degree polynomial in θ is a linear combination of θT �θ, θT γ5 �θ and θT �γ μ θ, μ = 0, 1, 2, 3 since if M is any symmetric matrix, then θT M θ = 0 and further that as noted above, �, γ5 � and �γ μ , μ = 0, 1, 2, 3 is a basis for

Quantum Mechanics for Scientists and Engineers 191

194

4 × 4 skew symmetric matrices. We now evaluate the change in the superfield S under a supersymmetry transformation αT L where α is a Fermionic 4-vector. We have αT L(C(x)) = αT (γ5 �∂/∂θ + γ μ θ∂μ )(C(x)) = αT γ μ θC,μ (x) αT L(θT �ω(x)) = α (−γ5 ω(x) + γ μ θθT �ω,μ (x)) T

αT L(θT �θM (x)) = = −2αT γ5 θM (x) + θT �θαT γ μ θM,μ (x))

Note that we have used the fact that θ �θ commutes with θ (two anticommutations make one commutation). T

αT L(θT γ5 �θN (x)) = −2αT θN (x) + θT γ5 �θαT γ μ θN,μ (x) αT L(θT �γ μ θVμ (x))

= αT (−2γ5 γ μ θVμ (x) + θT �γ μ θγ ν θVμ,ν (x)) = −2αT γ5 γ μ θVμ + θT �γ μ θαT γ ν θVμ,ν αT L(θT �θθT γ5 �β(x))

= −2αT γ5 θθT γ5 �β(x)) −θT �θαT β(x)

+θT �θγ μ θθT γ5 �β,μ (x) We now recall that

θθT = (−1/4)(θT �θ� + θT γ5 �θγ5 � + θT �γ μ θγμ �)

and hence αT L(θT �θθT γ5 �β(x)) = −2αT γ5 θθT γ5 �β(x)) −θT �θαT β(x)

−(1/4)θT �θαT γ μ (θT �θ� + θT γ5 �θγ5 � + θT �γ μ θγμ �)γ5 �β,μ (x) = −2αT γ5 θθT γ5 �β(x)) −θT �θαT β(x)

Now using

−(1/4)θT �θ(θT �θαT γ μ � + θT γ5 �θαT γ μ γ5 � + θT �γ ν θαT γ μ γν �)γ5 �β,μ (x) θT �θ.θT γ5 �θ = 0, θT �θ.θT �γ ν θ = 0

we get

αT L(θT �θθT γ5 �β(x)) = −2αT γ5 θθT γ5 �β(x)) −θT �θαT β(x)

+(1/4)(θT �θ)2 αT γ μ γ5 β,μ (x) Finally, we find that αT L((θT �θ)2 K(x)) = αT (−4γ5 θθT �θK(x))

Quantum Mechanics for Scientists and Engineers 192

195

Now denoting the inifnitesimal supersymmetry transformation by δ = αT L we obtain on comparing the coefficients of different powers of θ, the following transformation rules for the components of the superfield under a supersymmetry transformation: δC(x) = −αT γ5 ω(x) θT �δω(x) = αT γ μ θC,μ (x) −2αT γ5 θM (x) − 2αT θN (x) −2αT γ5 γ μ θVμ (x)

or equivalently,

δω(x) = −γ μ �αC,μ (x) +2γ5 �αM (x) +2�αN +2γ μ �γ5 αVμ (x)

where we have used the fact that γ μ � and � are skew symmetric. Further, θT �θδM (x) + θT γ5 �θδN (x) + θT �γ μ θδVμ (x) = αT γ μ θθT �ω,μ (x)) −2αT γ5 θθT γ5 �β(x)) from which we deduce by expanding θθ as earlier, T

−θT �θαT β(x) δM (x) =

(1/4)αT γ μ ω,μ (x) −(3/2)αT β(x),

δN (x) = (1/4)θT γ μ γ5 ω,μ (x) +(1/2)αT γ5 β(x) δVμ (x) = (1/4)αT γ ν γμ ω,ν (x)) (1/2)αT γμ β(x) where we have used the fact that γ5 commutes with � and anticommutes with γ μ . Note that if we have an equation of the form a(x)θθT =a quadratic polynomial in θ, then we can expand θθT as a linear combination of θT �θ, θT γ5 �θ and θT �γ μ θ and equate the corresponding coefficients. We also have the useful fact that if any two of these six distinct quadratic polynomials are multiplied, we get zero. Next, equating cubic terms in θ, (θT �θ)θT γ5 �δβ(x) = θT �θαT γ μ θM,μ (x)) +θT γ5 �θαT γ μ θN,μ (x) +θT �γ μ θαT γ ν θVμ,ν −4αT γ5 θθT �θK(x))

Quantum Mechanics for Scientists and Engineers 193

196

from which we deduce again by the above mentioned principles, (multiply both sides by θ, expand θθT as above and use the fact that the product of this quantity with θT γ5 �θ is zero and also its product with θT �γ μ θ is zero: (1/4)(θT �θ)2 γ5 δβ(x) = −(1/4)θT �θ)2 �γ μT αM,μ (x)

−(1/4)(θT γ5 �θ)2 γ5 �γ μT αN,μ (x)

(−1/4)(θT �γ μ )θ)(θT �γ ρ θ)γρ �γ νT αVμ,ν +(θT �θ)2 �γ5 αK(x) Equivalently,

γ5 δβ(x) = −�γ μT αM,μ (x) + 4�γ5 αK(x) +γ5 �γ μT αN,μ (x)

−η μρ γρ �γ νT αVμ,ν

Here, we make use of the following easily verifiable identities:

(θT �θ)2 = 8θ0 θ1 θ2 θ3 (θT γ5 �θ)2 = −8θ0 θ1 θ2 θ3 (θT �γ μ θ)(θT �γ ν θ) = = 8θ0 θ1 θ2 θ3 η μν Multiplying both sides of the above identity by γ5 gives us δβ(x) = −γ5 �γ

μT

αM,μ (x) + 4�αK(x)

+�γ μT αN,μ (x)

−γ5 γ μ �γ νT αVμ,ν

Finally, equating the fourth degree term in θ gives us

(θT �θ)2 δK(x) = (1/4)(θT �θ)2 αT γ μ γ5 β,μ (x) or equivalently, Now writing and

δK(x) = (1/4)αT γ μ γ5 β,μ (x) β(x) = λ(x) + a.γ μ ω,μ (x) K(x) = D(x) + b∂ μ ∂μ C(x)

and using {γ5 , γ μ } = 0 and {γ μ , γ ν } = η μν I4 , we get δD(x) + b.∂ μ ∂μ δC(x) = (1/4)αT γ ν γ5 (λ,ν + a.γ μ ω,μν ) So we would expect that

= (−1/4)αT γ5 γ ν λ,ν − (a/8)γ5 ∂ μ ∂μ ω δD(x) = (−1/4)αT γ5 γ ν λ,ν , bδC(x) = (−a/8)αT γ5 ω(x)

Quantum Mechanics for Scientists and Engineers 194

197

On the other hand, we have already seen that δC(x) = −αT γ5 ω(x) So we must choose

a = 8b

We note that δD(x) is a perfect four divergence and hence its integral over the whole of four dimensional space-time  is zero. Thus, D(x)d4 x is invariant under the supersymmetry transformation δ = αT L and hence is a candidate Lagrangian density for supersymmetric theories. We also have δβ(x) = δλ(x) + a.γ μ δω,μ (x) −γ5 �γ μT αM,μ (x) +�γ μT αN,μ (x)

−γ5 γ μ �γ νT αVμ,ν

+4�α(D(x) + b∂ μ ∂μ C(x)) So,

δλ(x) = −γ5 �γ μT αM,μ (x) +�γ μT αN,μ (x)

−γ5 γ μ �γ νT αVμ,ν

+4�α(D(x) + b∂ μ ∂μ C(x)) −a.γ μ (−γ ν �αC,ν (x)

+2γ5 �αM (x) + 2�αN +2γ ν �γ5 αVν (x)),μ or equivalently using

γ5 γ μ = −γ μ γ5 ,

and

�γ μT = γ μ �,

we get

δλ = (1 − 2a)(γ μ γ5 �αM,μ +(1 − 2a)γ μ �αN,μ

+(γ γ − 2aγ ν γ μ )�γ5 αVμ,ν μ ν

+4�α(D + b∂μ ∂ μ C) + aγ μ γ ν C,μν and hence, it follows by taking a = 1/2 and noting that (γ μ γ ν C,μν = (1/2)∂μ ∂ μ C, that δλ = [γ μ , γ ν ]�γ5 αVμ,ν + 4�αD Suppose we choose D = 0, λ = 0. Then, It follows then that if Vμ,ν − Vν,μ = 0, then

δD = 0, δλ = [γ μ , γ ν ]�γ5 αVμ,ν δλ = 0

This means that a superfield S[x, θ] with the constraints D = 0, λ = 0 after a supersymmetry transformation, again maintains the same constraints provided that Vμ,ν − Vν,μ = 0, ie, provided that Vμ (x) = ∂μ Z(x)

Quantum Mechanics for Scientists and Engineers 195

198

for some scalar field Z(x). A superfield S with these constraints, ie, D = 0, λ = 0 is called a Chiral field. We denote such a superfield by Φ(x, θ). Remark: Apparently, some errors in the signs of the variables occur above. These are easily rectified if we take into account that α is a Fermionic parameter vector which anticommutes with θ. We would then have αT θ = −θT α and more generally for a bosonic matrix A, we would have αT Aθ = −θT Aα. [18] Quantum Control in the sense of Belavkin and Luc Bouten: The HP equation describing unitary evolution in the tensor product of the system Hilbert space and the Boson Fock space of the noisy bath is given by dU (t) = (−i(H + L∗ L/2)dt + LdA∗ (t) − L∗ SdA(t) + (S − I)dΛ(t))U (t) where S ∗ S = I, H ∗ = H. Using quantum Ito’s formula dA.dA∗ = dt, dAdΛ = dA, dΛ.dA∗ = dA∗ , (dΛ)2 = dΛ∗ = dΛ with all the other product differentials zero (ie, of o(dt)). The corresponding quantum process equation describing noisy evolution of a system observable X coupled to the bath is given by jt (X) = U (t)∗ XU (t) = U (t)∗ (X ⊗ I)U (t) Application of quantum Ito’s formula gives djt (X) = jt (θ0 (X))dt + jt (θ1 (X))dA(t) + jt (θ2 (X))dA(t)∗ + jt (θ3 (X))dΛ(t) where the structure maps θk , k = 0, 1, 2, 3 of this Evans-Hudson flow are given by θ0 (X) = i[H, X] − (1/2)(L∗ LX + XL∗ L − 2L∗ XL) θ1 (X) = [L∗ , X]S, θ2 (X) = S ∗ [X, L], θ3 (X) = S ∗ XS − X

