Advanced Quantum Physics Lecture Handout

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Advanced Quantum Physics Lecture Handout

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Preface Quantum mechanics underpins a variety of broad subject areas within physics and the physical sciences from high energy particle physics, solid state and atomic physics through to chemistry. As such, the subject resides at the core of every physics programme. By building upon the conceptual foundations introduced in the IB Quantum Physics course, the aim of Part II Advanced Quantum Physics is to develop further conceptual insights and technical fluency in the subject. Although 24 lectures is a long course to prepare(!), it is still insufficient to cover all of the topics that different physicists will describe as “core”. For example, some will argue that concepts of quantum computation should already be included as an established component of the core. Simiilarly, in the field of quantum optics, some will say that a detailed knowledge of atomic and molecular spectroscopy is key. In the field of solid state physics, the concept of second quantization in many-body physics is also considered central. In all of these cases, we will be able to touch only the surface of the subject. However, the material included in this course has been chosen to cover the key conceptual foundations that provide access to these more advanced subjects, the majority of which will be covered in subequent optional courses in Part II and Part III. In the following, we list an approximate “lecture by lecture” synopsis of the different topics treated in this course. Those topics marked by a † will be covered if time permits. 1 Foundations of quantum physics: Overview of course structure and organization; brief revision of historical background: from wave mechanics to the Schr¨odinger equation. 2 Quantum mechanics in one dimension: Wave mechanics of unbound particles; potential step; potential barrier and quantum tunneling; bound states; rectangular well; δ-function potential well; † KronigPenney model of a crystal. 3 Operator methods in quantum mechanics: Operator methods; uncertainty principle for non-commuting operators; Ehrenfest theorem and the time-dependence of operators; symmetry in quantum mechanics; Heisenberg representation; postulates of quantum theory; quantum harmonic oscillator. 4 Quantum mechanics in more than one dimension: Rigid diatomic molecule; angular momentum; commutation relations; raising and lowering operators; representation of angular momentum states. 5 Quantum mechanics in more than one dimension: Central potential; atomic hydrogen; radial wavefunction. 6 Motion of charged particle in an electromagnetic field: Classical mechanics of a particle in a field; quantum mechanics of particle in a field; atomic hydrogen – normal Zeeman effect; † diamagnetic hydrogen and quantum chaos; gauge invariance and the Aharonov-Bohm effect; free electrons in a magnetic field – Landau levels. 7-8 Quantum mechanical spin: History and the Stern-Gerlach experiment; spinors, spin operators and Pauli matrices; relating the spinor to spin direction; spin precession in a magnetic field; parametric resonance; addition of angular momenta. Advanced Quantum Physics

ii 9 Time-independent perturbation theory: Perturbation series; first and second order expansion; degenerate perturbation theory; Stark effect; nearly free electron model. 10 Variational and † WKB method: Ground state energy and eigenfunctions; application to helium; excited states; † Wentzel-Kramers-Brillouin method. 11 Identical particles: Particle indistinguishability and quantum statistics; space and spin wavefunctions; consequences of particle statistics; ideal quantum gases; degeneracy pressure in neutron stars; Bose-Einstein condensation in ultracold atomic gases. 12-13 Atomic structure: Relativistic corrections; spin-orbit coupling; Darwin structure; Lamb shift; hyperfine structure; Multi-electron atoms; Helium; Hartree approximation and beyond; Hund’s rule; periodic table; coupling schemes LS and jj; atomic spectra; Zeeman effect. 14-15 Molecular structure: Born-Oppenheimer approximation; H2 +ion; H2 molecule; ionic and covalent bonding; molecular spectra; rotation; nuclear statistics; vibrational transitions. 16 Field theory of atomic chain: From particles to fields: classical field theory of the harmonic atomic chain; quantization of the atomic chain; phonons. 17 Quantum electrodynamics: Classical theory of the electromagnetic field; theory of waveguide; quantization of the electromagnetic field and photons. 18 Time-independent perturbation theory: Time-evolution operator; Rabi oscillations in two level systems; time-dependent potentials – general formalism; perturbation theory; sudden approximation; harmonic perturbations and Fermi’s Golden rule; second order transitions. 19 Radiative transitions: Light-matter interaction; spontaneous emission; absorption and stimulated emission; Einstein’s A and B coefficents; dipole approximation; selection rules; lasers. 20-21 Scattering theory I: Basics; elastic and inelastic scattering; method of particle waves; Born approximation; scattering of identical particles. 22-24 Relativistic quantum mechanics: History; Klein-Gordon equation; Dirac equation; relativistic covariance and spin; free relativistic particles and the Klein paradox; antiparticles and the positron; Coupling to EM field: gauge invariance, minimal coupling and the connection to nonrelativistic quantum mechanics; † field quantization.

Handout To accompany the course, a substantial handout has been prepared.1 In some cases, the handout contains material (usually listed as Info blocks) that goes beyond the scope of the lectures. Needless to say, the examination will be 1 I should note that, in preparing the handout, I have made use of some web-based material – particularly the excellent lecture notes by Fowler at Virginia – and notes prepared by David Ward in previous generations of the course. I have also included links to useful material on the course webpage.

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iii limited to material that is covered in lectures and not the handout. Since this handout is substantially new, it is inevitable that there will be some typographical errors – some of them may even be important... I would be most grateful if you could e-mail the errors that you find to bds10@cam. I will try to maintain a “corrected” set of notes on the web. The overheads used in lectures can also be recovered from the course web2 site along with other relevant and useful material.

Problem Sets The problem sets are a vital and integral part of the course providing the means to reinforce key ideas as well as practice techniques. Problems indicated by a † symbol are regarded as challenging. Throughout these notes, I have included a number of simpler exercises which may be completed “along the way”, and aim to reinforce some of the ideas developed in the text.

Books As a core of every undergraduate and graduate physics programme, there is a wealth of excellent textbooks on the subject. Choosing ones that suit is a subjective exercise. Apart from the handout, I am not aware of a text that addresses all of the material covered in this course: Most are of course more dense and far-reaching, and others are simply more advanced or imbalanced towards specialist topics. At the same time, I would not recommend relying solely on the handout. Apart from the range of additional examples they offer, the textbooks will often provide a more erudite and engaging discussion of the material. Although there are too many texts on the subject to discuss every one, I have included below some of the books that I believe to be particularly useful.

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http://www.tcm.phy.cam.ac.uk/~bds10/aqp.html

Advanced Quantum Physics

Bibliography [1] B. H. Bransden and C. J. Joachain, Quantum Mechanics, (2nd edition, Pearson, 2000). This is a classic text which covers core elements of advanced quantum mechanics. It is particularly strong in the area of atomic physics, but weaker on many-particle physics. [2] S. Gasiorowicz, Quantum Physics, (2nd edn. Wiley 1996, 3rd edition, Wiley, 2003). This is an excellent textbook that covers material at approximately the right level for the course. However, the published text (as opposed to the supplementary material available online) omits some topics which are addressed in this course. [3] H. Haken and H. C. Wolf, The Physics of Atoms and Quanta, (6th edn Springer, 2000). This is a more advanced text which addresses many aspects of atomic physics and quantum optics in a readable manner. [4] K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction, (OUP, 2009). This is a new text which includes some entertaining new topics within an old field. It also has a useful set of mathematica-based examples if you can get hold of the disk! [5] L. D. Landau and L. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Volume 3, (Butterworth-Heinemann, 3rd edition, 1981). This is a rich and classic text which covers the many of the core topics in this course in most cases at a level that goes well-beyond our target. [6] F. Schwabl, Quantum Mechanics, (Springer, 4th edition, 2007). This book provides a clear exposition of the core topics addressed at the same level as the current text. [7] R. Shankar, Principles of Quantum Mechanics, (Springer; 2nd edition, 1994). This is a very interesting and idiosyncratic text that I particularly like. It is not comprehensive enough to cover every topic in this course. But where there is overlap, it provides an excellent and interesting commentary.

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Chapter 1

Wave mechanics and the Schr¨ odinger equation William Thomson, 1st Baron Kelvin 1824-1907

Although this lecture course will assume a familiarity with the basic concepts of wave mechanics, to introduce more advanced topics in quantum theory, it makes sense to begin with a concise review of the foundations of the subject. In particular, in the first chapter of the course, we will begin with a brief discussion of the historical challenges that led to the development of quantum theory almost a century ago. The formulation of a consistent theory of statistical mechanics, electrodynamics and special relativity during the latter half of the 19th century and the early part of the 20th century had been a triumph of “unification”. However, the undoubted success of these theories gave an impression that physics was a mature, complete, and predictive science. Nowehere was confidence expressed more clearly than in the famous quote made at the time by Lord Kelvin: There is nothing new to be discovered in physics now. All that remains is more and more precise measurement. However, there were a number of seemingly unrelated and unsettling problems that challenged the prevailing theories.

1.1 1.1.1

Historical foundations of quantum physics Black-body radiation

In 1860, Gustav Kirchhoff introduced the concept of a “black body”, an object that absorbs all electromagnetic radiation that falls upon it – none passes through and none is reflected. Since no light is reflected or transmitted, the object appears black when it is cold. However, above absolute zero, a black body emits thermal radiation with a spectrum that depends on temperature. To determine the spectrum of radiated energy, it is helpful to think of a black body as a thermal cavity at a temperature, T . The energy radiated by the cavity can be estimated by considering the resonant modes. In three-dimensions, the number of modes, per unit frequency per unit volume is given by 8πν 2 N (ν)dν = 3 dν , c where, as usual, c is the speed of light.1 1 If we take the cavity to have dimension L3 , the modes of the cavity involve wave numbers k = πn/L where n = (nx , ny , nz ) denote the vector of integers nx = 0, 1, 2, · · · ∞, etc. The corresponding frequency of each mode is given by ν = c|k|/2π, where c is the velocity of light. The number of modes (per unit volume) having frequencies between ν and ν + dν is

Advanced Quantum Physics

Kelvin was educated at Glasgow and Cambridge. He became professor of natural philosophy at Glasgow in 1846. From 1846 to 1851 Kelvin edited the “Cambridge and Dublin Mathematical Journal,” to which he contributed several important papers. Some of his chief discoveries are announced in the “Secular Coating of the Earth,” and the Bakerian lecture, the “Electrodynamic Qualities of Metals.” He invented the quadrant, portable, and absolute electrometers, and other scientific instruments. In January 1892, he was raised to the peerage as Lord Kelvin.

Gustav Robert Kirchhoff 18241887 A German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coined the term “black body” radiation in 1862, and two sets of independent concepts in both circuit theory and thermal emission are named “Kirchhoff’s laws” after him.

John William Strutt, 3rd Baron Rayleigh OM (1842-1919) An English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize in 1904. He also discovered the phenomenon of Rayleigh scattering, explaining why the sky is blue, and predicted the existence of surface waves known as Rayleigh waves.

1.1. HISTORICAL FOUNDATIONS OF QUANTUM PHYSICS

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Figure 1.1: The COBE (Cosmic Background Explorer) satellite made careful measurements of the shape of the spectrum of the emission from the cosmic microwave background. As one can see, the behaviour at low-frequencies (longwavelengths) conforms well with the predicted Rayleigh-Jeans law translating to a temperature of 2.728K. However, at high frequencies, there is a departure from the predicted ν 2 dependence. The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. Within the framework of classical statistical mechanics, each of these modes have an equal chance of being excited, and the average energy in each mode is kB T (equipartition), where kB is the Boltzmann constant. The corresponding energy density is therefore given by the Rayleigh-Jeans law, ρ(ν, T ) =

8πν 2 kB T . c3

This result predicts that ρ(ν, T ) increases without bound at high frequencies, ν — the untraviolet (UV) catastrophe. However, such behaviour stood in contradiction with experiment which revealed that the short-wavelength dependence was quite benign (see, e.g., Fig. 1.1). To resolve difficulties presented by the UV catastrophe, Planck hypothesized that, for each mode ν, energy is quantized in units of hν, where h denotes the Planck constant. In this case, the energy of each mode is given by2 !∞ −nhν/kB T hν n=0 nhνe !ε(ν)" = ! = hν/k T , ∞ −nhν/kB T B e e −1 n=0 leading to the anticipated suppression of high frequency modes. From this result one obtains the celebrated Planck radiation formula, ρ(ν, T ) =

8πν 2 c3

!ε(ν)" =

8πhν 3

1

c3

ehν/kB T − 1

.

(1.1)

This result conforms with experiment (Fig. 1.1), and converges on the RayleighJeans law at low frequencies, hν/kB T → 0. Planck’s result suggests that electromagnetic energy is quantized: light of wavelength λ = c/ν is made up of quanta each of which has energy hν. The equipartion law fails for oscillation modes with high frequencies, hν % kB T . A 2

)dk therefore given by N (ν)dν = L13 × 2 × 18 × (4πk , where the factor of 2 accounts for the two (π/L)3 2 polarizations, the factor (4πk )dk is the volume of the shell from k to k + dk in reciprocal space, the factor of 1/8 accounts for the fact that only positive wavenumbers are involved in the closed cavity, and the factor of (π/L)3 denotes the volume of phase space occupied by each mode. Rearranging the equation, and noting that dk = 2πdν/c, we obtain the relation in the text. P∞ −βnhν 2 If we define the partition function, Z = , where β = 1/kBP T , #E$ = n=0 e ∞ n −∂β ln Z. Making use of the formula for the sum of a geometric progression, = n=0 r 1/(1 − r), we obtain the relation.

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Max Karl Ernst Ludwig Planck 1858-1947 German physicist whose work provided the bridge between classical and modern physics. Around 1900 Planck developed a theory explain the nature of black-body radiation. He proposed that energy is emitted and absorbed in discrete packets or “quanta,” and that these had a definite size – Planck’s constant. Planck’s finding didn’t get much attention until the idea was furthered by Albert Einstein in 1905 and Niels Bohr in 1913. Planck won the Nobel Prize in 1918, and, together with Einstein’s theory of relativity, his quantum theory changed the field of physics.

Albert Einstein 1879-1955 was a Germanborn theoretical physicist. He is best known for his theory of relativity and specifically mass-energy equivalence, expressed by the equation E = mc2 . Einstein received the 1921 Nobel Prize in Physics “for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.”

1.1. HISTORICAL FOUNDATIONS OF QUANTUM PHYSICS

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Figure 1.2:

Measurements of the photoelectric effect taken by Robert Millikan showing the variation of the stopping voltage, eV , with variation of the frequency of incident light. Figure reproduced from Robert A. Millikan’s Nobel Lecture, The Electron and the Light-Quanta from the Experimental Point of View, The Nobel Foundation, 1923.

quantum theory for the specific heat of matter, which takes into account the quantization of lattice vibrational modes, was subsequently given by Debye and Einstein.

1.1.2

Photoelectric effect

We turn now to the second ground-breaking experiment in the development of quantum theory. When a metallic surface is exposed to electromagnetic radiation, above a certain threshold frequency, the light is absorbed and electrons are emitted (see figure, right). In 1902, Philipp Eduard Anton von Lenard observed that the energy of individual emitted electrons increases with the frequency of the light. This was at odds with Maxwell’s wave theory of light, which predicted that the electron energy would be proportional to the intensity of the radiation. In 1905, Einstein resolved this paradox by describing light as composed of discrete quanta (photons), rather than continuous waves. Based upon Planck’s theory of black-body radiation, Einstein theorized that the energy in each quantum of light was proportional to the frequency. A photon above a threshold energy, the “work function” W of the metal, has the required energy to eject a single electron, creating the observed effect. In particular, Einstein’s theory was able to predict that the maximum kinetic energy of electrons emitted by the radiation should vary as k.e.max = hν − W . Later, in 1916, Millikan was able to measure the maximum kinetic energy of the emitted electrons using an evacuated glass chamber. The kinetic energy of the photoelectrons were found by measuring the potential energy of the electric field, eV , needed to stop them. As well as confirming the linear dependence of the kinetic energy on frequency (see Fig. 1.2), by making use of his estimate for the electron charge, e, established from his oil drop experiment in 1913, he was able to determine Planck’s constant to a precision of around 0.5%. This discovery led to the quantum revolution in physics and earned Einstein the Nobel Prize in 1921.

1.1.3

Compton Scattering

In 1923, Compton investigated the scattering of high energy X-rays and γ-ray from electrons in a carbon target. By measuring the spectrum of radiation at different angles relative to the incident beam, he found two scattering peaks. The first peak occurred at a wavelength which matched that of the incident beam, while the second varied with angle. Within the framework of a purely classical theory of the scattering of electromagnetic radiation from a charged Advanced Quantum Physics

Robert Andrews Millikan 18681953 He received his doctorate from Columbia University and taught physics at the University of Chicago (1896-1921) and the California Institute of Technology (from 1921). To measure electric charge, he devised the Millikan oil-drop experiment. He verified Albert Einstein’s photoelectric equation and obtained a precise value for the Planck constant. He was awarded the 1923 Nobel Prize in physics.

Arthur Holly Compton 18921962: An American physicist, he shared the 1927 Nobel Prize in Physics with C. T. R. Wilson for his discovery of the Compton effect. In addition to his work on X rays he made valuable studies of cosmic rays and contributed to the development of the atomic bomb.

1.1. HISTORICAL FOUNDATIONS OF QUANTUM PHYSICS

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particle – Thomson scattering – the wavelength of a low-intensity beam should remain unchanged.3 Compton’s observation demonstrated that light cannot be explained purely as a classical wave phenomenon. Light must behave as if it consists of particles in order to explain the low-intensity Compton scattering. If one assumes that the radiation is comprised of photons that have a well defined momentum as h well as energy, p = hν c = λ , the shift in wavelength can be understood: The interaction between electrons and high energy photons (ca. keV) results in the electron being given part of the energy (making it recoil), and a photon with the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is conserved. By taking into account both conservation of energy and momentum of the system, the Compton scattering formula describing the shift in the wavelength as function of scattering angle θ can be derived,4 ∆λ = λ# − λ =

h (1 − cos θ) . me c

The constant of proportionality h/me c = 0.002426 nm, the Compton wavelength, characterizes the scale of scattering. Moreover, as h → 0, one finds that ∆λ → 0 leading to the classical prediction.

1.1.4

Figure 1.3:

Variation of the wavelength of X rays scattered from a Carbon target. A. H. Compton, Phys. Rev. 21, 483; 22, 409 (1923)

Atomic spectra

The discovery by Rutherford that the atom was comprised of a small positively charged nucleus surrounded by a diffuse cloud of electrons lead naturally to the consideration of a planetary model of the atom. However, a classical theory of electrodynamics would predict that an accelerating charge would radiate energy leading to the eventual collapse of the electron into the nucleus. Moreover, as the electron spirals inwards, the emission would gradually increase in frequency leading to a broad continuous spectra. Yet, detailed studies of electrical discharges in low-pressure gases revealed that atoms emit light at discrete frequencies. The clue to resolving these puzzling observations lay in the discrete nature of atomic spectra. For the hydrogen atom, light emitted when the atom is thermally excited has a particular pattern: Balmer had discovered in 1885 that the emitted wavelengths follow the empirical law, λ = λ0 (1/4 − 1/n2 ) A (see Fig. 1.4). Neils Bohr realized where n = 3, 4, 5, · · · and λ0 = 3645.6˚ that these discrete vaues of the wavelength reflected the emission of individual photons having energy equal to the energy difference between two allowed orbits of the electron circling the nucleus (the proton), En − Em = hν, leading to the conclusion that the allowed energy levels must be quantized and varying H , where RH = 109678 cm−1 denotes the Rydberg constant. as En = − hcR n2 3

Classically, light of sufficient intensity for the electric field to accelerate a charged particle to a relativistic speed will cause radiation-pressure recoil and an associated Doppler shift of the scattered light. But the effect would become arbitrarily small at sufficiently low light intensities regardless of wavelength. 4 If we assume that the total energy and momentum are conserved in the scattering of a photon (γ) from an initially stationary target electron (e), we have Eγ + Ee = Eγ ! + Ee! and pγ = pγ ! + pe! . Here Eγ = hν and Ee = me c2 denote p the energy of the photon and electron before the collision, while Eγ ! = hν # and Ee! = (pe! c)2 + (mc2 )2 denote the energies after. From the equation for energy conservation, one obtains (pe! c)2 = (h(ν − ν # ) + me c2 )2 − (me c2 )2 . From the equation for momentum conservation, one obtains p2e! = p2γ + p2γ ! − 2|pγ ||pγ ! | cos θ. Then, noting that Eγ = pγ c, a rearrangement of these equations obtains the Compton scattering formula.

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Ernest Rutherford, 1st Baron Rutherford of Nelson, 18711937 was a New Zealand born British chemist and Physicist who became known as the father of nuclear physics. He discovered that atoms have a small charged nucleus, and thereby pioneered the Rutherford model (or planetary model of the atom, through his discovery of Rutherford scattering. He was awarded the Nobel Prize in Chemistry in 1908. He is widely credited as splitting the atom in 1917 and leading the first experiment to “split the nucleus” in a controlled manner by two students under his direction, John Cockcroft and Ernest Walton in 1932.

1.1. HISTORICAL FOUNDATIONS OF QUANTUM PHYSICS

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Figure 1.4: Schematic describ-

ing various transitions (and an image with the corresponding visible spectral lines) of atomic hydrogen.

How could the quantum hν restricting allowed radiation energies also restrict the allowed electron orbits? In 1913 Bohr proposed that the angular momentum of an electron in one of these orbits was quantized in units of Planck’s constant, L = me vr = n!, 2

!=

h . 2π

(1.2)

2

e )2 2πch3m . As a result, one finds that RH = ( 4π$ 0

( Exercise. Starting with the Bohr’s planetary model for atomic hydrogen, find how the quantization condition (1.2) restricts the radius of the allowed (circular) orbits. Determine the allowed energy levels and obtain the expression for the Rydberg constant above. But why should only certain angular momenta be allowed for the circling electron? A heuristic explanation was provided by de Broglie: just as the constituents of light waves (photons) are seen through Compton scattering to act like particles (of definite energy and momentum), so particles such as electrons may exhibit wave-like properties. For photons, we have seen that the relationship between wavelength and momentum is p = h/λ. de Broglie hypothesized that the inverse was true: for particles with a momentum p, the wavelength is λ=

h , p

i.e. p = !k ,

(1.3)

where k denotes the wavevector of the particle. Applied to the electron in the atom, this result suggested that the allowed circular orbits are standing waves, from which Bohr’s angular momentum quantization follows. The de Broglie hypothesis found quantitative support in an experiment by Davisson and Germer, and independently by G. P. Thomson in 1927. Their studies of electron diffraction from a crystalline array of Nickel atoms (Fig. 1.5) confirmed that the diffraction angles depend on the incident energy (and therefore momentum). This completes the summary of the pivotal conceptual insights that paved the way towards the development of quantum mechanics. Advanced Quantum Physics

Niels Henrik David Bohr 18851962 A Danish physicist who made fundamental contributions to the understanding atomic structure and quantum mechanics, for which he received the Nobel Prize in Physics in 1922. Bohr mentored and collaborated with many of the top physicists of the century at his institute in Copenhagen. He was also part of the team of physicists working on the Manhattan Project. Bohr married Margrethe Norlund in 1912, and one of their sons, Aage Niels Bohr, grew up to be an important physicist who, like his father, received the Nobel prize, in 1975.

Louis Victor Pierre Raymond, 7th duc de Broglie 1892-1987 A French physicist, de Broglie had a mind of a theoretician rather than one of an experimenter or engineer. de Broglie’s 1924 doctoral thesis Recherches sur la th´eorie des quanta introduced his theory of electron waves. This included the particle-wave property duality theory of matter, based on the work of Einstein and Planck. He won the Nobel Prize in Physics in 1929 for his discovery of the wave nature of electrons, known as the de Broglie hypothesis or m´ ecanique ondulatoire.

1.2. WAVE MECHANICS

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Figure 1.5: In 1927, Davisson and Germer bombarded a single crystal of nickel with a beam of electrons, and observed several beams of scattered electrons that were almost as well defined as the incident beam. The phenomenological similarities with X-ray diffraction were striking, and showed that a wavelength could indeed be associated with the electrons. The first figure shows the intensity of electron scattering against co-latitude angle for various bombarding voltages. The second figure shows the intensity of electron scattering vs. azimuth angle - 54V, co-latitude 50. Figures taken taken from C. Davisson and L. H. Germer, Reflection of electrons by a crystal of nickel, Nature 119, 558 (1927).

1.2

Wave mechanics

de Broglie’s doctoral thesis, defended at the end of 1924, created a lot of excitement in European physics circles. Shortly after it was published in the Autumn of 1925, Pieter Debye, a theorist in Zurich, suggested to Erwin Schr¨odinger that he give a seminar on de Broglie’s work. Schr¨odinger gave a polished presentation, but at the end, Debye remarked that he considered the whole theory rather childish: Why should a wave confine itself to a circle in space? It wasnt as if the circle was a waving circular string; real waves in space diffracted and diffused; in fact they obeyed three-dimensional wave equations, and that was what was needed. This was a direct challenge to Schr¨odinger, who spent some weeks in the Swiss mountains working on the problem, and constructing his equation. There is no rigorous derivation of Schr¨odinger’s equation from previously established theory, but it can be made very plausible by thinking about the connection between light waves and photons, and constructing an analogous structure for de Broglie’s waves and electrons (and, of course, other particles).

1.2.1

Maxwell’s wave equation

For a monochromatic wave in vacuum, with no currents or charges present, Maxwell’s wave equation, ∇2 E −

1 ¨ E = 0, c2

(1.4)

admits the plane wave solution, E = E0 ei(k·r−ωt) , with the linear dispersion relation ω = c|k| and c the velocity of light. Here, (and throughout the text) ¨ ≡ ∂ 2 E. We know from the photoelectric effect we adopt the convention, E t and Compton scattering that the photon energy and momentum are related to the frequency and wavelength of light through the relations E = hν = !ω, p = λh = !k. The wave equation tells us that ω = c|k| and hence E = c|p|. If we think of ei(k·r−ωt) as describing a particle (photon) it would be more natural to write the plane wave in terms of the energy and momentum of the Advanced Quantum Physics

James Clerk Maxwell 1831-1879 was a Scottish theoretical physicist and mathematician. His greatest achievement was the development of classical electromagnetic theory, synthesizing all previous unrelated observations of electricity, magnetism and optics into a consistent theory. Maxwell’s equations demonstrated that electricity, magnetism and light are all manifestations of electromagnetic field.

1.2. WAVE MECHANICS

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particle as E0 ei(p·r−Et)/!. Then, one may see that the wave equation applied to the plane wave describing particle propagation yields the familiar energymomentum relationship, E 2 = (cp)2 for a massless relativistic particle. This discussion suggests how one might extend the wave equation from the photon (with zero rest mass) to a particle with rest mass m0 . We require a wave equation that, when it operates on a plane wave, yields the relativistic energy-momentum invariant, E 2 = (cp)2 + m20 c4 . Writing the plane wave function φ(r, t) = Aei(p·r−Et)/!, where A is a constant, we can recover the energy-momentum invariant by adding a constant mass term to the wave operator, $ % " # (cp)2 − E 2 + m20 c4 i(p·r−Et)/! ∂t2 m20 c2 2 i(p·r−Et)/! ∇ − 2 − 2 e =− e = 0. c ! (!c)2 This wave equation is called the Klein-Gordon equation and correctly describes the propagation of relativistic particles of mass m0 . However, its form is seems inappropriate for non-relativistic particles, like the electron in hydrogen. Continuing along the same lines, let us assume that a non-relativistic electron in free space is also described by a plane wave of the form Ψ(x, t) = Aei(p·r−Et)/!. We need to construct an operator which, when applied to this wave function, just gives us the ordinary non-relativistic energy-momentum p2 . The factor of p2 can obviously be recovered from two relation, E = 2m derivatives with respect to r, but the only way we can get E is by having a single differentiation with respect to time, i.e. i!∂t Ψ(r, t) = −

!2 2 ∇ Ψ(r, t) . 2m

This is Schr¨odinger’s equation for a free non-relativistic particle. One remarkable feature of this equation is the factor of i which shows that the wavefunction is complex. How, then, does the presence of a spatially varying scalar potential effect the propagation of a de Broglie wave? This question was considered by Sommerfeld in an attempt to generalize the rather restrictive conditions in Bohr’s model of the atom. Since the electron orbit was established by an inversesquare force law, just like the planets around the Sun, Sommerfeld couldn’t understand why Bohr’s atom had only circular orbits as opposed to Keplerlike elliptical orbits. (Recall that all of the observed spectral lines of hydrogen were accounted for by energy differences between circular orbits.) de Broglie’s analysis of the allowed circular orbits can be formulated by assuming that, at some instant, the spatial variation of the wavefunction on going around the orbit includes a phase term of the form eipq/!, where here the parameter q measures the spatial distance around the orbit. Now, for an acceptable wavefunction, the total phase change on going around the orbit must be 2πn, where n is integer. For the usual Bohr circular orbit, where p = |p| is constant, this leads to quantization of the angular momentum L = pr = n!. Sommerfeld considered a general Keplerian elliptical orbit. Assuming that the de Broglie relation p = h/λ still holds, the wavelength must vary as the particle moves around the orbit, being shortest where the particle travels fastest, at its closest approach to the nucleus. Nevertheless, the phase change on moving a short distance ∆q should still be p∆q/!. Requiring the wavefunction to link up smoothly on going once around the orbit gives the Bohr-Sommerfeld Advanced Quantum Physics

Arnold Johannes Wilhelm Sommerfeld 1868-1951 A German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics. He introduced the fine-structure constant into quantum mechanics.

1.2. WAVE MECHANICS

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quantization condition &

p dq = nh ,

(1.5)

' where denotes the line integral around a closed orbit. Thus only certain elliptical orbits are allowed. The mathematics is non-trivial, but it turns out that every allowed elliptical orbit has the same energy as one of the allowed circular orbits. That is why Bohr’s theory gave the correct energy levels. This analysis suggests that, in a varying potential, the wavelength changes in concert with the momentum. ( Exercise. As a challenging exercise, try to prove Sommerfeld’s result for the elliptical orbit.

1.2.2

Schr¨ odinger’s equation

Following Sommerfeld’s considerations, let us then consider a particle moving in one spatial dimension subject to a “roller coaster-like” potential. How do we expect the wavefunction to behave? As discussed above, we would expect the wavelength to be shortest where the potential is lowest, in the minima, because that’s where the particle is going the fastest. Our task then is to construct a wave equation which leads naturally to the relation following from p2 +V (x). In contrast to the free particle (classical) energy conservation, E = 2m case discussed above, the relevant wavefunction here will no longer be a simple plane wave, since the wavelength (determined through the momentum via the de Broglie relation) varies with the potential. However, at a given position x, the momentum is determined by the “local wavelength”. The appropriate wave equation is the one-dimensional Schr¨odinger equation, i!∂t Ψ(x, t) = −

!2 ∂x2 Ψ(x, t) + V (x)Ψ(x, t) , 2m

(1.6)

with the generalization to three-dimensions leading to the Laplacian operator in place of ∂x2 (cf. Maxwell’s equation). So far, the validity of this equation rests on plausibility arguments and hand-waving. Why should anyone believe that it really describes an electron wave? Schr¨odinger’s test of his equation was the hydrogen atom. He looked for Bohr’s “stationary states”: states in which the electron was localized somewhere near the proton, and having a definite energy. The time dependence would be the same as for a plane wave of definite energy, e−iEt/!; the spatial dependence would be a time-independent function decreasing rapidly at large distances from the proton. From the solution of the stationary wave equation for the Coulomb potential, he was able to deduce the allowed values of energy and momentum. These values were exactly the same as those obtained by Bohr (except that the lowest allowed state in the “new” theory had zero angular momentum): impressive evidence that the new theory was correct.

1.2.3

Time-independent Schr¨ odinger equation

As with all second order linear differential equations, if the potential V (x, t) = V (x) is time-independent, the time-dependence of the wavefunction can be

Advanced Quantum Physics

Erwin Rudolf Josef Alexander Schrdinger 1887-1961 was an Austrian theoretical physicist who achieved fame for his contributions to quantum mechanics, especially the Schr¨ odinger equation, for which he received the Nobel Prize in 1933. In 1935, after extensive correspondence with personal friend Albert Einstein, he proposed the Schr¨ odinger’s cat thought experiment.

1.2. WAVE MECHANICS

9

separated from the spatial dependence. Setting Ψ(x, t) = T (t)ψ(x), and separating the variables, the Schr¨odinger equation takes the form, ( 2 2 ) ∂x − !2m ψ(x) + V (x)ψ(x) i!∂t T (t) = = const. = E . ψ(x) T (t) Since we have a function of only x set equal to a function of only t, they both must equal a constant. In the equation above, we call the constant E (with some knowledge of the outcome). We now have an equation in t set equal to a constant, i!∂t T (t) = ET (t), which has a simple general solution, T (t) = Ce−iEt/!, where C is some constant. The corresponding equation in x is then given by the stationary, or time-independent Schr¨ odinger equation, −

!2 ∂x2 ψ(x) + V (x)ψ(x) = Eψ(x) . 2m

The full time-dependent solution is given by Ψ(x, t) = e−iEt/!ψ(x) with definite energy, E. Their probability density |Ψ(x, t)|2 = |ψ(x)|2 is constant in time – hence they are called stationary states! The operator ˆ = − ! ∂x + V (x) H 2m 2 2

defines the Hamiltonian and the stationary wave equation can be written as ˆ ˆ the eigenfunction equation, Hψ(x) = Eψ(x), i.e. ψ(x) is an eigenstate of H with eigenvalue E.

1.2.4

Particle flux and conservation of probability

In analogy to the Poynting vector for the electromagnetic field, we may want to know the probability current. For example, for a free particle system, the probability density is uniform over all space, but there is a net flow along the direction of momentum. We can derive an equation showing conservation of probability by differentiating the probability density, P (x, t) = |ψ(x, t)|2 , and using the Schr¨odinger equation, ∂t P (x, t)+∂x j(x, t) = 0. This translates to the usual conservation equation if j(x, t) is identified as the probability current, i! ∗ [ψ ∂x ψ − ψ∂x ψ ∗ ] . (1.7) 2m *b *b If we integrate over some interval in x, a ∂t P (x, t)dx = − a ∂x j(x, t)dx it *b follows that ∂t a P (x, t)dx = j(x = a, t) − j(x = b, t), i.e. the rate of change of probability is equal to the net flux entering the interval. Extending this analysis to three space dimensions, we obtain the continuity equation, ∂t P (r, t) + ∇ · j(r, t) = 0, from which follows the particle flux, j(x, t) = −

j(r, t) = −

i! ∗ [ψ (r, t)∇ψ(r, t) − ψ(r, t)∇ψ ∗ (r, t)] . 2m

(1.8)

This completes are survey of the foundations and development of quantum theory. In due course, it will be necessary to develop some more formal mathematical aspects of the quantum theory. However, before doing, it is useful to acquire some intuition for the properties of the Schr¨odinger equation. Therefore, in the next chapter, we will explore the quantum mechanics of bound and unbound particles in a one-dimensional system turning to discuss more theoretical aspects of the quantum formulation in the following chapter. Advanced Quantum Physics

Chapter 2

Quantum mechanics in one dimension Following the rules of quantum mechanics, we have seen that the state of a quantum particle, subject to a scalar potential V (r), is described by the time-dependent Schr¨odinger equation, ˆ i!∂t Ψ(r, t) = HΨ(r, t) ,

(2.1)

ˆ = − ! ∇ + V (r) denotes the Hamiltonian. To explore its properwhere H 2m ties, we will first review some simple and, hopefully, familiar applications of the equation to one-dimensional systems. In addressing the one-dimensional geometry, we will divide our consideration between potentials, V (x), which leave the particle free (i.e. unbound), and those that bind the particle to some region of space. 2

2.1 2.1.1

2

Wave mechanics of unbound particles Free particle

In the absence of an external potential, the time-dependent Schr¨odinger equation (2.1) describes the propagation of travelling waves. In one dimension, the corresponding complex wavefunction has the form Ψ(x, t) = A ei(kx−ωt) , k represents the free particle where A is the amplitude, and E(k) = !ω(k) = !2m energy dispersion for a non-relativistic particle of mass, m, and wavevector k = 2π/λ with λ the wavelength. Each wavefunction describes a plane wave in which the particle has definite energy E(k) and, in accordance with the de Broglie relation, momentum p = !k = h/λ. The energy spectrum of a freely-moving particle is therefore continuous, extending from zero to infinity and, apart from the spatially constant state k = 0, has a two-fold degeneracy corresponding to right and left moving particles. For an infinite system, it makes no sense to fix the amplitude A by the normalization of the total probability. Instead, it is useful to fix the flux associated with the wavefunction. Making use of Eq. (1.7) for the particle current, the plane wave is associated with a constant (time-independent) flux, 2 2

j(x, t) = −

i! !k p (Ψ∗ ∂x Ψ − c.c.) = |A|2 = |A|2 . 2m m m

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2.1. WAVE MECHANICS OF UNBOUND PARTICLES For a given value of the!flux j, the amplitude is given, up to an arbitrary constant phase, by A = mj/!k. To prepare a wave packet which is localized to a region of space, we must superpose components of different wave number. In an open system, this may be achieved using a Fourier expansion. For any function,1 ψ(x), we have the Fourier decomposition,2 " ∞ 1 ψ(k) eikx dk , ψ(x) = √ 2π −∞ where the coefficients are defined by the inverse transform, " ∞ 1 ψ(k) = √ ψ(x) e−ikx dx . 2π −∞ The normalization of ψ(k) #follows automatically from the normalization of #∞ ∗ ∞ ψ(x), −∞ ψ (k)ψ(k)dk = −∞ ψ ∗ (x)ψ(x)dx = 1, and both can represent probability amplitudes. Applied to a wavefunction, ψ(x) can be understood as a wave packet made up of contributions involving definite momentum states, eikx , with amplitude set by the Fourier coefficient ψ(k). The probability for a particle to be found in a region of width dx around some value of x is given by |ψ(x)|2 dx. Similarly, the probability for a particle to have wave number k in a region of width dk around some value of k is given by |ψ(k)|2 dk. (Remember that p = !k so the momentum distribution is very closely related. Here, for economy of notation, we work with k.) The Fourier transform of a normalized Gaussian wave packet, ψ(k) = 1/4 e−α(k−k0 )2 , is also a Gaussian (exercise), ( 2α ) π ψ(x) =

$

1 2πα

%1/4

x2

eik0 x e− 4α .

From these representations, we can see that it is possible to represent a single particle, localized in real space as a superposition of plane wave states localized in Fourier space. But note that, while we have achieved our goal of finding localized wave packets, this has been at the expense of having some non-zero width in x and in k. For the Gaussian wave packet, we can straightforwardly obtain the width (as measured by the root mean square – RMS) of the probability distribution, √ ∆x = (#(x − #x$)2 $)1/2 ≡ (#x2 $ − #x$2 $)1/2 = α, and ∆k = √14α . We can again see that, as we vary the width in k-space, the width in x-space varies to keep the following product constant, ∆x∆k = 12 . If we translate from the wavevector into momentum p = !k, then ∆p = !∆k and ∆p ∆x =

! . 2

If we consider the width of the distribution as a measure of the “uncertainty”, we will prove in section (3.1.2) that the Gaussian wave packet provides the minimum uncertainty. This result shows that we cannot know the position of a particle and its momentum at the same time. If we try to localize a particle to a very small region of space, its momentum becomes uncertain. If we try to 1 More precisely, we can make such an expansion providing we meet some rather weak conditions of smoothness and differentiability of ψ(x) – conditions met naturally by problems which derive from physical systems! 2 Here we will adopt an ecomony of notation using the same symbol ψ to denote the wavefunction and its Fourier coefficients. Their identity will be disclosed by their argument and context.