The measurement process is assumed to be

Yout (t) = U (t)∗ Yin (t)U (t), Yin (t) = A(t) + A(t)∗ This is a non-demolition measurement in the sense of Belavakin: [Yout (t), Yout (s)] = [Yout (t), js (X)] = 0, s ≥ t We have by quantum Ito’s formula, dYout (t) = dYin (t) + jt (S − I)dA + jt (S ∗ − I)dA∗ + jt (L + L∗ )dt = jt (S)dA + jt (S ∗ )dA∗ + jt (L + L∗ )dt Let and

ηout (t) = {Yout (s) : s ≤ t} πt (X) = E(jt (X)|ηout (t))

where expectations are taken w.r.t. the state |f ⊗ ψ(β) > where |f >∈ h, < f, f >= 1 with h being the system Hilbert space in which the operators H, S, L, X are defined and |ψ(β) >= exp(− � β �2 /2)|e(β) > with |e(β) > being the exponential vector ∞  √ β ⊗n / n! |e(β) >= n=0

where β ∈ L2 (R+ ). If β = 0, then the bath is in the vacuum state. Belavkin’s filtering equation is dπt (X) = πt (Lt X) + (πt (XMt + Mt∗ X) − πt (Mt + Mt∗ )πt (X))dW (t) where

¯ dW (t) = dYout (t) − (πt (Mt + Mt∗ ) + β(t) + β(t))dt

Quantum Mechanics for Scientists and Engineers 196

and

199

Mt = L + (S − I)β(t),

2 ¯ Lt X = θ0 (X) + β(t)θ1 (X) + β(t)θ 2 (X) + |β(t)| θ3 (X)

In the special case when β = 0 (vacuum state of the bath), we get

Mt = L, Lt X = θ0 (X) = i[H, X] − (1/2)(L∗ LX + XL∗ L − 2L∗ XL) Writing

πt (X) = T r(ρ(t)X)

where ρ(t) can be regarded as a classical random process with values in the space of density operators in the system Hilbert space h, we can express the Belavkin equation as dρ(t) = L∗t (ρ(t))dt + ((Mt ρ(t) + ρ(t)Mt∗ ) − T r(ρ(t)(Mt + Mt∗ ))ρ(t))(dYout (t) − T r(ρ(t)(Mt + Mt∗ )) + 2Re(β(t)))dt) It is easy to see that where

∗ 2 ∗ ¯ L∗t (ρ) = θ0∗ (ρ) + β(t)θ1∗ (ρ) + β(t)θ 2 (ρ) + |β(t)| θ3 (ρ)

θ0∗ (ρ) = −i[H, ρ] − (1/2)(L∗ Lρ + ρL∗ L − 2LρL∗ ) θ1∗ (ρ) = SρL∗ − L∗ Sρ = [Sρ, L∗ ], θ3∗ (ρ) = SρS ∗ − ρ

Note that for any map θ in B(h), the Banach space of bounded operators in h, we define its dual map θ∗ by the prescription T r(Y θ(X))T r(θ∗ (Y )X) Sometimes it is more convenient to define the dual map by the prescription T r(Y ∗ θ(X)) = T r(θ∗ (Y )∗ X) This is true especially if the operators in question are non-Hermitian. This latter definition can be expressed as < Y, θ(X) >=< θ∗ (Y ), X > where < ., . > is the inner product on B(h) defined by < X, Y >= T r(X ∗ Y ) We note that

and hence

dW (t) = dYout (t) − (πt (Mt + Mt∗ ) + 2Re(β(t)))dt

= jt (S)dA(t) + jt (S ∗ )dA(t)∗ + jt (L + L∗ )dt − (πt (Mt + Mt∗ ) + 2Re(β(t))dt E(dW (t)|ηout (t)) = ¯ πt (S)β(t)dt + πt (S ∗ )β(t)dt − πt (L + L∗ )dt − πt (Mt + Mt∗ )dt − 2Re(β(t))dt =0

and further,

(dW (t))2 = jt (SS ∗ )dt = dt

by quantum Ito’s formula. Hence, the innovations process {W (t) : t ≥ 0} is a classical Brownian motion w.r.t. the filtration {ηout (t), t ≥ 0}. We now assume that t = 0 so that t + dt = dt. We choose a Hermitian system observable Z and apply the control unitary c = exp(iZdY (t)) Udt where Y = Yout . Since t = 0, jt = j0 = I and hence dY (t) = SdA(t) + S ∗ dA(t)∗ + (L + L∗ )dt − ((Mt + Mt∗ ) + 2Re(β(t))dt ¯ = S(dA(t) − β(t)dt) + S ∗ (dA(t)∗ − β(t)dt)

Quantum Mechanics for Scientists and Engineers 197

200

Note that in the general case, ie, t > 0, we have dW (t) = jt (S)dA(t) + jt (S ∗ )dA(t)∗ + jt (L + L∗ )dt − (πt (Mt + Mt∗ ) + 2Re(β(t))dt ¯ = (jt (S)dA − πt (S)β(t)dt) + (jt (S ∗ )dA∗ − πt (S ∗ )β(t)dt) +(jt (L + L∗ ) − πt (L + L∗ ))dt

We also note that

dYout (t) = dW (t)

in the special case when β = 0 and L = −L. The resulting state at time t + dt = dt after applying the control under the condition β = 0 (so that Mt = L) is given by ∗

ρc (dt) = Uc (dt)ρ(dt)dUc (dt)∗ (I + iZdY − Z 2 dt/2)(ρc (0) + L∗ (ρc (0))dt + ((Lρc (0) + ρc (0)L∗ ) − T r(ρc (0)(L + L∗ ))ρc (0))dW (t))(I − iZdY − Z 2 dt/2) [19] Proof of the Shannon Cq coding theorem. A is a finite alphabet and for each x ∈ A, ρ(x) is a density matrix in the finite dimensional Hilbert space H = CN . Let P be a probability distribution on A and define the set of all δ-typical sequences of length n:  T (P, n, δ) = {u ∈ An : |N (x|u) − nP (x)| ≤ δ nP (x)(1 − P (x))∀x ∈ A}

Here, N (x|u) is the number of times x occurs in the sequence u. It is clear that u ∈ T (P, n, δ) implies 2−nS(P )−Kδ where

S(P ) = −



√ n

≤ P ⊗n (u) = Πx∈A P (x)N (x|u) ≤ 2−nS(P )+Kδ

P (x)log(P (x)), K =

x∈A

√

n



P (x)(1 − P (x))log(P (x))

x∈A

The greedy algorithm: Choose the maximal integer M such that for a given probability distribution Pon A and M given � > 0 there exist sequences u1 , ..., uM ∈ T (P, n, δ) and operators D1 , ..., DM ≥ 0 in H such that i=1 Di ≤ I, T r(ρ(ui )Di ) ≥ 1 − �∀i and Di ≤ E(n, ui , δ)∀i. Here,  E(ρ(x)⊗N (x|u) , δ) E(n, u, δ) = x∈A

where if ρ is any density matrix and n any positive integer, then if  ρ= |i > p(i) < i| i

is the spectral representation of ρ, then 

E(ρ⊗n , δ) =

N (i|i1 ,...,in )−np(i)|≤δ

It is easily seen that where

2−nS(ρ)−δK1 S(ρ) = −

 i

This simple fact is based on the identity

√

n

√

np(i)(1−p(i))∀i

≤ ρ⊗n E(ρ⊗n , δ) ≤ 2−nS(ρ)+δK1

p(i)log(p(i)), K1 =

 i

√

n

p(i)(1 − p(i))log(p(i))



p(i1 )...p(in ) = Πi p(i)N (i|i1 ,...,in ) = 2 and

|i1 , ..., in >< i1 , ..., in |

i

N (i|i1 ,...,in )log(p(i))

ρ⊗n |i1 , ..., in >= p(i1 )...p(in )|i1 , ..., in >

Quantum Mechanics for Scientists and Engineers 198

201

It follows that on setting ρ⊗n (u) =



ρ(x)⊗N (x|u)

x∈A

we get 2−n



x∈A

Pu (x)S(ρ(x))−δ



x∈A

K1 (x)

√

N (x|u)

≤ ρ⊗n (u)E(n, u, δ) ≤ 2−n

where



x∈A

Pu (x)S(ρ(x))+δ



x∈A

K1 (x)

√

N (x|u)

Pu (x) = N (x|u)/n

and

 K1 (x) = Pρ(x) (i)(1 − Pρ(x) (i))log(Pρ(x) (i)) i

where

ρ(x) =

 i

is the spectral representation of ρ(x). Writing

|i > Pρ(x) (i) < i|

K0 =



K1 (x)

x∈A

it follows that

2−n



x∈A

Pu (x)S(ρ(x))−δK0

√

n

Remark: If u ∈ T (P, n, δ), then

≤ ρ⊗n (u)E(n, u, δ) ≤ 2−n

|N (x|u) − nP (x)| ≤ δ and hence and so

P (x) − δ







x∈A

Pu (x)S(ρ(x))+δK0

√

n

nP (x)(1 − P (x))∀x ∈ A

P (x)(1 − P (x))/n ≤ Pu (x) = N (x|u)/n ≤ P (x) + δ



P (x)(1 − P (x))/n, x ∈ A

√ √ P (x) − δ/ 2n ≤ Pu (x) ≤ P (x) + δ/ 2n, x ∈ A

In other words, for any � > 0, there exists n0 such that n ≥ n0 implies

|Pu (x) − P (x)| ≤ �, x ∈ A, u ∈ T (P, n, δ) where δ > 0 is given. It follows from this observation and the above inequality that if u ∈ T (P, n, δ), then there is a K2 independent of n such that 2−n



x∈A

P (x)S(ρ(x))−δK2

√

n

≤ ρ⊗n (u)E(n, u, δ) ≤ 2−n

For example, we can take K2 = K0 +





x∈A

P (x)S(ρ(x))+δK2

√

n

(P (x)(1 − P (x))1/2 S(ρ(x))

x∈A

A lower bound on M in the greedy algorithm: M Lemma: In the greedy algorithm, define i=1 Di = D. Then, T r(ρ(u)D) ≥ γ∀u ∈ T (P, n, δ). For suppose T r(ρ(u)D) ≤ γ for some u ∈ T (P, n, δ). Then since γ ≤ 1 − �, it follows that u ∈ / {u1 , ..., uM }. Now define D� = (1 − D)1/2 E(n, u, δ)(1 − D)1/2 Then, (writing ρ(u) for ρ⊗n (u) = ⊗x∈A ρ(x)⊗N (x|u) ), we have T r(ρ(u)D� ) = T r(ρ(u)E(n, u, δ)) − T r(ρ(u)(E(n, u, δ) − (1 − D)1/2 E(n, u, δ)(1 − D)1/2 )) ≥ T r(ρ(u)E(n, u, δ))− � ρ(u) − (1 − D)1/2 ρ(u)(1 − D)1/2 �1 ≥ T r(ρ(u)E(n, u, δ)) − α.T r(ρ(u)D) ≥ T r(ρ(u)E(n, u, δ)) − αγ

Now observe that for any density ρ,

  {|N (i|i1 , ..., in ) − nPρ (i)| ≤ δ nPρ (i)(1 − Pρ (i)))