Advanced Quantum Physics

11

2.1. WAVE MECHANICS OF UNBOUND PARTICLES make a particle with a definite momentum, its probability distribution spreads out over space. With this introduction, we now turn to consider the interaction of a particle with a non-uniform potential background. For non-confining potentials, such systems fall into the class of scattering problems: For a beam of particles incident on a non-uniform potential, what fraction of the particles are transmitted and what fraction are reflected? In the one-dimensional system, the classical counterpart of this problem is trivial: For particle energies which exceed the maximum potential, all particles are eventually transmitted, while for energies which are lower, all particles are reflected. In quantum mechanics, the situation is richer: For a generic potential of finite extent and height, some particles are always reflected and some are always transmitted. Later, in chapter 14, we will consider the general problem of scattering from a localized potential in arbitrary dimension. But for now, we will focus on the one-dimensional system, where many of the key concepts can be formulated.

2.1.2

Potential step

As we have seen, for a time-independent potential, the wavefunction can be factorized as Ψ(x, t) = e−iEt/!ψ(x), where ψ(x) is obtained from the stationary form of the Schr¨odinger equation, & 2 2 ' ! ∂x − + V (x) ψ(x) = Eψ(x) , 2m and E denotes the energy of the particle. As |Ψ(x, t)|2 represents a probablility density, it must be everywhere finite. As a result, we can deduce that the wavefunction, ψ(x), is also finite. Moreover, since E and V (x) are presumed finite, so must be ∂x2 ψ(x). The latter condition implies that ' both ψ(x) and ∂x ψ(x) must be continuous functions of x, even if V has a discontinuity. Consider then the influence of a potential step (see figure) on the propagation of a beam of particles. Specifically, let us assume that a beam of particles with kinetic energy, E, moving from left to right are incident upon a potential step of height V0 at position x = 0. If the beam has unit amplitude, the reflected and transmitted (complex) amplitudes are set by r and t. The corresponding wavefunction is given by ψ< (x) = eik< x + re−ik< x x < 0 x>0 ψ> (x) = teik> x (

(

0) where k< = and k> = 2m(E−V . Applying the continuity conditions !2 on ψ and ∂x ψ at the step (x = 0), one obtains the relations 1 + r = t and ik< (1 − r) = ik> t leading to the reflection and transmission amplitudes,

2mE !2

r=

k< − k> , k< + k>

t=

2k< . k< + k>

The reflectivity, R, and transmittivity, T , are defined by the ratios, R=

reflected flux , incident flux

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T =

transmitted flux . incident flux

12

2.1. WAVE MECHANICS OF UNBOUND PARTICLES With the incident, reflected, and transmitted fluxes given by |A|2 !km< , |Ar|2 !km< , and |At|2 !km> respectively, one obtains ) ) ) ) ) 2k< )2 k> ) k< − k> )2 4k< k> 2 k> ) = |r|2 , ) ) T = R = )) ) k< + k> ) k< = |t| k< = (k< + k> )2 . k< + k> ) From these results one can confirm that the total flux is, as expected, conserved in the scattering process, i.e. R + T = 1.

' Exercise. While E −V0 remains positive, show that the beam is able to propagate across the potential step (see figure). Show that the fraction of the beam that is reflected depends on the relative height of the step while the phase depends on the sign of V0 . In particular, show that for V0 > 0, the reflected beam remains in phase with the incident beam, while for V0 < 0 it is reversed. Finally, when E − V0 < 0, show that the beam is unable to propagate to the right (R = 1). Instead show that there is an evanescent decay of the wavefunction into the barrier region with a decay ! length set by 2π !2 /2m(V0 − E). If V0 → ∞, show that the system forms a standing wave pattern.

2.1.3

Potential barrier

Having dealt with the potential step, we now turn to consider the problem of a beam of particles incident upon a square potential barrier of height V0 (presumed positive for now) and width a. As mentioned above, this geometry is particularly important as it includes the simplest example of a scattering phenomenon in which a beam of particles is “deflected” by a local potential. Moreover, this one-dimensional geometry also provides a platform to explore a phenomenon peculiar to quantum mechanics – quantum tunneling. For these reasons, we will treat this problem fully and with some care. Since the barrier is localized to a region of size a, the incident and(trans-

, mitted wavefunctions have the same functional form, eik1 x , where k1 = 2mE !2 and differ only in their complex amplitude, i.e. after the encounter with the barrier, the transmitted wavefunction undergoes only a change of amplitude (some particles are reflected from the barrier, even when the energy of the incident beam, E, is in excess of V0 ) and a phase shift. To deterimine the relative change in amplitude and phase, we can parameterise the wavefunction as ψ1 (x) = eik1 x + re−ik1 x x≤0 ψ2 (x) = Aeik2 x + Be−ik2 x 0 ≤ x ≤ a a≤x ψ3 (x) = teik1 x

( 0) . Here, as with the step, r denotes the reflected ampliwhere k2 = 2m(E−V !2 tude and t the transmitted. Applying the continuity conditions on the wavefunction, ψ, and its derivative, ∂x ψ, at the barrier interfaces at x = 0 and x = a, one obtains * * 1+r =A+B k1 (1 − r) = k2 (A − B) , . Aeik2 a + Be−ik2 a = teik1 a k2 (Aeik2 a − Be−ik2 a ) = k1 teik1 a Together, these four equations specify the four unknowns, r, t, A and B. Solving, one obtains (exercise) t=

2k1 k2 e−ik1 a , 2k1 k2 cos(k2 a) − i(k12 + k22 ) sin(k2 a)

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13

2.1. WAVE MECHANICS OF UNBOUND PARTICLES

14

translating to a transmissivity of T = |t|2 =

1+

1 4

+

1 k1 k2

−

k2 k1

,2

, sin2 (k2 a)

and the reflectivity, R = 1 − T . As a consistency check, we can see that, when V0 = 0, k2 = k1 and t = 1, as expected. Moreover, T is restricted to the interval from 0 to 1 as required. So, for barrier heights in the range E > V0 > 0, the transmittivity T shows an oscillatory behaviour with k2 reaching unity when k2 a = nπ with n integer. At these values, there is a conspiracy of interference effects which eliminate altogether the reflected component of the wave leading to perfect transmission. Such a situation arises when the width of the barrier is perfectly matched to an integer or half-integer number of wavelengths inside the barrier. When the energy of the incident particles falls below the energy of the barrier, 0 < E < V0 , a classical beam would be completely reflected. However, in the quantum system, particles are able to tunnel through the barrier region and escape leading to a non-zero transmission coefficient. In this regime, k2 = iκ2 becomes pure imaginary leading to an evanescent decay of the wavefunction under the barrier and a suppression, but not extinction, of transmission probability, T = |t|2 =

1+

1 4

+

1 k1 κ2

+

κ2 k1

,2

Transmission probability of a √ finite potential barrier for 2mV0 a/! = 7. Dashed: classical result. Solid line: quantum mechanics.

. sinh2 κ2 a

For κ2 a ) 1 (the weak tunneling limit), the transmittivity takes the form T *

16k12 κ22 −2κ2 a e . (k12 + κ22 )2

Finally, on a cautionary note, while the phenomenon of quantum mechanical tunneling is well-established, it is difficult to access in a convincing experimental manner. Although a classical particle with energy E < V0 is unable to penetrate the barrier region, in a physical setting, one is usually concerned with a thermal distribution of particles. In such cases, thermal activation may lead to transmission over a barrier. Such processes often overwhelm any contribution from true quantum mechanical tunneling. ' Info. Scanning tunneling microscopy (STM) is a powerful technique for viewing surfaces at the atomic level. Its development in the early eighties earned its inventors, Gerd Binnig and Heinrich Rohrer (at IBM Z¨ urich), the Nobel Prize in Physics in 1986. STM probes the density of states of a material using the tunneling current. In its normal operation, a lateral resolution of 0.1 nm and a depth resolution of 0.01 nm is typical for STM. The STM can be used not only in ultra-high vacuum, but also in air and various other liquid or gas ambients, and at temperatures ranging from near zero kelvin to a few hundred degrees Celsius. The STM is based on the concept of quantum tunnelling (see Fig. 2.1). When a conducting tip is brought in proximity to a metallic or semiconducting surface, a bias between the two can allow electrons to tunnel through the vacuum between them. For low voltages, this tunneling current is a function of the local density of states at the Fermi level, EF , of the sample.3 Variations in current as the probe passes over the surface are translated into an image. STM can be a challenging technique, as it requires extremely clean surfaces and sharp tips. 3 Although the meaning of the Fermi level will be address in more detail in chapter 8, we mention here that it represents the energy level to which the electron states in a metal are fully-occupied.

Advanced Quantum Physics

Real part of the wavefunction for E/V0 = 0.6 (top), E/V0 = 1.6 (middle), and E/V0 = 1 + π 2 /2 (bottom), where mV0 a2 /!2 = 1. In the first case, the system shows tunneling behaviour, while in the third case, k2 a = π and the system shows resonant transmission.

STM image showing two point defects adorning the copper (111) surface. The point defects (possibly impurity atoms) scatter the surface state electrons resulting in circular standing wave patterns.

2.2. WAVE MECHANICS OF BOUND PARTICLES

Figure 2.1: Principle of scanning tunneling microscopy: Applying a negative sample voltage yields electron tunneling from occupied states at the surface into unoccupied states of the tip. Keeping the tunneling current constant while scanning the tip over the surface, the tip height follows a contour of constant local density of states.

2.1.4

The rectangular potential well

Finally, if we consider scattering from a potential well (i.e. with V0 < 0), while E > 0, we can apply the results of the previous section to find a continuum of unbound states with the corresponding resonance behaviour. However, in addition to these unbound states, for E < 0 we have the opportunity to find bound states of the potential. It is to this general problem that we now turn. ' Exercise. Explore the phase dependence of the transmission coefficient in this regime. Consider what happens to the phase as resonances (bound states) of the potential are crossed.

2.2

Wave mechanics of bound particles

In the case of unbound particles, we have seen that the spectrum of states is continuous. However, for bound particles, the wavefunctions satisfying the Schr¨odinger equation have only particular quantized energies. In the onedimensional system, we will find that all binding potentials are capable of hosting a bound state, a feature particular to the low dimensional system.

2.2.1

The rectangular potential well (continued)

As a starting point, let us consider a rectangular potential well similar to that discussed above. To make use of symmetry considerations, it is helpful to reposition the potential setting  x ≤ −a  0 V (x) = −V0 −a ≤ x ≤ a ,  0 a≤x

where the potential depth V0 is assumed positive. In this case, we will look for bound state solutions with energies lying in the range −V0 < E < 0. ˆ Pˆ ] = 0 Since the Hamiltonian is invariant under parity transformation, [H, ˆ ˆ (where P ψ(x) = ψ(−x)), the eigenstates of the Hamiltonian H must also be

Advanced Quantum Physics

15

2.2. WAVE MECHANICS OF BOUND PARTICLES

16

eigenstates of parity, i.e. we expect the eigenfunctions to separate into those symmetric and those antisymmetric under parity.4 For E < 0 (bound states), the wavefunction outside the well region must have the form ψ(x < −a) = Ceκx ,

with κ = the form

ψ(x > a) = De−κx ,

( − 2mE while in the central well region, the general solution is of !2 ψ(−a < x < a) = A cos(kx) + B sin(kx) ,

( 0) . Once again we have four equations in four unknowns. where k = 2m(E+V !2 The calculation shows that either A or B must be zero for a solution. This means that the states separate into solutions with even or odd parity. For the even states, the solution of the equations leads to the quantization condition, κ = tan(ka)k, while for the odd states, we find κ = − cot(ka)k. These are transcendental equations, and must be solved numerically. The 2 0a figure (right) compares κa = ( 2mV − (ka)2 )1/2 with ka tan(ka) for the even !2 states and to −ka cot(ka) for the odd states. Where the curves intersect, we have an allowed energy. From the structure of these equations, it is evident that an even state solution can always be found for arbitrarily small values of the binding potential V0 while, for odd states, bound states appear only at a critical value of the coupling strength. The wider and deeper the well, the more solutions are generated. ' Exercise. Determine the pressure exerted on the walls of a rectangular potential well by a particle inside. For a hint on how to proceed, see the discussion on degeneracy pressure on page 85.

2.2.2

The δ-function potential well

Let us now consider perhaps the simplest binding potential, the δ-function, V (x) = −aV0 δ(x). Here the parameter ‘a’ denotes some microscopic length scale introduced to make the product aδ(x) dimensionless.5 For a state to be bound, its energy must be negative. Moreover, the form of the potential demands that the wavefunction is symmetric under parity, x → −x. (A wavefunction which was antisymmetric must have ψ(0) = 0 and so could not be influenced by the δ-function potential.) We therefore look for a solution of the form * κx e x0 e

! where κ = −2mE/!2 . With this choice, the wavefunction remains everywhere continuous including at the potential, x = 0. Integrating the stationary form of the Schr¨odinger equation across an infinitesimal interval that spans the region of the δ-funciton potential, we find that ∂x ψ|+% − ∂x ψ|−% = − 4 5

2maV0 ψ(0) . !2

Later, in section 3.2, we will discuss the role of symmetries in quantum mechanics. Note that the dimenions of δ(x) are [Length−1 ].

Advanced Quantum Physics

2

0a Comparison of κa = ( 2mV − !2 2 1/2 with ka tan(ka) and (ka) ) 2 0a −ka cot(ka) for 2mV = 14. !2 For this potential, there are a total of three solutions labelled A, B and C. Note that, from the geometry of the curves, there is always a bound state no matter how small is the potential V0 .

2.2. WAVE MECHANICS OF BOUND PARTICLES From this result, we obtain that κ = maV0 /!2 , leading to the bound state energy E=−

ma2 V02 . 2!2

Indeed, the solution is unique. An attractive δ-function potential hosts only one bound state. ' Exercise. Explore the bound state properties of the “molecular” binding potential V (x) = −aV0 [δ(x + d) + δ(x − d)]. Show that it consists of two bound states, one bonding (nodeless) and one antibonding (single node). How does the energy of the latter compare with two isolated δ-function potential wells?

2.2.3

Info: The δ-function model of a crystal

Finally, as our last example of a one-dimensional quantum system, let us consider a particle moving in a periodic potential. The Kronig-Penney model provides a caricature of a (one-dimensional) crystalline lattice potential. The potential created by the ions is approximated as an infinite array of potential wells defined by a set of repulsive δ-function potentials, V (x) = aV0

∞ 0

n=−∞

δ(x − na) .

Since the potential is repulsive, it is evident that all states have energy E > 0. This potential has a new symmetry; a translation by the lattice spacing a leaves the protential unchanged, V (x + a) = V (x). The probability density must therefore exhibit the same translational symmetry, |ψ(x + a)|2 = |ψ(x)|2 , which means that, under translation, the wavefunction differs by at most a phase, ψ(x + a) = eiφ ψ(x). In the region from (n − 1)a < x < na, the general solution of the Schr¨odinger equation is plane wave like and can be written in the form, ψn (x) = An sin[k(x − na)] + Bn cos[k(x − na)] , ! where k = 2mE/!2 and, following the constraint on translational invariance, An+1 = eiφ An and Bn+1 = eiφ Bn . By applying the boundary conditions, one can derive a constraint on k similar to the quantized energies for bound states considered above. Consider the boundary conditions at position x = na. Continuity of the wavefunction, ψn |x=na = ψn+1 |x=na , translates to the condition, Bn = An+1 sin(−ka) + Bn+1 cos(−ka) or Bn+1 =

Bn + An+1 sin(ka) . cos(ka)

0 Similarly, the discontinuity in the first derivative, ∂x ψn+1 |x=na −∂x ψn |na = 2maV !2 ψn (na), 2maV0 leads to the condition, k [An+1 cos(ka) + Bn+1 sin(ka) − An ] = !2 Bn . Substituting the expression for Bn+1 and rearranging, one obtains

An+1 =

2maV0 Bn cos(ka) − Bn sin(ka) + An cos(ka) . !2 k

Similarly, replacing the expression for An+1 in that for Bn+1 , one obtains the parallel equation, Bn+1 =

2maV0 Bn sin(ka) + Bn cos(ka) + An sin(ka) . !2 k

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17

2.2. WAVE MECHANICS OF BOUND PARTICLES

18

With these two eqations, and the relations An+1 = eiφ An and Bn+1 = eiφ Bn , we obtain the quantization condition,6 cos φ = cos(ka) +

maV0 sin(ka) . !2 k

√ As !k = 2mE, this result relates the allowed values of energy to the real parameter, φ. Since cos φ can only take values between −1 and 1, there are a sequence of allowed bands of energy with energy gaps separating these bands (see Fig. 2.2). Such behaviour is characteristic of the spectrum of periodic lattices: In the periodic system, the wavefunctions – known as Bloch states – are indexed by a “quasi”momentum index k, and a band index n where each Bloch band is separated by an energy gap within which there are no allowed states. In a metal, electrons (fermions) populate the energy states starting with the lowest energy up to some energy scale known as the Fermi energy. For a partially-filled band, low-lying excitations associated with the continuum of states allow electrons to be accelerated by a weak electric field. In a band insulator, all states are filled up to an energy gap. In this case, a small electric field is unable to excite electrons across the energy gap – hence the system remains insulating.

' Exercise. In the Kronig-Penney model above, we took the potential to be repulsive. Consider what happens if the potential is attractive when we also have to consider the fate of the states that were bound for the single δ-function potential. In this case, you will find that the methodology and conclusions mirror the results of the repulsive potential: all states remain extended and the continuum of states exhibits a sequence of band gaps controlled by similar sets of equations.

6

Eliminating An and Bn from the equations, a sequence of cancellations obtains „ « 2maV0 e2iφ − eiφ sin(ka) + 2 cos(ka) + 1 = 0 . 2 ! k

Then multiplying by e−iφ , we obtain the expression for cos φ.

Advanced Quantum Physics

Figure 2.2: Solid line shows the variation of cos φ with ka over a range from −1 to 1 for V0 = 2 and ma2 !2 = 1. The blue line shows 0.01×E = (!k)2 /2m. The shaded region represents values of k and energy for which there is no solution.

Chapter 3

Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be represented through a wave-like description. For example, the electron spin degree of freedom does not translate to the action of a gradient operator. It is therefore useful to reformulate quantum mechanics in a framework that involves only operators. Before discussing properties of operators, it is helpful to introduce a further simplification of notation. One advantage of the operator algebra is that it ˆ = pˆ2 , does not rely upon a particular basis. For example, when one writes H 2m where the hat denotes an operator, we can equally represent the momentum operator in the spatial coordinate basis, when it is described by the differential operator, pˆ = −i!∂x , or in the momentum basis, when it is just a number pˆ = p. Similarly, it would be useful to work with a basis for the wavefunction which is coordinate independent. Such a representation was developed by Dirac early in the formulation of quantum mechanics. In the parlons of mathematics, square integrable functions (such as wavefunctions) are said form a vector space, much like the familiar three-dimensional vector spaces. In the Dirac notation, a state vector or wavefunction, ψ, is represented as a “ket”, |ψ". Just as we can express any three-dimensional vector in terms of the basis vectors, r = xˆ e1 + yˆ e2 + zˆ e3 , so we can expand any wavefunction as a superposition of basis state vectors, |ψ" = λ1 |ψ1 " + λ2 |ψ2 " + · · · . Alongside the ket, we can define the “bra”, #ψ|. Together, the bra and ket define the scalar product ! ∞ dx φ∗ (x)ψ(x) , #φ|ψ" ≡ −∞

from which follows the identity, #φ|ψ"∗ = #ψ|φ". In this formulation, the real space representation of the wavefunction is recovered from the inner product ψ(x) = #x|ψ" while the momentum space wavefunction is obtained from ψ(p) = #p|ψ". As with a three-dimensional vector space where a · b ≤ |a| |b|, the magnitude of the scalar product is limited by the magnitude of the vectors, " #ψ|φ" ≤ #ψ|ψ"#φ|φ" , a relation known as the Schwartz inequality.