T r(ρ⊗n E(ρ⊗n , δ)) = Pρ⊗n (

i

Quantum Mechanics for Scientists and Engineers 199

202

≥1−

 i

P (|N (i|i1 , ..., in ) − nPρ (i)| ≥ δ



nPρ (i)(1 − Pρ (i))

≥ 1 − N/δ 2 − − − (1)

where we have used the union bound, the Chebyshev inequality and the fact that for each i, N (i|i1 , ..., in ) is a random  variable having mean nPρ (i) and variance nPρ (i)(1 − Pρ (i)). Note that we also have  P (T (P, n, δ)c ) = P (u ∈ An : |N (x|u) − nP (x)| > δ nP (x)(1 − P (x))f orsomex ∈ A)   ≤ P (u ∈ An : |N (x|u) − nP (x)| > δ nP (x)(1 − P (x))) x∈A

by the union bound and Chebyshev’s inequality. Thus,

≤ a/δ 2

P (T (P, n, δ)) ≥ 1 − a/δ 2 From the above arguments (ie equn(1)), T r(ρ(u)E(n, u, δ)) = Πx∈A T r(ρ(x)⊗N (x|u) E(ρ(x)N (x|u) , δ)) ≥ (1 − N/δ 2 )a

Note:If δ = O(n ) where 0 < s < 1, then (1 − N/δ 2 )a , then it follows that T r(ρ(u)E(n, u, δ)) can be made as close  √ θ to unity as possible by choosing n sufficiently large and quantities like 2−nS(ρ)±K δ n are of the order 2−nS(ρ)±K”n where 0 < θ < 1. Equivalently, the logarithm of these quantities divided by n would converge to −S(ρ) as n → ∞ with the probabilities T r(ρ(u)E(n, u, δ)) converging to zero. We have now derived the inequality s/2

T r(ρ(u)D� ) ≥ T r(ρ(u)E(n, u, δ)) − αγ ≥ (1 − N/δ 2 )a − αγ ≥ 1 − � if for sufficiently large n provided δ is allowed to vary with n as in the above note and γ < �/α is assumed. We thus get a contradiction to the maximality of M on using the fact that 0 ≤ D� ≤ 1 − D ie

D + D� ≤ 1

This completes the proof of the lemma. A lower bound on M: Lemma: Let 0 ≤ Z, T ≤ be operators and ρ a density operator in a Hilbert space such that for some θ > 0, we have ρT ≤ θT . Then, T r(Z) ≥ θ−1 (T r(ρZ) − T r(ρ(1 − T )))

An intuitive interpretation of this inequality: Z, T are viewed as events. If Z occurs, then T may or may not occur. If Z and T occur, then the probability of this is the trace of ρT ∩ Z which is smaller than the trace of θT ∩ Z which in turn is smaller than the trace of θZ. If Z occurs and T does not occur then the probability of this is the trace of ρZ ∩ (1 − T ) which is smaller than the trace of ρ(1 − T ). Combining these two facts, we get T r(ρZ) ≤ θT r(Z) + T r(ρ(1 − T )) and the proof of the Lemma is complete. Now define

ρ¯ =



P (x)ρ(x)

x∈A

Then, we have where

¯ 3δ ρ⊗n , δ) ≤ 2−nS(ρ)+K ρ¯⊗n E(¯

K3 =

√ n

E(¯ ρ⊗n , δ)

 Pρ (i)(1 − Pρ (i))log(Pρ (i)) j

Quantum Mechanics for Scientists and Engineers 200

203

Hence by the above lemma, ¯ 3δ T r(D) ≥ 2nS(ρ)−K

√

n

(T r(¯ ρ⊗n D) − T r(¯ ρ(¯ ρ⊗n (1 − E(¯ ρ⊗n , δ)))

Now, since



ρ¯⊗n =

P ⊗n (u)ρ(u)

u∈An

it follows that

and further,

Thus, On the other hand,

T r(¯ ρ⊗n D) ≥



P ⊗n (u)T r(ρ(u)D)

u∈T (P,n,δ)

≥ γP ⊗n (T (P, n, δ)) ≥ γ(1 − N/δ 2 ) T r(¯ ρ⊗n E(¯ ρ⊗n , δ)) = Pρ¯⊗n (T (Pρ¯, n, δ)) ≥ 1 − N/δ 2 ¯ 3δ T r(D) ≥ 2nS(ρ)−K

T r(D) =

M 

T r(Di ) ≤

i=1

≤ M.2n

Combining the above two inequalities gives us



x∈A

¯ M ≥ (1 − 2N/δ 2 )2n(S(ρ)−



√

n

(1 − 2N/δ 2 )

M 

T r(E(n, ui , δ))

i=1

√ P (x)S(ρ(x))+K5 δ n

x∈A

√ Pu (x)S(ρ(x)))−K5 δ n

− − − (2)

which is the desired lower bound on M . Remark: Here we have made used of the following fact:  ρ(x)⊗N (x|u) E(ρ(x)⊗N (x|u) , δ) ρ(u)E(n, u, δ) = x∈A

≥ (Πx∈A 2

−N (x|u)S(ρ(x))−δ

where as defined earlier

K1 (x) =

√

N (x|u))K1 (x)

E(n, u, δ)

 Pρ(x) (i)(1 − Pρ(x) (i))log(Pρ(x) (i) i

It follows that

T r(E(n, u, δ)) ≤ 2n



x∈A

√ Pu (x)S(ρ(x))+K5 δ n

and hence (2) is obtained. Note that Pui (x) differs from P (x) by an amount of order  K1 (x) K5 =

√

n since ui ∈ T (P, n, δ). Here,

x∈A

An upper bound on M: Consider the spectral representation of ρ¯:  ρ¯ = |i > Pρ¯(i) < i| i

and with respect to the onb {|i >: 1 ≤ i ≤ N } of this representation, define the density operators  |i >< i|ρ(x)|i >< i|, x ∈ A ρ˜(x) = i

˜ Then, ρ˜(x), x ∈ A form a commuting family of operators in H = CN . Define the typical projections E(n, u, δ) in the usual way corresponding to this family ρ˜(x), x ∈ A. In other words, if |j1 , ..., jN (x|u) > are the eigenvectors of ρ˜(x)⊗N (x|u) , then  ρ˜(x)⊗N (x|u) ρ˜(u) = x∈A

Quantum Mechanics for Scientists and Engineers 201

204

˜ u, δ) = E(n,



E(˜ ρ(x)⊗N (x|u) , δ)

x∈A

=





x∈A (j1 ,...,jN (x|u) )∈T (j,x)∀j

where with

|j1 , ..., jN (x|u) >< j1 , ..., jN (x|u) |

T (j, x) = {(j1 , ..., jN (x|u) ) : N (j|j1 , ..., jN (x|u) ) − N (x|u)p(j|x)| ≤ δ

ie the probability of the j

p(j|x) =< j|ρ(x)|j > th



N (x|u)p(j|x)(1 − p(j|x))}

eigenvector of ρ¯ occurring given that the system is in the state ρ(x). Note that we can write  ˜ u, δ) = E(n, |v >< v|

where the summation is over all n-length sequences v of the form  v= (j1 , ..., jN (x|u) ) x∈A

with

(j1 , ..., jN (x|u) ) ∈ T (j, x)∀j = 1, 2, ..., N

It immediately follows that

N (j|v) =



N (j|j1 , ..., jN (x|u) )

x∈A

Suppose Pu (x) = P (x), x ∈ A. Then |N (j|v) − nPρ¯(j)| = | ≤



x∈A

≤



x∈A

N (j|j1 , ..., jN (x|u) ) −



nP (x)p(j|x)|

x∈A

|N (j|j1 , ..., jN (x|u) − p(j|x)N (x|u)|

  δ N (x|u)p(j|x)(1 − p(j|x))

x∈A

√  P (x)p(j|x)(1 − p(j|x)) =δ n x∈A

 √  P (x))1/2 .( p(j|x)(1 − p(j|x)))1/2 ≤ δ n( x

x

√ ≤ δ an

From this, it follows that

√ v ∈ T (Pρ¯, n, δ a)

Thus, we have proved that

√ ˜ u, δ) ≤ E(¯ E(n, ρ⊗n , δ a)

whenever Pu (x) = P (x)∀x ∈ A. Now define

˜ ui , δ)Di E(n, ˜ ui , δ), i = 1, 2, ..., M Di� = E(n, Then, Di� ≤

M 

√ √ E(¯ ρ⊗n , δ a)Di E(¯ ρ⊗n , δ a)

i=1

√ √ ρ⊗n , δ a) = E(¯ ρ⊗n , a δ)DE(¯ ≤I

Quantum Mechanics for Scientists and Engineers

205

Now, we observe that



ρ(u)E(n, u, δ) =

ρ(x)⊗N (x|u) E(ρ(x)⊗N (x|u) , δ)

x∈A

≤ (Πx∈A 2−N (x|u)S(ρ(x))+δ

where as defined earlier

K1 (x) =

√

N (x|u))K1 (x)

E(n, u, δ)

 Pρ(x) (i)(1 − Pρ(x) (i))log(Pρ(x) (i) i

and hence

ρ(u)E(n, u, δ) ≤ 2−n



x

√ Pu (x)S(ρ(x))+K5 δ n

E(n, u, δ)

and if we assume further that u ∈ T (P, n, δ), then we get ρ(u)E(n, u, δ) ≤ 2−n since for such u,



x

√ P (x)S(ρ(x))+K6 δ n

E(n, u, δ) − − − (3)

√ |Pu (x) − P (x)| ≤ δ/ 2n

as observed earlier. It follows from (3) and the above lemma that T r(Di� ) ≥ 2n Now we have



x

√ P (x)S(ρ(x))−K6 δ n

(T r(ρ(ui )Di� ) − T r(ρ(ui )(1 − E(n, ui , δ)))

˜ ui , δ)ρ(ui )E(n, ˜ ui , δ))Di ) T r(ρ(ui )Di� ) = T r(ρ(ui )Di ) − T r((ρ(ui ) − E(n, ˜ ui , δ)ρ(ui )E(n, ˜ ui , δ) �1 ≥ 1 − �− � ρ(ui ) − E(n, ˜ ui , δ))) ≥ 1 − � − βT r(ρ(ui )(1 − E(n,

Let π denote any specific rank one projection of the form |i1 , ..., in >< i1 , ..., in | where |i1 , ..., , in > are eigenvectors of ρ¯⊗n obtained by tensoring the orthonormal basis of eigenvectors of ρ¯. Then, we have  ˜ ui , δ)) = ˜ ui , δ)π) T r(ρ(ui )π E(n, T r(ρ(ui )E(n, π

=



˜ ui , δ)) T r(πρ(ui )π E(n,

π

˜ ui , δ)) = T r(˜ ρ(ui )E(n, ≥ 1 − N/δ 2

Thus,

˜ ui , δ)) ≤ N/δ 2 T r(ρ(ui )E(n,

and also Thus, we get

T r(ρ(ui )E(n, ui , δ)) ≤ N/δ 2 T r(Di� ) ≥ 2n = 2n

It follows that T r(

M  i=1





x

x

√ P (x)S(ρ(x))−K6 δ n √

P (x)S(ρ(x))−K6 δ n

Di� ) ≥ M.2n



x

(1 − � − βN/δ 2 − N/δ 2 )