Advanced Quantum Physics

3.1. OPERATORS

3.1

20

Operators

An operator Aˆ is a “mathematical object” that maps one state vector, |ψ", ˆ into another, |φ", i.e. A|ψ" = |φ". If ˆ A|ψ" = a|ψ" , with a real, then |ψ" is said to be an eigenstate (or eigenfunction) of Aˆ with eigenvalue a. For example, the plane wave state ψp (x) = #x|ψp " = A eipx/! is an eigenstate of the momentum operator, pˆ = −i!∂x , with eigenvalue p. For a free particle, the plane wave is also an eigenstate of the Hamiltonian, ˆ = pˆ2 with eigenvalue p2 . H 2m 2m In quantum mechanics, for any observable A, there is an operator Aˆ which acts on the wavefunction so that, if a system is in a state described by |ψ", the expectation value of A is ˆ #A" = #ψ|A|ψ" =

!

∞

ˆ dx ψ ∗ (x)Aψ(x) .

(3.1)

−∞

Every operator corresponding to an observable is both linear and Hermitian: That is, for any two wavefunctions |ψ" and |φ", and any two complex numbers α and β, linearity implies that ˆ ˆ ˆ A(α|ψ" + β|φ") = α(A|ψ") + β(A|φ") . ˆ the Hermitian conjugate operator Moreover, for any linear operator A, (also known as the adjoint) is defined by the relation ˆ = #φ|Aψ"

!

ˆ = dx φ∗ (Aψ)

!

dx ψ(Aˆ† φ)∗ = #Aˆ† φ|ψ" .

(3.2)

ˆ From the definition, #Aˆ† φ|ψ" = #φ|Aψ", we can prove some useful rela† ˆ ˆ and tions: Taking the complex conjugate, #A φ|ψ"∗ = #ψ|Aˆ† φ" = #Aψ|φ", † ˆ then finding the Hermitian conjugate of A , we have ˆ #ψ|Aˆ† φ" = #(Aˆ† )† ψ|φ" = #Aψ|φ",

i.e. (Aˆ† )† = Aˆ .

Therefore, if we take the Hermitian conjugate twice, we get back to the same ˆ † = Aˆ† + B ˆ † just ˆ † = λ∗ Aˆ† and (Aˆ + B) operator. Its easy to show that (λA) † ˆ ˆ ˆ † Aˆ† from the properties of the dot product. We can also show that (AB) = B † † † ˆ ˆ ˆ ˆ ˆ ˆ from the identity, #φ|ABψ" = #A φ|Bψ" = #B A φ|ψ". Note that operators are associative but not (in general) commutative, ˆ ˆ B|ψ") ˆ ˆ ˆ A|ψ" ˆ AˆB|ψ" = A( = (AˆB)|ψ" &= B . A physical variable must have real expectation values (and eigenvalues). This implies that the operators representing physical variables have some special properties. By computing the complex conjugate of the expectation value of a physical variable, we can easily show that physical operators are their own Hermitian conjugate, #! ∞ $∗ ! ∞ ∗ ∗ ∗ ˆ ˆ ˆ ˆ #ψ|H|ψ" = ψ (x)Hψ(x)dx = ψ(x)(Hψ(x)) dx = #Hψ|ψ" . −∞

−∞

ˆ ˆ ˆ † ψ|ψ", and H ˆ † = H. ˆ Operators that are their i.e. #Hψ|ψ" = #ψ|Hψ" = #H own Hermitian conjugate are called Hermitian (or self-adjoint). Advanced Quantum Physics

3.1. OPERATORS

21

ˆ = −i!∇ is Hermitian. Fur' Exercise. Prove that the momentum operator p ther show that the parity operator, defined by Pˆ ψ(x) = ψ(−x) is also Hermitian.

ˆ = Ei |i" form an orthonormal (i.e. Eigenfunctions of Hermitian operators H|i" #i|j" = δij ) complete basis: For % a complete set of states |i", we can expand in a coordinate repa state function |ψ" as |ψ" = i |i"#i|ψ". % Equivalently, % resentation, we have ψ(x) = #x|ψ" = i #i|ψ"φi (x), where i #x|i"#i|ψ" = φi (x) = #x|i". ' Info. Projection operators and completeness: A ‘ket’ state vector followed by a ‘bra’ state vector is an example of an operator. The operator which projects a vector onto the jth eigenstate is given by |j"#j|. First the bra vector dots into the state, giving the coefficient of |j" in the state, then its multiplied by the unit vector |j", turning it back into a vector, with the right length to be a projection. An operator maps one vector into another vector, so this is an operator. If we sum over a complete set of states, like the eigenstates of a Hermitian operator, we obtain the (useful) resolution of identity & |i"#i| = I . i

% Again, in coordinate form, we can write i φ∗i (x)φi (x" ) = δ(x − x" ). Indeed, we can % form a projection operator into a subspace, Pˆ = subspace |i"#i|.

As in a three-dimensional vector % space, the expansion of the vectors |φ" and % allows the dot product to be taken |ψ", as |φ" = i bi |i" and |ψ" = i ci |i",% by multiplying the components, #φ|ψ" = i b∗i ci . ' Example: The basis states can be formed from any complete set of orthogonal states. In particular, they can' be formed from' the basis states of the position or ∞ ∞ the momentum operator, i.e. −∞ dx|x"#x| = −∞ dp|p"#p| = I. If we apply these definitions, we can then recover the familiar Fourier representation, ! ∞ ! ∞ 1 dp #x|p" #p|ψ" = √ dp eipx/! ψ(p) , ψ(x) ≡ #x|ψ" = ( )* + 2π! −∞ −∞ √ eipx/! / 2π!

where #x|p" denotes the plane wave state |p" expressed in the real space basis.

3.1.1

Time-evolution operator

The ability to develop an eigenfunction expansion provides the means to explore the time evolution of a general wave packet, |ψ" under the action of a Hamiltonian. Formally, we can evolve a wavefunction forward in time by applying the time-evolution operator. For a Hamiltonian which is timeˆ |ψ(0)", where indepenent, we have |ψ(t)" = U ˆ ˆ = e−iHt/! , U 1 denotes % the time-evolution operator. By inserting the resolution of identity, I = i |i"#i|, where the states |i" are eigenstates of the Hamiltonian with eigenvalue Ei , we find that & & ˆ |i"#i|ψ(0)" = |i"#i|ψ(0)"e−iEi t/! . |ψ(t)" = e−iHt/! i

i

ˆ This equation follows from integrating the time-dependent Schr¨ odinger equation, H|ψ! = i!∂t |ψ!. 1

Advanced Quantum Physics

3.1. OPERATORS

22

ˆ = ' Example: Consider the harmonic oscillator Hamiltonian H

pˆ2 2m

+ 12 mω 2 x2 . Later in this chapter, we will see that the eigenstates, |n", have equally-spaced eigenvalues, En = !ω(n + 1/2), for n = 0, 1, 2, · · ·. Let us then consider the time-evolution of a general wavepacket, |ψ(0)", under % the action of the Hamiltonian. From the equation above, we find that |ψ(t)" = n |n"#n|ψ(0)"e−iEn t/! . Since the eigenvalues are equally spaced, let us consider what happens when t = tr ≡ 2πr/ω, with r integer. In this case, since e2πinr = 1, we have & |ψ(tr )" = |n"#n|ψ(0)"e−iωtr /2 = (−1)r |ψ(0)" . n

From this result, we can see that, up to an overall phase, the wave packet is perfectly reconstructed at these times. This recurrence or “echo” is not generic, but is a manifestation of the equal separation of eigenvalues in the harmonic oscillator.

' Exercise. Using the symmetry of the harmonic oscillator wavefunctions under parity show that, at times tr = (2r + 1)π/ω, #x|ψ(tr )" = e−iωtr /2 #−x|ψ(0)". Explain the origin of this recurrence. The time-evolution operator is an example of a unitary operator. The latter are defined as transformations which preserve the scalar product, #φ|ψ" = ˆ ψ" =! #φ|ψ", i.e. ˆ φ|U ˆ ψ" = #φ|U ˆ †U #U ˆ †U ˆ = I. U

3.1.2

Uncertainty principle for non-commuting operators

ˆ B] ˆ &= 0, it is straightforward to For non-commuting Hermitian operators, [A, establish a bound on the uncertainty in their expectation values. Given a state |ψ", the mean square uncertainty is defined as ˆ 2 ψ" = #ψ|U ˆ 2 ψ" (∆A)2 = #ψ|(Aˆ − #A") ˆ − #B") ˆ 2 ψ" = #ψ|Vˆ 2 ψ" , (∆B)2 = #ψ|(B ˆ = Aˆ − #ψ|Aψ" ˆ and Vˆ = B ˆ − #ψ|Bψ". ˆ where we have defined the operators U ˆ ˆ ˆ ˆ ˆ ˆ Since #A" and #B" are just constants, [U , V ] = [A, B]. Now let us take the ˆ |ψ" + iλVˆ |ψ" with itself to develop some information about scalar product of U the uncertainties. As a modulus, the scalar product must be greater than or ˆ ψ|Vˆ ψ" − ˆ 2 ψ" + λ2 #ψ|Vˆ 2 ψ" + iλ#U equal to zero, i.e. expanding, we have #ψ|U ˆ ˆ iλ#V ψ|U ψ" ≥ 0. Reorganising this equation in terms of the uncertainties, we thus find ˆ , Vˆ ]|ψ" ≥ 0 . (∆A)2 + λ2 (∆B)2 + iλ#ψ|[U If we minimise this expression with respect to λ, we can determine when the inequality becomes strongest. In doing so, we find ˆ , Vˆ ]|ψ" = 0, 2λ(∆B)2 + i#ψ|[U

λ=−

ˆ , Vˆ ]|ψ" i #ψ|[U . 2 (∆B)2

Substiuting this value of λ back into the inequality, we then find, 1 ˆ , Vˆ ]|ψ"2 . (∆A)2 (∆B)2 ≥ − #ψ|[U 4

Advanced Quantum Physics

3.1. OPERATORS

23

We therefore find that, for non-commuting operators, the uncertainties obey the following inequality, i ˆ ˆ B]" . ∆A ∆B ≥ #[A, 2 If the commutator is a constant, as in the case of the conjugate operators [ˆ p, x] = −i!, the expectation values can be dropped, and we obtain the relaˆ B]. ˆ For momentum and position, this result recovers tion, (∆A)(∆B) ≥ 2i [A, Heisenberg’s uncertainty principle, ! i p, x]" = . ∆p ∆x ≥ #[ˆ 2 2 ˆ t] = i!, Similarly, if we use the conjugate coordinates of time and energy, [E, we have ∆E ∆t ≥

3.1.3

! . 2

Time-evolution of expectation values

Finally, to close this section on operators, let us consider how their expectation values evolve. To do so, let us consider a general operator Aˆ which may itself involve time. The time derivative of a general expectation value has three terms. d ˆ ˆ ˆ ˆ t |ψ") . #ψ|A|ψ" = ∂t (#ψ|)A|ψ" + #ψ|∂t A|ψ" + #ψ|A(∂ dt If we then make use of the time-dependent Schr¨odinger equation, i!∂t |ψ" = ˆ H|ψ", and the Hermiticity of the Hamiltonian, we obtain d i, ˆ ˆ A|ψ" ˆ ˆ ˆ +#ψ|∂t A|ψ" #ψ|A|ψ" = #ψ|H − #ψ|AˆH|ψ" . dt ! )* + ( i ˆ A]|ψ" ˆ #ψ|[H, ! This is an important and general result for the time derivative of expectation values which becomes simple if the operator itself does not explicitly depend on time, d i ˆ ˆ A]|ψ" ˆ #ψ|A|ψ" = #ψ|[H, . dt ! From this result, which is known as Ehrenfest’s theorem, we see that expectation values of operators that commute with the Hamiltonian are constants of the motion. ' Exercise. Applied to the non-relativistic Schr¨odinger operator for a single

ˆ = pˆ2 + V (x), show that #x" ˙ = particle moving in a potential, H 2m Show that these equations are consistent with the relations, . / . / d d ∂H ∂H #x" = , #ˆ p" = − , dt ∂p dt ∂x the counterpart of Hamilton’s classical equations of motion.

Advanced Quantum Physics

%p& ˆ m ,

#pˆ˙ " = −#∂x V ".

Paul Ehrenfest 1880-1933 An Austrian physicist and mathematician, who obtained Dutch citizenship in 1922. He made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition and the Ehrenfest theorem.

3.2. SYMMETRY IN QUANTUM MECHANICS

3.2

Symmetry in quantum mechanics

Symmetry considerations are very important in quantum theory. The structure of eigenstates and the spectrum of energy levels of a quantum system reflect the symmetry of its Hamiltonian. As we will see later, the transition probabilities between different states under a perturbation, such as that imposed by an external electromagnetic field, depend in a crucial way on the transformation properties of the perturbation and lead to “selection rules”. Symmetries can be classified into two types, discrete and continuous, according to the transformations that generate them. For example, a mirror symmetry is an example of a discrete symmetry while a rotation in three-dimensional space is continuous. Formally, the symmetries of a quantum system can be represented by a ˆ , that act in the Hilbert group of unitary transformations (or operators), U 2 space. Under the action of such a unitary transformation, operators corresponding to observables Aˆ of the quantum model will then transform as, ˆ. ˆ † AˆU Aˆ → U ˆ −1 . ˆ = I, i.e. U ˆ† = U ˆ †U For unitary transformations, we have seen that U Under what circumstances does such a group of transformations represent a symmetry group? Consider a Schr¨odinger particle in three dimensions:3 ˆ . We The basic observables are the position and momentum vectors, ˆr and p can always define a transformation of the coordinate system, or the observˆ = ˆr or p ˆ 4 If R is an element ˆ is mapped to R[A]. ables, such that a vector A of the group of transformations, then this transformation will be represented ˆ (R), such that by a unitary operator U ˆU ˆ . ˆ †A ˆ = R[A] U Such unitary transformations are said to be symmetries of a general opˆ p, ˆr) if erator O(ˆ ˆ †O ˆU ˆ = O, ˆ U

i.e.

ˆ U ˆ] = 0 . [O,

ˆ p, ˆr) ≡ H, ˆ the quantum Hamiltonian, such unitary transformations are If O(ˆ said to be symmetries of the quantum system.

3.2.1

Observables as generators of transformations

ˆ and ˆr for a Schr¨odinger particle are themselves generaThe vector operators p tors of space-time transformations. From the standard commutation relations 2

In quantum mechanics, the possible states of a system can be represented by unit vectors (called “state vectors”) residing in “state space” known as the Hilbert space. The precise nature of the Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions. 3 In the following, we will focus our considerations on the realm of “low-energy” physics where the relevant space-time transformations belong to the Galilei group, leaving our discussion of Lorentz invariance to the chapter on relativistic quantum mechanics. 4 e.g., for a clockwise spatial rotation by an angle θ around ez , we have, 0 1 cos θ sin θ 0 ˆj , R = @ − sin θ cos θ 0 A . R[r] = Rij x 0 0 1

Similarly, for a spatial translation by a vector a, R[r] = r + a. (Exercise: construct representations for transformations corresponding to spatial reflections, and inversion.)

Advanced Quantum Physics

24

3.2. SYMMETRY IN QUANTUM MECHANICS

25

one can show that, for a constant vector a, the unitary operator $ # i ˆ ˆ , U (a) = exp − a · p ! acting in the Hilbert space of a Schr¨odinger particle performs a spatial transˆ † (a)f (r)U ˆ (a) = f (r + a), where f (r) denotes a general algebraic lation, U function of r. ˆ = −i!∇, ' Info. The proof runs as follows: With p ˆ † (a) = ea·∇ = U

∞ & 1 ai · · · ain ∇i1 · · · ∇in , n! 1 n=0

where summation on the repeated indicies, im is assumed. Then, making use of the Baker-Hausdorff identity (exercise) ˆ ˆ ˆ −A ˆ [A, ˆ B]] ˆ + ··· , ˆ + [A, ˆ B] ˆ + 1 [A, eA Be =B 2!

(3.3)

it follows that ˆ † (a)f (r)U ˆ (a) = f (r) + ai (∇i f (r)) + 1 ai ai (∇i ∇i f (r)) + · · · = f (r + a) , U 1 1 1 2 2! 1 2 where the last identity follows from the Taylor expansion.

' Exercise. Prove the Baker-Hausdorff identity (3.3). Therefore, a quantum system has spatial translation as an invariance group if and only if the following condition holds, ˆ (a)H ˆ =H ˆU ˆ (a), U

i.e.

ˆ =H ˆp ˆH ˆ. p

ˆ = H(ˆ ˆ p), This demands that the Hamiltonian is independent of position, H as one might have expected! Similarly, the group of unitary transformaˆ (b) = exp[− i b · ˆr], performs translations in momentum space. tions, U ! ˆ (b) = Moreover, spatial rotations are generated by the transformation U i ˆ ˆ ˆ denotes the angular momentum operator. exp[− ! θen · L], where L = ˆr × p ' Exercise. For an infinitesimal rotation by an angle θ by a fixed axis, eˆn , 2 ˆ = I − i θˆ construct R[r] and show that U ! en · L + O(θ ). Making use of the identity a N −a limN →∞ (1 − N ) = e , show that “large” rotations are indeed generated by the 1 0 ˆ = exp − i θˆ unitary transformations U ! en · L .

As we have seen, time translations are generated by the time evolution opˆ (t) = exp[− i Ht]. ˆ erator, U Therefore, every observable which commutes with ! the Hamiltonian is a constant of the motion (invariant under time translations), ˆ ˆ ˆ ˆ −iHt/! ˆ Aˆ = AˆH ˆ ⇒ eiHt/! = A, Ae H

∀t .

We now turn to consider some examples of discrete symmetries. Amongst these, perhaps the most important in low-energy physics are parity and timereversal. The parity operation, denoted Pˆ , involves a reversal of sign on all coordinates. Pˆ ψ(r) = ψ(−r) . Advanced Quantum Physics

3.2. SYMMETRY IN QUANTUM MECHANICS

26

This is clearly a discrete transformation. Application of parity twice returns the initial state implying that Pˆ 2 = 1. Therefore, the eigenvalues of the parity operation (if such exist) are ±1. A wavefunction will have a defined parity if and only if it is an even or odd function. For example, for ψ(x) = cos(x), Pˆ ψ = cos(−x) = cos(x) = ψ; thus ψ is even and P = 1. Similarly ψ = sin(x) is odd with P = −1. Later, in the next chapter, we will encounter the spherical harmonic functions which have the following important symmetry under parity, Pˆ Y!m = (−1)! Ylm . Parity will be conserved if the Hamiltonian is invariant under the parity operation, i.e. if the Hamiltonian is invariant under a reversal of sign of all the coordinates.5 In classical mechanics, the time-reversal operation involves simply “running the movie backwards”. The time-reversed state of the phase space coordinates (x(t), p(t)) is defined by (xT (t), pT (t)) where xT (t) = x(t) and pT (t) = −p(t). Hence, if the system evolved from (x(0), p(0)) to (x(t), p(t)) in time t and at t we reverse the velocity, p(t) → −p(t) with x(t) → x(t), at time 2t the system would have returned to x(2t) = x(0) while p(2t) = −p(0). If this happens, we say that the system is time-reversal invariant. Of course, this is just the statement that Newton’s laws are the same if t → −t. A notable case where this is not true is that of a charged particle in a magnetic field. As with classical mechanics, time-reversal in quantum mechanics involves the operation t → −t. However, referring to the time-dependent Schr¨odinger ˆ t), we can see that the operation t → −t is equation, i!∂t ψ(x, t) = Hψ(x, ˆ ∗ = H. ˆ equivalent to complex conjugation of the wavefunction, ψ → ψ ∗ if H Let us then consider the time-evolution of ψ(x, t), i

ˆ

c.c.

i

ˆ ∗ (x)t

ψ(x, 0) → e− ! H(x)t ψ(x, 0) → e+ ! H

evolve

i

ˆ

i

ˆ ∗ (x)t

ψ ∗ (x, 0) → e− ! H(x)t e+ ! H

ψ ∗ (x, 0) .