(1 − � − (β + 1)N/δ 2 )

√ P (x)S(ρ(x))−K6 δ n

(1 − � − (β + 1)N/δ 2 )

on the one hand and on the other,

T r(

M  i=1

Di� ) =

M 

˜ ui , δ)Di E(n, ˜ ui , δ)) T r(E(n,

i=1

We have already noted that if P = Pu , then √ ˜ u, δ) ≤ E(¯ E(n, ρ⊗n , δ a)

Quantum Mechanics for Scientists and Engineers 203

206

so if we assume that Pui = P ∀i, then we get T r(

M  i=1

and hence we get

√ √ Di� ) ≤ T r(E(¯ ρ⊗n , δ a)DE(¯ ρ⊗n , δ a) √ ≤ T r(E(¯ ρ⊗n , δ a)) ¯ 7δ ≤ 2nS(ρ)+K

¯ 7 2nS(ρ)+K

√

an

≥ M.2n



x

√

an

√ P (x)S(ρ(x))−K6 δ n

(1 − � − (β + 1)N/δ 2 )

from which we get the upper bound on M in the special case when Pui = P ∀i: 

¯ M ≤ (1 − � − (β + 1)N/δ 2 )−1 2n(S(ρ)−

x

P (x)S(ρ(x)))+(K6 +K7

√

√ a)δ n

Suppose now that u1 , ..., uM , D1 , ..., DM are as in the greedy algorithm. Let Q be any empirical probability distribution on A corresponding to the integer n. This means that there is a sequence u ∈ An such that Q(x) = N (x|u)/n, x ∈ A. It is clear that the number of empirical probability distributions on A corresponding to n cannot exceed (n + 1)a . (Each symbol x ∈ A can occur in a sequence of length n, k times where k = 0, 1, ..., n). We now consider the subset FQ of all integers 1, 2, ..., M such that ui is of empirical type Q, ie, N (x|ui )/n = Q(x), ∀x ∈ A. Let MQ denote the cardinality of FQ . Then it is clear that  M= MQ Q

a where the summation is over all empirical distributions Q, there being  atmost (n + 1) of them. By the above inequality, we have since T r(ρ(ui )Di ) ≥ 1 − �, Di ≤ E(n, ui , δ), ∀i ∈ FQ and i∈FQ Di ≤ I,

MQ ≤ (1 − � − (β + 1)N/δ 2 )−1 2n(S(ρ¯Q )−



x

Q(x)S(ρ(x)))+(K6 +K7

≤ 1 − � − (β + 1)N/δ 2 )−1 2nC+(K6 +K7

where



ρQ =

√

√

√ a)δ n

√ a)δ n

Q(x)ρ(x)

x

and



C = maxP (S(

x

P (x)ρ(x)) −



P (x)S(ρ(x)))

x

the maximum being taken over all probability distributions P on A. Summing this over all empirical distributions Q of length n on A gives us  √ √ M= MQ ≤ (n + 1)a .(1 − � − (β + 1)N/δ 2 )−1 2nC+(K6 +K7 a)δ n Q

which is the desired upper bound on M . √ A remark: Now suppose we do not assume Pui = P ∀i. Since ui ∈ T (P, n, δ), we have |Pui (x) − P (x)| ≤ δ/ 2n∀i, x. ˜ u, δ) in the same way as above except that we replace P by Pu . Then, by the same arguments For any u, we define E(n, as above, we have √ ˜ u, δ) ≤ E(¯ E(n, ρ⊗n u , δ a) where

ρ¯u =



Pu (x)ρ(x)

x∈A

˜ leading to the results: We proceed in the same way as above by defining Di� in terms of the new E   ˜ ui , δ)Di E(n, ˜ ui , δ) D� = Di� = E(n, i

i

= E(¯ ρ

⊗n

√ √ ρ⊗n , δ a)+ , δ a)DE(¯

Quantum Mechanics for Scientists and Engineers

 i

where

207

¯ i (Ei − E) ¯ + (Ei − E)D

 i

¯ iE ¯ ¯ + ED ¯ i (Ei − E) (Ei − E)D

√ ˜ ui , δ), E ¯ = E(¯ Ei = E(n, ρ⊗n , δ a)

Thus, in terms of norms,

¯ + 2.maxi � Ei − E ¯ �2 +maxi � Ei − E �1 T r(D� ) ≤ T r(E) 1

˜ u, δ) Note that ui ∈ T (P, n, δ). Now if u ∈ T (P, n, δ), then Pu is close to P , hence ρ¯u will be close to ρ¯ and so E(n, ¯ and we would get that T r(D� ) cannot exceed T r(E) ¯ by an order of an ¯ Thus, Ei will be close to E will be close E. ¯ �2 cannot grow faster than an exponential of np where exponential of n provided that we are able to show that � Ei − E  √ ¯ x P (x)S(ρ(x))+Kδ n would follow. To make this more p < 1). Hence, the desired lower bound of the form M ≤ 2n(S(ρ)− precise, we observe that u ∈ T (P, n, δ) implies |Pu⊗n (v) − P ⊗n (v)| = |Πx∈A Pu (x)N (x|v) − Πx∈A P (x)N (x|v) | 

= |e

x



=≤ |exp( We can write

N (x|v)log(Pu (x))

x



−e

x

N (x|v)log(P (x))

|

N (x|v)(log(Pu (x)) − log(P (x))) − 1|

 √ Pu (x) = P (x) + f (x)/ n, x ∈ A, |f (x)| ≤ 1, f (x) = 0 x

Hence, 

|exp(

x

|Pu⊗n (v) − P ⊗n (v)| ≤  √ N (x|v)log(1 + f (x)/P (x) n)) − 1| = |exp( N (x|v)(f (x)/P (x) n − f (x)2 /2P (x)2 n + ...)) − 1| √

x

√ If v ∈ / T (P, n, δ), then N (x|v) < nP (x) + O( n) in which case we see that  √ √ N (x|v)(f (x)/P (x) n − f (x)2 /2P (x)2 n + ...) < O(1/ n) x

since



√ f (x) = 0. If v ∈ T (P, n, δ), then N (x|v) = nP (x) + g(x) n where |g(x)| < 1. Hence, in this case,   √ √ N (x|v)(f (x)/P (x) n − f (x)2 /2P (x)2 n + ..) = (g(x)f (x)/P (x) − f (x)2 /2P (x)) + O(1/ n) x

x

√ = c0 + O(1/ n)

Other useful inequalities are =|

 x

|log(Pu⊗n (v)) − log(P ⊗n (v))| =  N (x|v)log(Pu (x)) − N (x|v)log(P (x))| = |



x

√ N (x|v)(log(1 + f (x)/P (x) n)|

x

[20] Restricted quantum gravity in one spatial dimension and one time dimension. The metric is dτ 2 = (1 + 2U (t, x))dt2 − (1 + 2V (t, x))dx2 The position fields are U (t, x) and V (t, x) and to find the momentum fields, we must first evaluate the Lagrangian density √ β β α L = g μν −g(Γα μν Γαβ − Γμβ Γαβ )

This Lagrangian density is a function of U, V, U,μ , V,μ . Define the position fields as U, V and the canonical momentum fields as ∂L ∂L , πV = πU = ∂U,0 ∂V,0

Quantum Mechanics for Scientists and Engineers

208

Then, apply the Legendre transformation after solving for U,0 , V,0 in terms of U, V, ∇U, ∇V to get the Hamiltonian density as H(U, V, ∇U, ∇V, πU , πV ) = πU U,0 + πV V,0 − L Then, set up the Schrodinger wave equation  ( H(U (x), V (x), ∇U (x), ∇V (x), −iδ/δU (x), −iδ/δV (x))dx)ψt ({U (x), V (x) : x ∈ R) =i

∂ ψt ({U (x), V (x) : x ∈ R) ∂t

References [1] Rohit Singh, Naman Garg and H.Parthasarathy, ”Statistical modeling of transmission lines using stochastic differential equations with random loading effects along the line”, Technical report, NSIT, 2016. [2] Rohit Singh, Naman Garg and H.Parthasarathy, ”Quantization of transmission line equations using the GKSL formalism”, Technical report, NSIT, 2016. [3] Naman Garg, H.Parthasarathy and D.K.Upadhyay, ”Real time simulation of Hudson-Parthasarathy noisy Schrodinger equation and Belavkin equation”, Technical report, NSIT, 2016. [4] Naman Garg, H.Parthasarathy and D.K.Upadhyay, ”Real time control of the the Hudson-Parthasarathy noisy Schrodinger equation with observer obtained from Belavkin’s filter, Technical report, NSIT, 2016. [5] D.W.Stroock and S.R.S.Varadhan, Multidimensional Diffusion Processes, Springer, 1997. [6] L.D.Landau and E.M.Lifshitz, ”The classical theory of fields”, Butterworth and Heinemann. [7] Rohit Singh, Jyotsna Singh and H.Parthasarathy, ”Application of wavelets to partial differential equation based modeling of images, Technical Report, NSIT, 2016. [8] H.Parthasarathy, ”Time travel in the special and general theories of relativity and the notions of space and time in quantum gravity, Lecture delivered at the G.B.Pant University, 15th September, 2016. [9] Rohit Singh, Jyotsna Singh and H.Parthasarathy, ”Image parameter estimation using Edgeworth models for nonGaussian noise and nonlinear filtering theory in discrete time”, Technical Report, NSIT, 2016. [10] Rohit Singh, Naman Garg, Kumar Gautam and H.Parthasarathy, ”Design of quantum gates using scattering theory”, Technical Report, NSIT, 2016. [11] Lalit Kumar and H.Parthasarathy, ”The magnetic field produced by a non-uniform transmission line carrying current when nonlinear hysteresis and nonlinear capacitive effects are taken into account, Technical report, NSIT, 2016. [12] K.R.Parthasarathy, ”An introduction to quantum stochastic calculus”, Birkhauser, 1992. [13] K.R.Parthasarathy, ”Coding theorems of classical and quantum information theory”, Hindustan Book Agency. [14] K.R.Parthasarathy, ”Mathematical foundations of quantum mechanics”, Hindustan Book Agency. [15] J.Gough and Koestler, Quantum Filtering in Coherent States”. [16] S.Weinberg, The quantum theory of fields, Vols.1,2,3, Cambridge University Press. [17] Leonard Schiff, Quantum mechanics, TMH. [18] P.A.M.Dirac, ”The principles of quantum mechanics”, Oxford University Press. [19] S.Weinberg, Gravitation and Cosmology:Principles and applications of the general theory of relativity. [20] Mandel and Wolf, Optical coherence and quantum optics. [21] W.O.Amrein, A Hilbert space approach to quantum mechanics, CRC press. [22] T.Kato, Perturbation theory for linear operators, Springer.