ˆ ∗ (x) = H(x). ˆ Therefore, If we require that ψ(x, 2t) = ψ ∗ (x, 0), we must have H ˆ ˆ H is invariant under time-reversal if and only if H is real. ' Info. Although the group of space-transformations covers the symmetries that pertain to “low-energy” quantum physics, such as atomic physics, quantum optics, and quantum chemistry, in nuclear physics and elementary particle physics new observables come into play (e.g. the isospin quantum numbers and the other quark charges in the standard model). They generate symmetry groups which lack a classical counterpart, and they do not have any obvious relation with space-time transformations. These symmetries are often called internal symmetries in order to underline this fact.

3.2.2

Consequences of symmetries: multiplets

Having established how to identify whether an operator belongs to a group of symmetry transformations, we now consider the consequences. Consider ˆ in the Hilbert space, and an observable Aˆ a single unitary transformation U ˆ ˆ ˆ which commutes with U , [U , A] = 0. If Aˆ has an eigenvector |a", it follows ˆ |a" will be an eigenvector with the same eigenvalue, i.e. that U ˆ A|a" ˆ = AU ˆ |a" = aU |a" . U This means that either: 5

In high energy physics, parity is a symmetry of the strong and electromagnetic forces, but does not hold for the weak force. Therefore, parity is conserved in strong and electromagnetic interactions, but is violated in weak interactions.

Advanced Quantum Physics

3.3. THE HEISENBERG PICTURE

27

ˆ , or 1. |a" is an eigenvector of both Aˆ and U 2. the eigenvalue a is degenerate: the linear space spanned by the vectors ˆ n |a" (n integer) are eigenvectors with the same eigenvalue. U This mathematical argument leads to the conclusion that, given a group G of ˆ (x), x ∈ G, for any observable which is invariant under unitary operators U these transformations, i.e. ˆ (x), A] ˆ = 0 ∀x ∈ G , [U its discrete eigenvalues and eigenvectors will show a characteristic multiplet structure: there will be a degeneracy due to the symmetry such that the eigenvectors belonging to each eigenvalue form an invariant subspace under the group of transformations. ' Example: For example, if the Hamiltonian commutes with the angular moˆ i , i = x, y, z, i.e. it is invariant under three-dimensional rotamentum operators, L tions, an energy level with a given orbital quantum number , is at least (2, + 1)-fold degenerate. Such a degeneracy can be seen as the result of non-trivial actions of ˆ x and L ˆ y on an energy (and L ˆ z ) eigenstate |E, ,, m" (where m is the the operator L ˆ z ). magnetic quantum number asssociated with L

3.3

The Heisenberg Picture

Until now, the time dependence of an evolving quantum system has been placed within the wavefunction while the operators have remained constant – this is the Schr¨ odinger picture or representation. However, it is sometimes useful to transfer the time-dependence to the operators. To see how, let ˆ us consider the expectation value of some operator B, ˆ ˆ ˆ ˆ ˆ ˆ −iHt/! ˆ −iHt/! #ψ(t)|B|ψ(t)" = #e−iHt/!ψ(0)|B|e ψ(0)" = #ψ(0)|eiHt/!Be |ψ(0)" .

According to rules of associativity, we can multiply operators together beˆ ˆ ˆ −iHt/! ˆ , the timefore using them. If we define the operator B(t) = eiHt/!Be dependence of the expectation values has been transferred from the wavefunction. This is called the Heisenberg picture or representation and in it, the operators evolve with time while the wavefunctions remain constant. In this representation, the time derivative of the operator itself is given by ˆ = ∂t B(t)

ˆ ˆ ˆ iH ˆ ˆ ˆ ˆ −iHt/! ˆ iH e−iHt/! − eiHt/!B eiHt/!Be ! ! i ˆ i ˆ ˆ ˆ ˆ B]e ˆ −iHt/! = eiHt/![H, = [H, B(t)] . ! !

ˆ = ' Exercise. For the general Hamiltonian H

pˆ2 2m

+ V (x), show that the position and momentum operators obey Hamilton’s classical equation of motion.

3.4

Quantum harmonic oscillator

As we will see time and again in this course, the harmonic oscillator assumes a priveledged position in quantum mechanics and quantum field theory finding Advanced Quantum Physics

Werner Heisenberg 1901-76 A German physicist and one of the founders of the quantum theory, he is best known for his uncertainty principle which states that it is impossible to determine with arbitrarily high accuracy both the position and momentum of a particle. In 1926, Heisenberg developed a form of the quantum theory known as matrix mechanics, which was quickly shown to be fully equivalent to Erwin Schr¨ odinger’s wave mechanics. His 1932 Nobel Prize in Physics cited not only his work on quantum theory but also work in nuclear physics in which he predicted the subsequently verified existence of two allotropic forms of molecular hydrogen, differing in their values of nuclear spin.

3.4. QUANTUM HARMONIC OSCILLATOR

28

numerous and somtimes unexpected applications. It is useful to us now in that it provides a platform for us to implement some of the technology that has been developed in this chapter. In the one-dimensional case, the quantum harmonic oscillator Hamiltonian takes the form, 2 ˆ = pˆ + 1 mω 2 x2 , H 2m 2 where pˆ = −i!∂x . To find the eigenstates of the Hamiltonian, we could look for solutions of the linear second order differential equation correspondˆ = Eψ, where H ˆ = ing to the time-independent Schr¨odinger equation, Hψ 1 !2 2 2 2 − 2m ∂x + 2 mω x . The integrability of the Schr¨odinger operator in this case allows the stationary states to be expressed in terms of a set of orthogonal functions known as Hermite polynomials. However, the complexity of the exact eigenstates obscure a number of special and useful features of the harmonic oscillator system. To identify these features, we will instead follow a method based on an operator formalism. The form of the Hamiltonian as the sum of the squares of momenta and position suggests that it can be recast as the “square of an operator”. To this end, let us introduce the operator 2 2 3 4 3 4 mω mω pˆ pˆ † x+i , a = x−i , a= 2! mω 2! mω

where, for notational convenience, we have not drawn hats on the operators a and its Hermitian conjuate a† . Making use of the identity, a† a =

ˆ i H 1 mω 2 pˆ x + + [x, pˆ] = − 2! 2!mω 2! !ω 2

and the parallel relation, aa† = commutation relations

ˆ H !ω

+ 12 , we see that the operators fulfil the

[a, a† ] ≡ aa† − a† a = 1 . Then, setting n ˆ = a† a, the Hamiltonian can be cast in the form ˆ = !ω(ˆ H n + 1/2) . Since the operator n ˆ = a† a must lead to a positive definite result, we see that the eigenstates of the harmonic oscillator must have energies of !ω/2 or higher. Moreover, the ground state |0" can be identified by finding the state for which a|0" = 0. Expressed in the coordinate basis, this translates to the equation,6 2 3 4 mω −mωx2 /2! ! ∂x ψ0 (x) = 0, x+ ψ0 (x) = #x|0" = . e mω π 1/2 ! Since n ˆ |0" = a† a|0" = 0, this state is an eigenstate with energy !ω/2. The higher lying states can be found by acting upon this state with the operator ˆ |n" = n|n", we have a† . The proof runs as follows: If n aa† |n" = (a† ()*+ a† a +a† )|n" = (n + 1)a† |n" n ˆ (a† |n") = a† ()*+ a† a+1

n ˆ

6 " ! Formally, in coordinate basis, we have #x" |a|x! R = δ(x − x)(a + mω ∂x ) and #x|0! = ψ0 (x). Then making use of the resolution of identity dx|x!#x| = I, we have „ « Z ! #x|a|0! = 0 = dx #x|a|x" !#x" |0! = x + ∂x ψ0 (x) . mω

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First few states of the quantum harmonic oscillator. Not that the parity of the state changes from even to odd through consecutive states.

3.4. QUANTUM HARMONIC OSCILLATOR or, equivalently, [ˆ n, a† ] = a† . In other words, if |n" is an eigenstate of n ˆ with eigenvalue n, then a† |n" is an eigenstate with eigenvalue n + 1. From this result, we can deduce that the eigenstates for a “tower” |0", |1" = C1 a† |0", |2" = C2 (a† )2 |0", etc., where Cn denotes the normalization. If #n|n" = 1 we have n + 1)|n" = (n + 1) . #n|aa† |n" = #n|(ˆ 1 a† |n" the state |n + 1" is also normalized, Therefore, with |n + 1" = √n+1 #n + 1|n + 1" = 1. By induction, we can deduce the general normalization,

1 |n" = √ (a† )n |0" , n! ˆ with #n|n% " = δnn" , H|n" = !ω(n + 1/2)|n" and a† |n" =

√ n + 1|n + 1",

a|n" =

√

n|n − 1" .

The operators a and a† represent ladder operators and have the effect of lowering or raising the energy of the state. In fact, the operator representation achieves something quite remarkable and, as we will see, unexpectedly profound. The quantum harmonic oscillator describes the motion of a single particle in a one-dimensional potential well. It’s eigenvalues turn out to be equally spaced – a ladder of eigenvalues, separated by a constant energy !ω. If we are energetic, we can of course translate our results into a coordinate representation ψn (x) = #x|n".7 However, the operator representation affords a second interpretation, one that lends itself to further generalization in quantum field theory. We can instead interpret the quantum harmonic oscillator as a simple system involving many fictitious particles, each of energy !ω. In this representation, known as the Fock space, the vacuum state |0" is one involving no particles, |1" involves a single particle, |2" has two and so on. These fictitious particles are created and annihilated by the action of the raising and lowering operators, a† and a with canonical commutation relations, [a, a† ] = 1. Later in the course, we will find that these commutation relations are the hallmark of bosonic quantum particles and this representation, known as the second quantization underpins the quantum field theory of the electromagnetic field. ' Info. There is evidently a huge difference between a stationary (Fock) state of the harmonic oscillator and its classical counterpart. For the classical system, the equations of motion are described by Hamilton’s equations of motion, P X˙ = ∂P H = , m

P˙ = −∂X H = −∂x U = −mω 2 X ,

where we have used capital letters to distinguish them from the arguments used to describe the quantum system. In the phase space, {X(t), P (t)}, these equations describe a clockwise rotation along an elliptic trajectory specified by the initial conditions {X(0), P (0)}. (Normalization of momentum by mω makes the trajectory circular.) 7 Expressed in real space, the harmonic oscillator wavefunctions are in fact described by the Hermite polynomials, r « » – „r mωx2 1 mω ψn (x) = #x|n! = H x exp − , n 2n n! ! 2! 2

where Hn (x) = (−1)n ex

dn −x2 e . dxn

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29

3.4. QUANTUM HARMONIC OSCILLATOR

30

On the other hand, the time dependence of the Fock space state, as of any stationary state, is exponential, ψn (x, t) = #x|n"e−iEn t/! , and, as a result, gives time-independent expectation values of x, p, or any function thereof. The best classical image for such a state on the phase plane is a circle of radius r = x0 (2n + 1)1/2 , where x0 = (!/mω)1/2 , along which the wavefunction is uniformly spread as a standing wave. It is natural to ask how to form a wavepacket whose properties would be closer to the classical trajectories. Such states, with the centre in the classical point {X(t), P (t)}, and the smallest possible product of quantum uncertainties of coordinate and momentum, are called Glauber states.8 Conceptually the simplest way to present the Glauber state |α" is as the Fock ground state |0" with the centre shifted from the origin to the classical point {X(t), P (t)}. (After such a shift, the state automatically rotates, following the classical motion.) Let us study how this shift may be implemented in quantum mechanics. The mechanism for such shifts are called the translation operators. Previously, we have seen that the space and momentum translation operators are given by $ $ # # i i ˆ ˆ FX = exp − pˆX , FP = exp − P x ˆ . ! ! A shift by both X and P is then given by $ # † ∗ i ˆ − pˆX) = eαa −α a , Fˆα = exp (P x !

∗ † Fˆα† = eα a−αa ,

where α is the (normalised) complex amplitude of the classical oscillations we are 1 P (X + i mω ). The Glauber state is then defined trying to approximate, i.e. α = √2x 0 by |α" = Fˆα |0". Working directly with the shift operator is not too convenient because of its exponential form. However, it turns out that a much simpler representation for the Glauber state is possible. To see this, let us start with the following general property ˆ B] ˆ = µ (where Aˆ and B ˆ are operators, and µ is a of exponential operators: if [A, c-number), then (exercise – cf. Eq. (3.3)), ˆ + µ. ˆ −A = B eA Be ˆ

ˆ

(3.4)

ˆ ˆ ˆ = I, we If we define Aˆ = α∗ a − αa† , then Fˆα = e−A and Fˆα† = eA . If we then take B † ˆ ˆ have µ = 0, and Fα Fα = I. This merely means that the shift operator is unitary not a big surprise, because if we shift the phase point by (+α) and then by (−α), we certainly come back to the initial position. ˆ = a, using the commutation relations, If we take B

ˆ B] ˆ = [α∗ a − αa† , a] = −α[a† , a] = α , [A,

so that µ = α, and Fˆα† aFˆα = a + α. Now let us consider the operator Fˆα Fˆα† aFˆα . From the unitarity condition, this must equal aFˆα , while application of Eq. (3.4) yields Fˆα a + αFˆα , i.e. aFˆα = Fˆα a + αFˆα .

Applying this equality to the ground state |0" and using the following identities, a|0" = 0 and Fˆα |0" = |α", we finally get a very simple and elegant result: a|α" = α|α" . 8 After R. J. Glauber who studied these states in detail in the mid-1960s, though they were known to E. Schr¨ odinger as early as in 1928. Another popular name, coherent states, does not make much sense, because all the quantum states we have studied so far (including the Fock states) may be presented as coherent superpositions.

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3.5. POSTULATES OF QUANTUM THEORY Thus the Glauber state is an eigenstate of the annihilation operator, corresponding to the eigenvalue α, i.e. to the (normalized) complex amplitude of the classical process approximated by the state. This fact makes the calculations of the Glauber state properties much simpler. Presented as a superposition of Fock states, the Glauber state takes the form (exercise – try making use of the BCH identity (3.3).) |α" =

∞ &

n=0

αn |n",

αn = e−|α|

2

/2

αn . (n!)1/2

This means that the probability of finding the system in level n is given by the Poisson distribution, Pn = |αn |2 = #n"n e−%n& /n! where #n" = |α|2 . More importantly, δn = #n"1/2 / #n" when #n" 0 1 – the Poisson distribution approaches the Gaussian distribution when #n" is large. The time-evolution of Glauber states may be described most easily in the Schr¨odinger representation when the time-dependence is transferred to the wavefunction. In this (t) 1 case, α(t) ≡ √2x (X(t) + i Pmω ), where {X(t), P (t)} is the solution to the classical 0 equations of motion, α(t) ˙ = −iωα(t). From the solution, α(t) = α(0)e−iωt , one may show that the average position and momentum evolve classically while their fluctuations remain stationary, 41/2 41/2 3 3 x0 mωx0 ! !mω ∆x = √ = , ∆p = √ . = 2mω 2m 2 2 In the quantum theory of measurements these expressions are known as the “standard quantum limit”. Notice that their product ∆x ∆p = !/2 corresponds to the lower bound of the Heisenberg’s uncertainty relation.

' Exercise. Show that, in position space, the Glauber state takes the form $ # mω Px #x|α" = ψα (x) = C exp − (x − X)2 + i . 2! !

This completes our abridged survey of operator methods in quantum mechanics. With this background, we are now in a position to summarize the basic postulates of quantum mechanics.

3.5

Postulates of quantum theory

Since there remains no “first principles” derivation of the quantum mechanical equations of motion, the theory is underpinned by a set of “postulates” whose validity rest on experimental verification. Needless to say, quantum mechanics remains perhaps the most successful theory in physics. ' Postulate 1. The state of a quantum mechanical system is completely specified by a function Ψ(r, t) that depends upon the coordinates of the particle(s) and on time. This function, called the wavefunction or state function, has the important property that |Ψ(r, t)|2 dr represents the probability that the particle lies in the volume element dr ≡ dd r located at position r at time t. The wavefunction must satisfy certain mathematical conditions because of this probabilistic interpretation. For the case of a single particle, the net probability of finding it at some ' ∞ point in space must be unity leading to the normalization condition, −∞ |Ψ(r, t)|2 dr = 1. It is customary to also normalize many-particle wavefunctions to unity. The wavefunction must also be single-valued, continuous, and finite. Advanced Quantum Physics

31

3.5. POSTULATES OF QUANTUM THEORY ' Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. If we require that the expectation value of an operator Aˆ is real, then it follows that Aˆ must be a Hermitian operator. If the result of a measurement of an operator Aˆ is the number a, then a must be one of the ˆ = aΨ, where Ψ is the corresponding eigenfunction. This eigenvalues, AΨ postulate captures a central point of quantum mechanics – the values of dynamical variables can be quantized (although it is still possible to have a continuum of eigenvalues in the case of unbound states). ' Postulate 3. If a system is in a state described by a normalized wavefunction Ψ, then the average value of the observable corresponding to Aˆ is given by ! ∞ ˆ Ψ∗ AΨdr #A" = . −∞

If the system is in an eigenstate of Aˆ with eigenvalue a, then any measurement of the quantity A will yield a. Although measurements must always yield an eigenvalue, the state does not have to be an eigenstate of Aˆ initially. An arbitrary state can be expanded in the complete set ˆ i = ai Ψi ) as Ψ = %n ci Ψi , where n may go to of eigenvectors of Aˆ ( AΨ i infinity. In this case, the probability of obtaining the result ai from the measurement of Aˆ is given by P (ai ) = |#Ψi |Ψ"|2 = |ci |2 . The expectation value of Aˆ for the state Ψ is the sum over all possible values of the measurement and given by & & #A" = ai |#Ψi |Ψ"|2 = ai |ci |2 . i

i

Finally, a measurement of Ψ which leads to the eigenvalue ai , causes the wavefunction to “collapses” into the corresponding eigenstate Ψi . (In the case that ai is degenerate, then Ψ becomes the projection of Ψ onto the degenerate subspace). Thus, measurement affects the state of the system. ' Postulate 4. The wavefunction or state function of a system evolves in time according to the time-dependent Schr¨odinger equation i!

∂Ψ ˆ = HΨ(r, t) , ∂t

ˆ is the Hamiltonian of the system. If Ψ is an eigenstate of H, ˆ where H it follows that Ψ(r, t) = Ψ(r, 0)e−iEt/!.

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32

Chapter 4

Quantum mechanics in more than one-dimension Previously, we have explored the manifestations of quantum mechanics in one spatial dimension and discussed the properties of bound and unbound states. The concepts developed there apply equally to higher dimension. However, for a general two or three-dimensional potential, without any symmetry, the solutions of the Schr¨odinger equation are often inaccessible. In such situations, we may develop approximation methods to address the properties of the states (see chapter 7). However, in systems where there is a high degree of symmetry, the quantum mechanics of the system can often be reduced to a tractable lowdimensional theory.

4.1

Rigid diatomic molecule

As a pilot example let us consider the quantum mechanics of a rigid diatomic molecule with nuclear masses m1 and m2 , and (equilibrium) bond length, Re (see figure). Since the molecule is rigid, its coordinates can be specified by 2 r2 its centre of mass, R = m1mr11 +m +m2 , and internal orientation, r = r2 − r1 (with |r| = Re ). Defining the total mass M = m1 + m2 , and moment of inertia, I = µRe2 , where µ = m1 m2 /(m1 + m2 ) denotes the reduced mass, the corresponding Hamiltonian can be then separated into the kinetic energy associated with the centre of mass motion and the rotational kinetic energy, ˆ2 ˆ2 ˆ = P +L , H 2M 2I

(4.1)

ˆ =r×p ˆ = −i!∇R and L ˆ denotes the angular momentum associated where P with the internal degrees of freedom. Since the internal and centre of mass degrees of freedom separate, the wavefunction can be factorized as ψ(r, R) = eiK·R Y (r), where the first factor accounts for the free particle motion of the body, and the second factor relates to the internal angular degrees of freedom. As a result of the coordinate separation, we have reduced the problem of the rigid diatomic molecule to the study of the quantum mechanics of a particle moving on a sphere – the rigid rotor, ˆ2 ˆ rot = L . H 2I The eigenstates of this component of the Hamiltonian are simply the states of the angular momentum operator. Indeed, in any quantum mechanical system involving a radial potential, the angular momentum will be conserved, i.e. Advanced Quantum Physics

4.2. ANGULAR MOMENTUM

34

ˆ = 0 meaning that the angular component of the wavefunction can be ˆ L] [H, indexed by the states of the angular momentum operator. We therefore now digress to discuss the quantum mechanics of angular momentum.

4.2

Angular momentum

4.2.1

Commutation relations

Following the usual canonical quantization procedure, the angular momentum ˆ = r׈ ˆ obey the commutation operator is defined by L p where, as usual, r and p relations, [ˆ pi , rj ] = −i!δij .1 Using this relation, one may then show that the angular momentum operators obey the spin commutation relations, (exercise) ˆ i, L ˆ j ] = i!#ijk L ˆk . [L

(4.2)

where, as usual, #ijk denotes the totally antisymmetric tensor — the LeviCivita symbol.2 $ Exercise. Show that the angular momentum operator commutes with the ˆ = pˆ 2 + V (r). Show that Hamiltonian of a particle moving in a central potential, H 2m the Hamiltonian of a free particle of mass m confined to a sphere of radius R is given ˆ = Lˆ 2 2 . by H 2mR

4.2.2

Eigenvalues of angular momentum

In the following, we will construct a basis set of angular momentum states. Since the angular momentum is a vector quantity, it may be characterized by its magnitude and direction. For the former, let us define the operator ˆ2 = L ˆ2 + L ˆ2 + L ˆ 2 . With the latter, since the separate components of the L x y z angular momentum are all mutually non-commuting, we cannot construct a common set of eigenstates for any two of them. They do, however, commute ˆ 2 (exercise). Therefore, in the following, we will look for an eigenbasis with L ˆ 2 and one direction, say L ˆz, of L ˆ 2 |a, b$ = a|a, b$, L

ˆ z |a, b$ = b|a, b$ . L

To find |a, b$, we could simply proceed by looking for a suitable coordinate ˆ 2 and L ˆ z in terms of differential operators. However, basis to represent L although we will undertake such a programme in due course, before getting to this formalism, we can make substantial progress without resorting to an explicit coordinate representation. $ Info. Raising and lowering operators for angular momentum: The set of eigenvalues a and b can be obtained by making use of a trick based on a “ladder operator” formalism which parallels that used in the study of the quantum harmonic oscillator in section 3.4. Specifically, let us define the raising and lowering operators, ˆ x ± iL ˆy . ˆ± = L L 1 In this chapter, we will index the angular momentum operators with a ‘hat’. Later, we will become lazy and the hat may well disappear. 2 Recall that !ijk = 1 if (i, j, k) is an even permutation of (1,2,3), −1 if it is an odd permutation, and 0 if any index is repeated.

Advanced Quantum Physics

4.2. ANGULAR MOMENTUM

35

With this definition, one may then show that (exercise) ˆz, L ˆ ± ] = ±!L ˆ± . [L

ˆ 2 , we can deduce Since each component of the angular momentum commutes with L ˆ that the action of L± on |a, b$ cannot affect the value of a relating to the magnitude of the angular momentum. However, they do effect the projection: ˆzL ˆ±L ˆz, L ˆ ± ]|a, b$ = (b ± !)L ˆ ± |a, b$ . ˆ ± |a, b$ = L ˆ z |a, b$ + [L L

ˆ z with eigenvalue b, L ˆ ± |a, b$ is either zero, Therefore, if |a, b$ is an eigenstate of L ˆ ˆ or an eigenstate of Lz with eigenvalue b ± !, i.e. L± |a, b$ = C± (a, b)|a, b ± !$ where C± (a, b) is a normalisation constant. To fix the normalisation, we may note that the norm, !! !! 2 !! ˆ !! ˆ† L ˆ ˆ ˆ !!L± |a, b$!! = %a, b|L ± ± |a, b$ = %a, b|L∓ L± |a, b$ ,

ˆ ∓ . Then, making use of the relation L ˆ∓L ˆ† = L ˆ± = where we have used the identity L ± 2 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Lx + Ly ± i[Lx Ly ] = L − Lz ± !Lz , and the presumed normalisation, %a, b|a, b$ = 1, one finds !! !!2 " # !! ˆ !! ˆ2 − L ˆ 2 ± !L ˆ z |a, b$ = a − b2 ∓ !b . (4.3) !!L± |a, b$!! = %a, b| L z

As a represents the eigenvalue of a sum of squares of Hermitian operators, it is necessarily non-negative. Moreover, b is real. Therefore, for a given a, b must be bounded: there must be a bmax and a (negative or zero) bmin . In particular, !!2 !! !! !! ˆ !!L+ |a, bmax $!! = a − b2max − !bmax !! !!2 !! ˆ !! !!L− |a, bmin $!! = a − b2min + !bmin ,

For a given a, bmax and bmin are determined uniquely — there cannot be two states ˆ + . It also follows immediately that with the same a but different b annihilated by L a = bmax (bmax +!) and bmin = −bmax . Furthermore, we know that if we keep operating ˆ z eigenvalues bmin + !, ˆ + , we generate a sequence of states with L on |a, bmin $ with L bmin + 2!, bmin + 3!, · · ·. This series must terminate, and the only possible way for that to happen is for bmax to be equal to bmin + n! with n integer, from which it follows that bmax is either an integer or half an odd integer times ! At this point, we switch to the standard notation. We have established that the ˆ z form a finite ladder, with spacing !. We write them as m!, and % eigenvalues of L ˆ 2 , a = %(% + 1)!2 . is used to denote the maximum value of m, so the eigenvalue of L Both % and m will be integers or half odd integers, but the spacing of the ladder of m values is always unity. Although we have been writing |a, b$ with a = %(% + 1)!2 , b = m! we shall henceforth follow convention and write |%, m$.

ˆ 2 and L ˆ z have a common set of orthonormal In summary, the operators L eigenstates |%, m$ with ˆ 2 |%, m$ = %(% + 1)!2 |%, m$, L

ˆ z |%, m$ = m!|%, m$ , L

(4.4)

where %, m are integers or half-integers. The allowed quantum numbers m form a ladder with step spacing unity, the maximum value of m is %, and the minimum value is −%. With these results, we may then return to the normalization of the raising and lowering operators. In particular, making use of Eq. (4.3), we have $ ˆ + |%, m$ = %(% + 1) − m(m + 1)!|l, m + 1$ L $ ˆ − |%, m$ = %(% + 1) − m(m − 1)!|l, m − 1$ . L

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(4.5)

Figure 4.1: The following is a

schematic showing the angular momentum $ scheme for % = √ 2 2(2 + 1)! = 6! with L2 = and the five possible values for the Lz projection.

4.2. ANGULAR MOMENTUM

36

The use of m to denote the component of angular momentum in one direction came about because a Bohr-type electron in orbit is a current loop, with a magnetic moment parallel to its angular momentum. So the m measured the component of magnetic moment in a chosen direction, usually along an external magnetic field. For this reason, m is often termed the magnetic quantum number.

4.2.3

Representation of the angular momentum states

Having established expressions for the eigenvalues of the angular momentum operators, it is now necessary to establish coordinate representations for the corresponding eigenstates, Y!m (θ, φ) = %θ, φ|%, m$. Here the angles θ and φ denote the spherical coordinates parameterising the unit sphere (see figure). Previously, we obtained the eigenvalues of the angular momentum operator by making use of the raising and lowering operators in a manner that parallelled the study of the quantum harmonic oscillator. Similarly, to obtain explicit expressions for the eigenstates, we must make use of the coordinate representation of these operators. With r = rˆ er , the gradient operator can be written in spherical polar coordinates as 1 1 ˆθ ∂θ + e ˆφ ˆr ∂r + e ∂φ . ∇=e r r sin θ From this result, we thus obtain ˆ ± = !e±iφ (±∂θ + i cot θ∂φ ) , L

ˆ z = −i!∂φ , L

(4.6)

and, at least formally, ˆ 2 = −!2 L

%

& 1 1 2 ∂θ (sin θ∂θ ) + ∂ . sin θ sin2 θ φ

ˆ z , the eigenvalue equation (4.4), and Beginning with the eigenstates of L making use of the expression above, we have −i!∂φ Y!m (θ, φ) = m!Y!m (θ, φ) . Since the left hand side depends only on φ, the solution is separable and takes the form Y!m (θ, φ) = F (θ)eimφ . Note that, since m is integer, the continuity of the wavefunction, Y!m (θ, φ + 2π) = Y!m (θ, φ), is ensured. To determine the second component of the eigenstates, F (θ), we could immediately turn to the eigenvalue equation involving the differential operator ˆ 2, for L % & 1 m2 2 ∂ ∂θ (sin θ∂θ ) − F (θ) = %(% + 1)F (θ) . sin θ sin2 θ φ However, to construct the states, it is easier to draw upon the properties of the angular momentum raising and lowering operators (much in the same way that the Hermite polynomials are generated by the action of ladder operators in the harmonic oscillator problem). ˆ + |%, %$ = 0. MakConsider then the state of maximal m, |%, %$, for which L ing use of the coordinate representation of the raising operator above together with the separability of the wavefunction, this relation implies that ˆ + |%, %$ = !eiφ (∂θ + i cot θ∂φ ) Y!! (θ, φ) = !ei(!+1)φ (∂θ − % cot θ) F (θ) . 0 = %θ, φ|L Advanced Quantum Physics

4.2. ANGULAR MOMENTUM

37

From this result it follows that ∂θ F (θ) = % cot θF (θ) with the solution F (θ) = C sin! θ, and C a constant of normalization. States with values of m lower than % can then be obtained simply by repeated application of the angular ˆ − to the state |%, %$. This amounts to the momentum lowering operator L relation ' ( ˆ − )!−m sin! θei!φ Y!m (θ, φ) = C(L ' ( = C (−∂θ + i cot θ∂φ )!−m sin! θei!φ . The eigenfunctions produced by this procedure are well known and referred to as the spherical harmonics. In particular, one finds that the normalized eigenstates take the form, Y!m (θ, φ) = (−1)

m+|m|

%

2% + 1 (% − |m|)! 4π (% + |m|)!

&1/2

|m|

P! (cos θ)eimφ ,

(4.7)

where P!m (ξ) =

(1 − ξ 2 )m/2 dm+! 2 (ξ − 1)! , 2! %! dξ m+!

represent the associated Legendre polynomials. In particular, for the first few angular momentum states, we have Y00 = Y10 = Y20 =

√1

)4π

) 3 iφ Y11 = − 8π e sin θ ) ) 5 15 iφ 15 2iφ 2 θ − 1), Y (3 cos = − e sin θ cos θ, Y = sin2 θ 21 22 16π 8π 32π e

3 ) 4π

cos θ,

Figure 4.2 shows a graphical representation of the states for the lowest spherical harmonics. From the colour coding of the states, the symmetry, Y!,−m = ∗ is manifest. (−1)m Y!m As a complete basis set, the spherical harmonics can be used as a resolution of the identity ∞ * ! *

!=0 m=−!

|%, m$%%, m| = I .

Equivalently, expressed in the coordinate basis, we have ∞ * ! *

∗ Y!,m (θ% , φ% )Y!,m (θ, φ) =

!=0 m=−!

1 δ(θ − θ% )δ(φ − φ% ) , sin θ

where the prefactor sin θ derives from the measure. Similarly, we have the orthogonality condition, + π + 2π ∗ Y!,m (θ, φ)Y!! ,m! (θ, φ) = δ!!! δmm! . dθ sin θ 0

0

After this lengthy digression, we may now return to the problem of the quantum mechanical rotor Hamiltonian and the rigid diatomic molecule. From the analysis above, we have found that the eigenstates of the Hamiltonian (4.1) are given by ψ(R, r) = √12π eiK·R Y!,m (θ, φ) with eigenvalues EK,! = Advanced Quantum Physics

!2 !2 K2 + %(% + 1) , 2M 2I

4.3. THE CENTRAL POTENTIAL

Figure 4.2: First four groups of spherical harmonics, Y!m (θ, φ) shown as a function of spherical angular coordinates. Specifically, the plots show the surface generated by |Re Y!m (θ, φ)| to fix the radial coordinate and the colours indicate the relative sign of the real part. where each K, % value has a 2% + 1-fold degeneracy. $ Exercise. Using this result, determine the dependence of the heat capacity of a gas of rigid diatomic molcules on the angular degrees of freedom. How would this result change if the diatomic gas was constrained to just two spatial dimensions, i.e. the axis of rotation was always perpendicular to the plane in which the molecules can move?

4.3

The central potential

In a system where the central force field is entirely radial, the potential energy depends only on r ≡ |r|. In this case, a general non-relativistic Hamiltonian for a single particle is given by ˆ2 ˆ = p + V (r) . H 2m In the classical system, L2 = (r × p)2 = r2 p2 − (r · p)2 . As a result, we can 2 set p2 = Lr2 + p2r , where pr ≡ er · p denotes the radial component of the momentum. In the quantum system, since the space and position coordinates do not commute, we have (exercise) ˆ 2 = r2 p ˆ )2 + i!r · p ˆ. ˆ 2 − (r · p L ˆ = −i!r · ∇ = −i!r∂r , it follows that In spherical coordinates, since r · p 2 ˆ2 ˆ 2 = Lr2 − !r2 [(r∂r )2 + r∂r ]. Equivalently, noting that (r∂r )2 + r∂r = ∂r2 + 2r ∂r , p we can set % & ˆ2 L 2 2 2 2 ˆ = 2 − ! ∂r + ∂r . p r r Advanced Quantum Physics

38

4.4. ATOMIC HYDROGEN

39

Finally, substituted into the Schr¨odinger equation, we obtain the eigenvalue equation / , . ˆ2 2 L !2 ∂r2 + ∂r + (4.8) + V (r) ψ(r) = Eψ(r) . − 2m r 2mr2 ˆ 2 , we can immediately take adSince we already know the eigenstates of L vantage of the separability of the Hamiltonian to find that ψ(r) = R(r)Y!,m (θ, φ), where the radial part of the wavefunction is set by . & % 2 !2 !2 2 ∂r + ∂r + %(% + 1) + V (r) R(r) = ER(r) . − 2m r 2mr2 Finally, we can further simplify this expression by setting R(r) = u(r)/r, whereupon we obtain the “one-dimensional” equation % 2 2 & ! ∂r − + Veff (r) u(r) = u(r) , (4.9) 2m ! where the effective potential, Veff (r) = 2mr 2 %(% + 1) + V (r), acquires an additional component due to the centrifugal component of the force. Here the equation must be solved subject to the boundary condition u(0) = 0. From the normalization condition, + + ∞ 1 3 2 d r |ψ(r)| = drr2 2 |u(r)|2 = 1 , r 0 2

for a bound state to exist, limr→∞ |u(r)| ≤ a/r1/2+% with # > 0. From this one-dimensional form of the Hamiltonian, the question of the existence of bound states in higher dimension becomes clear. Since the wavefunction u(r) vanishes at the origin, we may map the Hamiltonian from the half-line to the full line with the condition that we admit only antisymmetric wavefunctions. The question of bound states can then be related back to the one-dimensional case. Previously, we have seen that a symmetric attractive potential always leads to a bound state in one-dimension. However, odd parity states become bound only at a critical strength of the interaction.

4.4

Atomic hydrogen

The Hydrogen atom consists of an electron bound to a proton by the Coulomb potential, V (r) = −

1 e2 . 4π#0 r

We can generalize the potential to a nucleus of charge Ze without complication of the problem. Since we are interested in finding bound states of the proton-electron system, we are looking for solutions with E negative. At large separations, the wave equation (4.9) simplifies to −

!2 ∂r2 u(r) * Eu(r) , 2m

√ having approximate solutions eκr and e−κr , where !κ = −2mE. (Here, strictly speaking, m should denote the reduced mass of electron-proton system.) The bound states we are looking for, of course, have exponentially decreasing wavefunctions at large distances. Advanced Quantum Physics

4.4. ATOMIC HYDROGEN

40

To further simplify the wave equation, it is convenient to introduce the dimensionless variable ρ = κr, leading to the equation . 2ν %(% + 1) 2 ∂ρ u(ρ) = 1 − u(ρ) , + ρ ρ2 where (for reasons which will become apparent shortly) we have introduced Ze2 κ . Notice that in transforming from r the dimensionless parameter ν = 4π% 0 2E to the dimensionless variable ρ, the scaling factor depends on energy, so will be different for different energy bound states! Consider now the behaviour of the wavefunction near the origin. The dominant term for sufficiently small ρ is the centrifugal component, so ∂ρ2 u(ρ) *

%(% + 1) u(ρ) , ρ2

for which the solutions are u(ρ) ∼ ρ−! and u(ρ) ∼ ρ!+1 . Since the wavefunction cannot be singular, we must choose the second solution. We have established that the wavefunction decays as e−κr = e−ρ at large distances, and goes as ρ!+1 close to the origin. Factoring out these two asymptotic behaviours, let us then define w(ρ) such that u(ρ) = e−ρ ρ!+1 w(ρ). We leave it as a tedious but straightforward exercise to show that ρ∂ρ2 w(ρ) + 2(% + 1 − ρ)∂ρ w(ρ) + 2(ν − (% + 1))w(ρ) = 0 . Substituting the trial series solution, w(ρ) = rence relation between successive coefficients:

0∞

k k=0 wk ρ ,

we obtain a recur-

wk+1 2(k + % + 1 − ν) . = wk (k + 1)(k + 2(% + 1)) For large values of k, wk+1 /wk → 2/k, so wk ∼ 2k /k! and therefore w(ρ) ∼ e2ρ . This means we have found the diverging radial wavefunction, u(ρ) ∼ eρ , which is in fact the correct behaviour for general values of the energy. To find the bound states, we must choose energies such that the series is not an infinite one. As long as the series stops somewhere, the exponential decrease will eventually take over, and yield a finite (bound state) wavefunction. Just as for the simple harmonic oscillator, this can only happen if for some k, wk+1 = 0. Inspecting the ratio wk+1 /wk , evidently the condition for a bound state is that ν = n,

integer ,

in which case the series for w(ρ) terminates at k = n − % − 1. From now on, since we know that for the functions we’re interested in ν is an integer, we replace ν by n. Finally, making use of the definitions of ν and κ above, we obtain the bound state energies, En = −

-

Ze2 4π#0

.2

m 1 Z2 ≡ − Ry . 2!2 n2 n2

Remarkably, this is the very same series of bound state energies found by Bohr from his model! Of course, this had better be the case, since the series of energies Bohr found correctly accounted for the spectral lines emitted by hot hydrogen atoms. Notice, though, that there are some important differences Advanced Quantum Physics

4.4. ATOMIC HYDROGEN

41

with the Bohr model: the energy here is determined entirely by n, called the principal quantum number, but, in contrast to Bohr’s model, n is not the angular momentum. The true ground state of the hydrogen atom, n = 1, has zero angular momentum: since n = k + % + 1, n = 1 means both l = 0 and k = 0. The ground state wavefunction is therefore spherically symmetric, and the function w(ρ) = w0 is just a constant. Hence u(ρ) = ρe−ρ w0 and the actual radial wavefunction is this divided by r, and of course suitably normalized. To write the wavefunction in terms of r, we need to find κ. Putting together $ ρ = κn r, κn = −2mEn /!2 and the expression for En , we find that κn = Ze2 m Z 4π%0 !2 n = na0 , where a0 =

4π#0 !2 = 0.529 × 10−10 m me2

denotes the Bohr radius – the radius of the lowest orbit in Bohr’s model.2 1 (Ze) 1 With this definition, the energy levels can then be expressed as En = − 4π% 2. 0 2a0 n Moving on to the excited states: for n = 2, we have a choice: either the radial function w(ρ) can have one term, as before, but now the angular momentum % = 1 (since n = k + % + 1); or w(ρ) can have two terms (so k = 1), and % = 0. Both options give the same energy, 0.25 Ry, since n is the same, and the energy only depends on n. In fact, there are four states at this energy, since % = 1 has states with m = 1, m = 0 and m = 1, and % = 0 has the one state m = 0. For n = 3, there are 9 states altogether: % = 0 gives one, % = 1 gives 3 and % = 2 gives 5 different m values. In fact, for principal quantum number n there are n2 degenerate states (n2 being the sum of the first n odd integers). From now on, we label the wavefunctions with the quantum numbers, ψn!m (r, θ, φ), so the ground state is the spherically symmetric ψ100 (r). For this state R(r) = u(r)/r, where u(ρ) = e−ρ ρ!+1 w(ρ) = e−ρ ρw0 , with w0 a constant and ρ = κ1 r = Zr/a0 . So, as a function of r, R10 (r) = N e−Zr/a0 with N the normalization constant: - .3 Z 2 −Zr/a0 e . R10 = 2 a0 For n = 2, % = 1 the function w(ρ) is still a single term, a constant, but now u(ρ) = e−ρ ρ!+1 w(ρ) = e−ρ ρ2 w0 , and, for n = 2, ρ = κ2 r = Zr/2a0 , remembering the energy-dependence of κ. After normalization, we find - .3/2 - . Zr Z 1 R21 = √ e−Zr/2a0 . a0 2 6 a0 The other n = 2 state has % = 0. So from n = k + % + 1, we have k = 1 and the series for w has two terms, k = 0 and k = 1, the ratio being wk+1 2(k + % + 1 − n) = −1 , = wk (k + 1)(k + 2(% + 1)) for the relevant values: k = 0, % = 0, n = 2. So w1 = −w0 , w(ρ) = w0 (1 − ρ). For n = 2, ρ = r/2a0 , the normalized wavefunction is given by 1 R20 = √ 2

-

Z a0

.3/2 -

1−

1 Zr 2 a0

.

e−Zr/2a0 .

Note that the zero angular momentum wavefunctions are non-zero and have non-zero slope at the origin. This means that the full three-dimensional wavefunctions have a slope discontinuity there! But this is fine - the potential is Advanced Quantum Physics

4.4. ATOMIC HYDROGEN

42

infinite at the origin. (Actually, the proton is not a point charge, so really the kink will be smoothed out over a volume of the size of the proton - a very tiny effect.) In practice, the first few radial functions w(ρ) can be constructed fairly easily using the method presented above, but it should be noted that the differential equation for w(ρ), ρ∂ρ2 w(ρ) + 2(% + 1 − ρ)∂ρ w(ρ) + 2(n − (% + 1))w(ρ) = 0 , is in fact Laplace’s equation, usually written (z∂z2 + (k − 1 − z)∂z + p)Lkp (z) = 0 , where k, p are integers, and Lkp (z) is a Laguerre polynomial. The two equations are the same if z = 2ρ, and the solution to the radial equation is therefore, wn! (ρ) = L2!+1 n−!−1 (2ρ) . The Laguerre polynomials L0p (z), and associated Laguerre polynomials Lkp (z) are given by: L0p (z) = ez

dp −z p (e z ), dz p

Lkp (z) = (−1)k

dp 0 L (z) . dz p p+k

(These representations can be found neatly by solving Laplace’s equation using - surprise - a Laplace transform.) The polynomials satisfy the orthonormality relations (with the mathematicians’ normalization convention) + ∞ [(p + k)!]3 δpq . e−z z k Lkp Lkq dz = p! 0 But what do the polynomials look like? The function e−z z p is zero at the origin (apart from the trivial case p = 0) and zero at infinity, always positive and having non-zero slope except at its maximum value, z = p. The p derivatives bring in p separated zeroes, easily checked by sketching the curves generated by successive differentiation. Therefore, L0p (z), a polynomial of degree p, has p real positive zeroes, and value at the origin L0p (0) = p!, since the only non-zero term at z = 0 is that generated by all p differential operators acting on z p . The associated Laguerre polynomial Lkp (z) is generated by differentiating 0 Lp+k (z) k times. Now L0p+k (z) has p + k real positive zeroes, differentiating it gives a polynomial one degree lower, with zeroes which must be one in each interval between the zeroes of L0p+k (z). This argument remains valid for successive derivatives, so Lkp (z) must have p real separate zeroes. Putting all this together, and translating back from ρ to r, the radial solutions are given by, . Zr ! 2!+1 Rn! (r) = N e−Zr/na0 Ln−!−1 (2Zr/na0 ) , na0 with N the normalization constant. For a given principle quantum number n,the largest % radial wavefunction is given by Rn,n−1 ∝ rn−1 e−Zr/na0 . $ Info. The eigenvalues of the Hamiltonian for the hydrogen exhibit an unexpectedly high degeneracy. The fact that En!m is independent of m is common to all central potentials – it is just a reflection of rotational invariance of the Hamiltonian. Advanced Quantum Physics

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43

However, the degeneracy of different % values with the same principle quantum number is considered “accidental”, a peculiarity of the 1/r potential. To understand the origin of the degeneracy for atomic hydrogen, it is helpful to reflect first on the classical dynamics. In classical mechanics, central forces also lead to conservation of angular momentum leaving orbits restricted to lie in a plane. However, for 1/r type potentials, these orbits are also closed, i.e. they do not precess. In classical mechanics, this implies that there is an additional conserved vector, since the direction of the major axis of the elliptical orbit is a constant of the motion. This direction is determined by the Runge-Lenz vector, R=

e2 r 1 p×L− . m 4π#0 r

In quantum theory, up to an operator ordering prescription, R becomes an operator, e2 r ˆ −L ˆ ×p ˆ = 1 (ˆ ˆ) − p×L . R 2m 4π#0 r ˆ = 0 (exercise). ˆ R] With this definition, one may confirm that [H, ˆ exhibits the following commutation relations, [R ˆj ] = ˆi, L As a vector operator, R ˆ (−2H) ˆ ˆ ˆ ˆ i!#ijk Rk . Similarly, [Ri , Nj ] = i! m #ijk Lk (exercise). Moreover, one may confirm that - 2 .2 ˆ e 2H ˆ 2 + !2 ) , ˆ2 = (L + R 4π#0 m ˆ 2. ˆ 2 and R ˆ can be written in terms of the two constants of motion, L showing that H ) −m ˆ ˆ = Focussing on the bound states, if we consider the Hermitian operator, K ˆ R, 2H ˆ j ] = i!#ijk L ˆ i, L ˆj ] = ˆ k , [K ˆ i, K which fulfil the following commutation relations, [K ˆ ˆ ˆ ˆ i!#ijk Kk , and [Li , Lj ] = i!#ijk Lk , we find that ˆ =− H

-

e2 4π#0

.2

m . 2 ˆ ˆ 2 + !2 ) 2(K + L

ˆ = If we now define the “raising and lowering” operators, M following commutation relations emerge (exercise),

ˆ K ˆ L+ 2 ,

ˆ = N

ˆ K ˆ L− 2

the

ˆ j ] = i!#ijk L ˆ i, M ˆk [M ˆ ˆ ˆ [Ni , Nj ] = i!#ijk Kk ˆi , M ˆj] = 0 , [N i.e. we have obtained two commuting angular momentum algebras(!) and - 2 .2 e m ˆ H=− . 2 ˆ ˆ 2 + !2 ) 4π#0 2(2M + 2N ˆ 2 and N ˆ 2, M ˆz, N ˆz , We can simultaneously diagonalize the operators, M ˆ 2 |m, n, µ, ν$ = !2 m(m + 1)|m, n, µ, ν$, M ˆ 2 |m, n, µ, ν$ = !2 m(m + 1)|m, n, µ, ν$, N

ˆ z |m, n, µ, ν$ = !µ|m, n, µ, ν$ M ˆz |m, n, µ, ν$ = !ν|m, n, µ, ν$ . N

where m, n = 0, 1/2, 1, 3/2, · · ·, µ = −m, −m + 1, · · · m and ν = −n, −n + 1, · · · n. ˆ ·L ˆ =L ˆ ·R ˆ = 0, then K ˆ ·L ˆ =L ˆ ·R ˆ = 0 and the only relevant states are those Since R ˆ 2 = 0, i.e. m = n. Therefore, ˆ 2−N for which M - 2 .2 e m ˆ |m, m, µ, ν$ H|m, m, µ, ν$ = − 4π#0 2!2 (4m(m + 1) + 1) - 2 .2 m e =− |m, m, µ, ν$ . 4π#0 2!2 (2m + 1)2

From this result, we can identify 2m + 1 = 1, 2, · · · as the principle quantum number. For a given (2m + 1) value, the degeneracy of the state is (2m + 1)2 as expected.

Advanced Quantum Physics

Chapter 5

Motion of a charged particle in a magnetic field Hitherto, we have focussed on applications of quantum mechanics to free particles or particles confined by scalar potentials. In the following, we will address the influence of a magnetic field on a charged particle. Classically, the force on a charged particle in an electric and magnetic field is specified by the Lorentz force law: F = q (E + v × B) , where q denotes the charge and v the velocity. (Here we will adopt a convention in which q denotes the charge (which may be positive or negative) and e ≡ |e| denotes the modulus of the electron charge, i.e. for an electron, the charge q = −e = −1.602176487 × 10−19 C.) The velocity-dependent force associated with the magnetic field is quite different from the conservative forces associated with scalar potentials, and the programme for transferring from classical to quantum mechanics - replacing momenta with the appropriate operators - has to be carried out with more care. As preparation, it is helpful to revise how the Lorentz force arises in the Lagrangian formulation of classical mechanics.

5.1

Classical mechanics of a particle in a field

For a system with m degrees of freedom specified by coordinates q1 , · · · qm , the classical action is determined from the Lagrangian L(qi , q˙i ) by S[qi ] =

!

dt L(qi , q˙i ) .

The action is said to be a functional of the coordinates qi (t). According to Hamilton’s extremal principle (also known as the principle of least action), the dynamics of a classical system is described by the equations that minimize the action. These equations of motion can be expressed through the classical Lagrangian in the form of the Euler-Lagrange equations, d (∂q˙ L(qi , q˙i )) − ∂qi L(qi , q˙i ) = 0 . dt i

(5.1)

" Info. Euler-Lagrange equations: According to Hamilton’s extremal principle, for any smooth set of curves wi (t), the variation of the action around the classical solution qi (t) is zero, i.e. lim!→0 1! (S[qi + #wi ] − S[qi ]) = 0. Applied to the action, Advanced Quantum Physics

Hendrik Antoon Lorentz 18531928 A Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the transformation equations subsequently used by Albert Einstein to describe space and time.

Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia 1736-1813 An Italian-born mathematician and astronomer, who lived most of his life in Prussia and France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. On the recommendation of Euler and D’Alembert, in 1766 Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, where he stayed for over twenty years, producing a large body of work and winning several prizes of the French Academy of Sciences. Lagrange’s treatise on analytical mechanics, written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century.

5.1. CLASSICAL MECHANICS OF A PARTICLE IN A FIELD

45

" the variation implies that, for any i, dt (wi ∂qi L(qi , q˙i ) + w˙ i ∂q˙i L(qi , q˙i )) = 0. Then, integrating the second term by parts, and droping the boundary term, one obtains $ # ! d dt wi ∂qi L(qi , q˙i ) − ∂q˙i L(qi , q˙i ) = 0 . dt Since this equality must follow for any function wi (t), the term in parentheses in the integrand must vanish leading to the Euler-Lagrange equation (5.1).

The canonical momentum is specified by the equation pi = ∂q˙i L, and the classical Hamiltonian is defined by the Legendre transform, H(qi , pi ) =

% i

pi qi − L(qi , q˙i ) .

(5.2)

It is straightforward to check that the equations of motion can be written in the form of Hamilton’s equations of motion, q˙i = ∂pi H,

p˙i = −∂qi H .

From these equations it follows that, if the Hamiltonian is independent of a particular coordinate qi , the corresponding momentum pi remains constant. For conservative forces,1 the classical Lagrangian and Hamiltonian can be written as L = T − V , H = T + V , with T the kinetic energy and V the potential energy. " Info. Poisson brackets: Any dynamical variable f in the system is some function of the phase space coordinates, the qi s and pi s, and (assuming it does not depend explicitly on time) its time-development is given by: d f (qi , pi ) = ∂qi f q˙i + ∂pi f p˙i = ∂qi f ∂pi H − ∂pi f ∂qi H ≡ {f, H}. dt The curly brackets are known as Poisson brackets, and are defined for any dynamical variables as {A, B} = ∂qi A ∂pi B − ∂pi A ∂qi B. From Hamilton’s equations, we have shown that for any variable, f˙ = {f, H}. It is easy to check that, for the coordinates {qi , pj } = δij . This was the and canonical momenta, {qi , qj } = 0 = {pi , pj }, classical mathematical structure that led Dirac to link up classical and quantum mechanics: He realized that the Poisson brackets were the classical version of the commutators, so a classical canonical momentum must correspond to the quantum differential operator in the corresponding coordinate.2

With these foundations revised, we now return to the problem at hand; the infleunce of an electromagnetic field on the dynamics of the charged particle. As the Lorentz force is velocity dependent, it can not be expressed simply as the gradient of some potential. Nevertheless, the classical path traversed by a charged particle is still specifed by the principle of least action. The electric and magnetic fields can be written in terms of a scalar and a vector potential ˙ The corresponding Lagrangian takes the form:3 as B = ∇×A, E = −∇φ− A. 1 L = mv2 − qφ + qv · A. 2

1

i.e. forces that conserve mechanical energy. For a detailed discussion, we refer to Paul A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series Number 2, 1964. 3 In a relativistic formulation, the interaction term R here looks less arbitrary: the relativistic version would have the relativistically invariant q Aµ dxµ added to the action integral, where the four-potential Aµ = (φ, A) and dxµ = (ct, dx1 , dx2 , dx3 ). This is the simplest possible invariant interaction between the R electromagnetic field and R the particle’s four-velocity. Then, in the non-relativistic limit, q Aµ dxµ just becomes q (v · A − φ)dt. 2

Advanced Quantum Physics

Sim´ eon Denis Poisson 17811840 A French mathematician, geometer, and physicist whose mathematical skills enabled him to compute the distribution of electrical charges on the surface of conductors. He extended the work of his mentors, Pierre Simon Laplace and Joseph Louis Lagrange, in celestial mechanics by taking their results to a higher order of accuracy. He was also known for his work in probability.

5.2. QUANTUM MECHANICS OF A PARTICLE IN A FIELD In this case, the general coordinates qi ≡ xi = (x1 , x2 , x3 ) are just the Cartesian coordinates specifying the position of the particle, and the q˙i are the three components x˙ i = (x˙ 1 , x˙ 2 , x˙ 3 ) of the particle velocities. The important point is that the canonical momentum pi = ∂x˙ i L = mvi + qAi , is no longer simply given by the mass × velocity – there is an extra term! Making use of the definition (5.2), the corresponding Hamiltonian is given by % 1 1 H(qi , pi ) = (mvi + qAi ) vi − mv2 + qφ − qv · A = mv2 + qφ . 2 2 i

Reassuringly, the Hamiltonian just has the familiar form of the sum of the kinetic and potential energy. However, to get Hamilton’s equations of motion, the Hamiltonian has to be expressed solely in terms of the coordinates and canonical momenta, i.e. H=

1 (p − qA(r, t))2 + qφ(r, t) . 2m

Let us now consider Hamilton’s equations of motion, x˙ i = ∂pi H and p˙i = −∂xi H. The first equation recovers the expression for the canonical momentum while second equation yields the Lorentz force law. To understand how, we must first keep in mind that dp/dt is not the acceleration: The A-dependent term also varies in time, and in a quite complicated way, since it is the field at a point moving with the particle. More precisely, ' & p˙i = m¨ xi + q A˙ i = m¨ xi + q ∂t Ai + vj ∂xj Ai ,

where we have assumed a summation over repeated indicies. The right-hand ∂H side of the second of Hamilton’s equation, p˙i = − ∂x , is given by i

1 (p − qA(r, t))q∂xi A − q∂xi φ(r, t) = qvj ∂xi Aj − q∂xi φ . m ' & Together, we obtain the equation of motion, m¨ xi = −q ∂t Ai + vj ∂xj Ai + qvj ∂xi Aj − q∂xi φ. Using the identity, v × (∇ × A) = ∇(v · A) − (v · ∇)A, and the expressions for the electric and magnetic fields in terms of the potentials, one recovers the Lorentz equations −∂xi H =

m¨ x = F = q (E + v × B) . With these preliminary discussions of the classical system in place, we are now in a position to turn to the quantum mechanics.

5.2

Quantum mechanics of a particle in a field

To transfer to the quantum mechanical regime, we must once again implement ˆ = −i!∇, so that [ˆ the canonical quantization procedure setting p xi , pˆj ] = vi . This leads to the novel situation that i!δij . However, in this case, pˆi %= mˆ the velocities in different directions do not commute.4 To explore influence of the magnetic field on the particle dynamics, it is helpful to assess the relative weight of the A-dependent contributions to the quantum Hamiltonian, ˆ = 1 (ˆ p − qA(r, t))2 + qφ(r, t) . H 2m 4

With mˆ vi = −i!∂xi − qAi , it is easy (and instructive) to verify that [ˆ vx , vˆy ] =

Advanced Quantum Physics

i!q B. m2

46

5.3. ATOMIC HYDROGEN: NORMAL ZEEMAN EFFECT

47

Expanding the square on the right hand side of the Hamiltonian, the cross-term (known as the paramagnetic term) leads to the contribution q! − 2im (∇ · A + A · ∇) = iq! m A · ∇, where the last equality follows from the Coulomb gauge condition, ∇ · A = 0.5 Combined with the diamagnetic (A2 ) contribution, one obtains the Hamiltonian, 2 2 ˆ = − ! ∇2 + iq! A · ∇ + q A2 + qφ . H 2m m 2m For a constant magnetic field, the vector potential can be written as A = −r × B/2. In this case, the paramagnetic component takes the form

iq! iq! q A·∇= (r × ∇) · B = − L · B, m 2m 2m

where L denotes the angular momentum operator (with the hat not shown for brevity!). Similarly, the diamagnetic term leads to ' q2B 2 2 q2 2 q2 & 2 2 A = r B − (r · B)2 = (x + y 2 ) , 2m 8m 8m where, here, we have chosen the magnetic field to lie along the z-axis.

5.3

Atomic hydrogen: Normal Zeeman effect

Before addressing the role of these separate contributions in atomic hydrogen, let us first estimate their relative magnitude. With &x2 + y 2 ' ( a20 , where a0 denotes the Bohr radius, and &Lz ' ( !, the ratio of the paramagnetic and diamagnetic terms is given by e a20 B 2 (q 2 /8me )&x2 + y 2 'B 2 = ( 10−6 B/T . (q/2me )&Lz 'B 4 !B

Therefore, while electrons remain bound to atoms, for fields that can be achieved in the laboratory (B ( 1 T), the diamagnetic term is negligible as compared to the paramagnetic term. Moreover, when compared with the Coulomb energy scale,

2

eB!/2me e! B/T = B( , me c2 α2 /2 (me cα)2 2.3 × 105

e 1 1 where α = 4π# ( 137 denotes the fine structure constant, one may see 0 !c that the paramagnetic term provides only a small perturbation to the typical atomic splittings. 5

The electric field E and magnetic field B of Maxwell’s equations contain only “physical” degrees of freedom, in the sense that every mathematical degree of freedom in an electromagnetic field configuration has a separately measurable effect on the motions of test charges in the vicinity. As we have seen, these “field strength” variables can be expressed in terms of the scalar potential φ and the vector potential A through the relations: E = −∇φ − ∂t A and B = ∇ × A. Notice that if A is transformed to A + ∇Λ, B remains unchanged, since B = ∇ × [A + ∇Λ] = ∇ × A. However, this transformation changes E as E = −∇φ − ∂t A − ∇∂t Λ = −∇[φ + ∂t Λ] − ∂t A . If φ is further changed to φ − ∂t Λ, E remains unchanged. Hence, both the E and B fields are unchanged if we take any function Λ(r, t) and simultaneously transform A → A + ∇Λ

φ → φ − ∂t Λ .

A particular choice of the scalar and vector potentials is a gauge, and a scalar function Λ used to change the gauge is called a gauge function. The existence of arbitrary numbers of gauge functions Λ(r, t), corresponds to the U(1) gauge freedom of the theory. Gauge fixing can be done in many ways.

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Splitting of the sodium D lines due to an external magnetic field. The multiplicity of the lines and their “selection rule” will be discussed more fully in chapter 9. The figure is taken from the original paper, P. Zeeman, The effect of magnetization on the nature of light emitted by a substance, Nature 55, 347 (1897).

5.4. GAUGE INVARIANCE AND THE AHARONOV-BOHM EFFECT

48

However, there are instances when the diamagnetic contriubution can play an important role. Leaving aside the situation that may prevail on neutron stars, where magnetic fields as high as 108 T may exist, the diamagnetic contribution can be large when the typical “orbital” scale &x2 + y 2 ' becomes macroscopic in extent. Such a situation arises when the electrons become unbound such as, for example, in a metal or a synchrotron. For a further discussion, see section 5.5 below. Retaining only the paramagnetic contribution, the Hamiltonian for a “spinless” electron moving in a Coulomb potential in the presence of a constant magnetic field then takes the form, ˆ =H ˆ 0 + e BLz , H 2m ˆ 0 , Lz ] = 0, the eigenstates of the unperturbed ˆ 0 = pˆ − e . Since [H where H 2m 4π#0 r ˆ and the corresponding energy Hamiltonian, ψl$m (r) remain eigenstates of H levels are specified by 2

2

En$m = −

Ry + !ωL m n2

eB where ωL = 2m denotes the Larmor frequency. From this result, we expect that a constant magnetic field will lead to a splitting of the (2)+1)-fold degeneracy of the energy levels leading to multiplets separated by a constant energy shift of !ωL . The fact that this behaviour is not recapitulated generically by experiment was one of the key insights that led to the identification of electron spin, a matter to which we will turn in chapter 6.

5.4

Gauge invariance and the Aharonov-Bohm effect

Our derivation above shows that the quantum mechanical Hamiltonian of a charged particle is defined in terms of the vector potential, A. Since the latter is defined only up to some gauge choice, this suggests that the wavefunction is not a gauge invariant object. Indeed, it is only the observables associated with the wavefunction which must be gauge invariant. To explore this gauge freedom, let us consider the influence of the gauge transformation, A )→ A" = A + ∇Λ,

φ )→ φ" − ∂t Λ ,

where Λ(r, t) denotes a scalar function. Under the gauge transformation, one may show that the corresponding wavefunction gets transformed as ) ( q ψ " (r, t) = exp i Λ(r, t) ψ(r, t) . (5.3) ! " Exercise. If wavefunction ψ(r, t) obeys the time-dependent Schr¨odinger equaˆ φ]ψ, show that ψ " (r, t) as defined by (5.3) obeys the equation tion, i!∂t ψ = H[A, ˆ " [A" , φ" ]ψ " . i!∂t ψ " = H The gauge transformation introduces an additional space and time-dependent phase factor into the wavefunction. However, since the observable translates to the probability density, |ψ|2 , this phase dependence seems invisible. " Info. One physical manifestation of the gauge invariance of the wavefunction is found in the Aharonov-Bohm effect. Consider a particle with charge q travelling Advanced Quantum Physics

Sir Joseph Larmor 1857-1942 A physicist and mathematician who made innovations in the understanding of electricity, dynamics, thermodynamics, and the electron theory of matter. His most influential work was Aether and Matter, a theoretical physics book published in 1900. In 1903 he was appointed Lucasian Professor of Mathematics at Cambridge, a post he retained until his retirement in 1932.

5.4. GAUGE INVARIANCE AND THE AHARONOV-BOHM EFFECT

49

Figure 5.1: (Left) Schematic showing the geometry of an experiment to observe the

Aharonov-Bohm effect. Electrons from a coherent source can follow two paths which encircle a region where the magnetic field is non-zero. (Right) Interference fringes for electron beams passing near a toroidal magnet from the experiment by Tonomura and collaborators in 1986. The electron beam passing through the center of the torus acquires an additional phase, resulting in fringes that are shifted with respect to those outside the torus, demonstrating the Aharonov-Bohm effect. For details see the original paper from which this image was borrowed see Tonomura et al., Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave, Phys. Rev. Lett. 56, 792 (1986).

along a path, P , in which the magnetic field, B = 0 is identically zero. However, a vanishing of the magnetic field does not imply that the vector potential, A is zero. Indeed, as we have seen, any Λ(r) such that A = ∇Λ will translate to this condition. In traversing the path, the wavefunction of the particle will acquire the phase factor " ϕ = !q P A · dr, where the line integral runs along the path. If we consider now two separate paths P and P " which share the same initial and final points, the relative phase of the wavefunction will be set by ! ! * ! q q q q A · dr = A · dr = A · dr − B · d2 r , ∆ϕ = ! P ! P! ! ! A " + where the line integral runs over the loop involving paths P and P " , and A runs over the area enclosed by the loop. The last relation follows from the application of Stokes’ theorem. This result shows that the" relative phase ∆ϕ is fixed by the factor q/! multiplied by the magnetic flux Φ = A B · d2 r enclosed by the loop.6 In the absence of a magnetic field, the flux vanishes, and there is no additional phase. However, if we allow the paths to enclose a region of non-vanishing magnetic field (see figure 5.1(left)), even if the field is identically zero on the paths P and P " , the wavefunction will acquire a non-vanishing relative phase. This flux-dependent phase difference translates to an observable shift of interference fringes when on an observation plane. Since the original proposal,7 the Aharonov-Bohm effect has been studied in several experimental contexts. Of these, the most rigorous study was undertaken by Tonomura in 1986. Tomomura fabricated a doughnut-shaped (toroidal) ferromagnet six micrometers in diameter (see figure 5.1b), and covered it with a niobium superconductor to completely confine the magnetic field within the doughnut, in accordance with the Meissner effect.8 With the magnet maintained at 5 K, they measured the phase difference from the interference fringes between one electron beam passing though the hole in the doughnut and the other passing on the outside of the doughnut. The results are shown in figure 5.1(right,a). Interference fringes are displaced with just half a fringe of spacing inside and outside of the doughnut, indicating the existence of the Aharonov-Bohm effect. Although electrons pass through regions free of any electromagnetic field, an observable effect was produced due to the existence of vector potentials. 6 Note that the phase difference depends on the magnetic flux, a function of the magnetic field, and is therefore a gauge invariant quantity. 7 Y. Aharonov and D. Bohm, Significance of electromagnetic potentials in quantum theory, Phys. Rev. 115, 485 (1959). 8 Perfect diamagnetism, a hallmark of superconductivity, leads to the complete expulsion of magnetic fields – a phenomenon known as the Meissner effect.

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Sir George Gabriel Stokes, 1st Baronet 1819-1903 A mathematician and physicist, who at Cambridge made important contributions to fluid dynamics (including the NavierStokes equations), optics, and mathematical physics (including Stokes’ theorem). He was secretary, and then president, of the Royal Society.

5.5. FREE ELECTRONS IN A MAGNETIC FIELD: LANDAU LEVELS

50

The observation of the half-fringe spacing reflects the constraints imposed by the superconducting toroidal shield. When a superconductor completely surrounds a magnetic flux, the flux is quantized to an integral multiple of quantized flux h/2e, the factor of two reflecting that fact that the superconductor involves a condensate of electron pairs. When an odd number of vortices are enclosed inside the superconductor, the relative phase shift becomes π (mod. 2π) – half-spacing! For an even number of vortices, the phase shift is zero.9

5.5

Free electrons in a magnetic field: Landau levels

Finally, to complete our survey of the influence of a uniform magnetic field on the dynamics of charged particles, let us consider the problem of a free quantum particle. In this case, the classical electron orbits can be macroscopic and there is no reason to neglect the diamagnetic contribution to the Hamiltonian. Previously, we have worked with a gauge in which A = (−y, x, 0)B/2, giving a constant field B in the z-direction. However, to address the Schr¨odinger equation for a particle in a uniform perpendicular magnetic field, it is convenient to adopt the Landau gauge, A(r) = (−By, 0, 0). " Exercise. Construct the gauge transformation, Λ(r) which connects these two representations of the vector potential.

In this case, the stationary form of the Schr¨odinger equation is given by 1 , ˆ (ˆ px + qBy)2 + pˆ2y + pˆ2z ψ(r) = Eψ(r) . Hψ(r) = 2m

ˆ commutes with both pˆx and pˆz , both operators have a common set of Since H eigenstates reflecting the fact that px and pz are conserved by the dynamics. The wavefunction must therefore take the form, ψ(r) = ei(px x+ipz z)/!χ(y), with χ(y) defined by the equation, $ / # . 2 pˆy 1 p2 + mω 2 (y − y0 )2 χ(y) = E − z χ(y) . 2m 2 2m Here y0 = −px /qB and ω = |q|B/m coincides with the cyclotron frequency of the classical charged particle (exercise). We now see that the conserved canonical momentum px in the x-direction is in fact the coordinate of the centre of a simple harmonic oscillator potential in the y-direction with frequency ω. As a result, we can immediately infer that the eigenvalues of the Hamiltonian are comprised of a free particle component associated with motion parallel to the field, and a set of harmonic oscillator states, En,pz = (n + 1/2)!ω +

p2z . 2m

The quantum numbers, n, specify states known as Landau levels. Let us confine our attention to states corresponding to the lowest oscillator (Landau level) state, (and, for simplicity, pz = 0), E0 = !ω/2. What is the degeneracy of this Landau level? Consider a rectangular geometry of area A = Lx × Ly and, for simplicity, take the boundary conditions to be periodic. The centre of the oscillator wavefunction, y0 = −px /qB, must lie 9

The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by Fritz London in 1948 using a phenomenological theory.

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Lev Davidovich Landau 19081968 A prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics. His accomplishments include the co-discovery of the density matrix method in quantum mechanics, the quantum mechanical theory of diamagnetism, the theory of superfluidity, the theory of second order phase transitions, the GinzburgLandau theory of superconductivity, the explanation of Landau damping in plasma physics, the Landau pole in quantum electrodynamics, and the two-component theory of neutrinos. He received the 1962 Nobel Prize in Physics for his development of a mathematical theory of superfluidity that accounts for the properties of liquid helium II at a temperature below 2.17K.

5.5. FREE ELECTRONS IN A MAGNETIC FIELD: LANDAU LEVELS between 0 and Ly . With periodic boundary conditions eipx Lx /! = 1, so that px = n2π!/Lx . This means that y0 takes a series of evenly-spaced discrete values, separated by ∆y0 = h/qBLx . So, for electron degrees of freedom, q = −e, the total number of states N = Ly /|∆y0 |, i.e. νmax =

Lx Ly B =A , h/eB Φ0

(5.4)

where Φ0 = e/h denotes the “flux quantum”. So the total number of states in the lowest energy level coincides with the total number of flux quanta making up the field B penetrating the area A. The Landau level degeneracy, νmax , depends on field; the larger the field, the more electrons can be fit into each Landau level. In the physical system, each Landau level is spin split by the Zeeman coupling, with (5.4) applying to one spin only. Finally, although we treated x and y in an asymmetric manner, this was merely for convenience of calculation; no physical quantity should dierentiate between the two due to the symmetry of the original problem. " Exercise. Consider the solution of the Schr¨odinger equation when working in the symmetric gauge with A = −r × B/2. Hint: consider the velocity commutation relations, [vx , vy ] and how these might be deployed as conjugate variables.

" Info. It is instructive to infer y0 from purely classical considerations: Writing ˙ m¨ y = − qB ˙ and m¨ z = 0. mv˙ = qv × B in component form, we have m¨ x = qB c y, c x, Focussing on the motion in the xy-plane, these equations integrate straightforwardly qB to give, mx˙ = qB c (y − y0 ), my˙ = − c (x − x0 ). Here (x0 , y0 ) are the coordinates of the centre of the classical circular motion (known as the “guiding centre”) – the velocity vector v = (x, ˙ y) ˙ always lies perpendicular to (r − r0 ), and r0 is given by y0 = y − mvx /qB = −px /qB,

x0 = x + mvy /qB = x + py /qB .

(Recall that we are using the gauge A(x, y, z) = (−By, 0, 0), and px = ∂x˙ L = mvx + qAx , etc.) Just as y0 is a conserved quantity, so is x0 : it commutes with the Hamiltonian since [x + cˆ py /qB, pˆx + qBy] = 0. However, x0 and y0 do not commute with each other: [x0 , y0 ] = −i!/qB. This is why, when we chose a gauge in which y0 was sharply defined, x0 was spread over the sample. If we attempt to localize the point (x0 , y0 ) as much as possible, it is smeared out over an area corresponding to one flux quantum. The0natural length scale of the problem is therefore the magnetic ! . length defined by ) = qB

" Info. Integer quantum Hall effect: Until now, we have considered the impact of just a magnetic field. Consider now the Hall effect geometry in which we apply a crossed electric, E and magnetic field, B. Taking into account both contributions, the total current flow is given by $ # j×B , j = σ0 E − ne

where σ0 denotes the conductivity, and n is the electron density. With the electric field oriented along y, and the magnetic field along z, the latter equation may be rewritten as $ # $ # $# σ0 B 0 1 jx ne = σ0 . σ0 B jy Ey − ne 1

Inverting these equations, one finds that jx =

−σ02 B/ne Ey , 1 + (σ0 B/ne)2 1 23 4 σxy

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jy =

σ0 Ey . 1 + (σ0 B/ne)2 1 23 4 σyy

51

Klaus von Klitzing, 1943German physicist who was awarded the Nobel Prize for Physics in 1985 for his discovery that under appropriate conditions the resistance offered by an electrical conductor is quantized. The work was first reported in the following reference, K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980).

5.5. FREE ELECTRONS IN A MAGNETIC FIELD: LANDAU LEVELS

Figure 5.2: (Left) A voltage V drives a current I in the positive x direction. Normal

Ohmic resistance is V /I. A magnetic field in the positive z direction shifts positive charge carriers in the negative y direction. This generates a Hall potential and a Hall resistance (V H/I) in the y direction. (Right) The Hall resistance varies stepwise with changes in magnetic field B. Step height is given by the physical constant h/e2 (value approximately 25 kΩ) divided by an integer i. The figure shows steps for i = 2, 3, 4, 5, 6, 8 and 10. The effect has given rise to a new international standard for resistance. Since 1990 this has been represented by the unit 1 klitzing, defined as the Hall resistance at the fourth step (h/4e2 ). The lower peaked curve represents the Ohmic resistance, which disappears at each step. These provide the classical expressions for the longitudinal and Hall conductivities, σyy and σxy in the crossed field. Note that, for these classical expressions, σxy is proportional to B. How does quantum mechanics revised this picture? For the classical model – Drude theory, the random elastic scattering of electrons impurities leads to a con2 τ stant drift velocity in the presence of a constant electric field, σ0 = ne me , where τ denotes the mean time between collisions. Now let us suppose the magnetic field is chosen so that number of electrons exactly fills all the Landau levels up to some N , i.e. nLx Ly = N νmax ⇒ n = N

eB . h

The scattering of electrons must lead to a transfer between quantum states. However, if all states of the same energy are filled,10 elastic (energy conserving) scattering becomes impossible. Moreover, since the next accessible Landau level energy is a distance !ω away, at low enough temperatures, inelastic scattering becomes frozen out. As a result, the scattering time vanishes at special values of the field, i.e. σyy → 0 and σxy →

e2 ne =N . B h

At critical values of the field, the Hall conductivity is quantized in units of e2 /h. Inverting the conductivity tensor, one obtains the resistivity tensor, $ # $−1 # σxx σxy ρxx ρxy = , −ρxy ρxx −σxy σxx where ρxx =

σxx , 2 + σ2 σxx xy

ρxx = −

σxy , 2 + σ2 σxx xy

So, when σxx = 0 and σxy = νe2 /h, ρxx = 0 and ρxy = h/νe2 . The quantum Hall state describes dissipationless current flow in which the Hall conductance σxy is quantized in units of e2 /h. Experimental measurements of these values provides the best determination of fundamental ratio e2 /h, better than 1 part in 107 . 10

Note that electons are subject to Pauli’s exclusion principle restricting the occupancy of each state to unity.

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Chapter 6

Spin Until we have focussed on the quantum mechanics of particles which are “featureless”, carrying no internal degrees of freedom. However, a relativistic formulation of quantum mechanics shows that particles can exhibit an intrinsic angular momentum component known as spin. However, the discovery of the spin degree of freedom marginally predates the development of relativistic quantum mechanics by Dirac and was acheived in a ground-breaking experiement by Stern and Gerlach (1922). In their experiment, they passed a well-collimated beam of silver atoms through a region of inhomogeneous field before allowing the particles to impact on a photographic plate (see figure). The magnetic field was directed perpendicular to the beam, and has a strong gradient, ∂z Bz != 0 so that a beam comprised of atoms with a magnetic moment would be bent towards the z or -z axis. As the magnetic moment will be proportional to the total angular momentum, such an experiment can be thought of as a measurement of its projection along z. At the time of the experiment, there was an expectation that the magnetic moment of the atom was generated in its entirety by the orbital angular momentum. As such, one would expect that there would be a minimum of three possible values of the z-component of angular momentum: the lowest non-zero orbital angular momentum is " = 1, with allowed values of the z-component m!, m = 1, 0, −1. Curiously, Stern and Gerlach’s experiment (right) showed that the beam of silver atoms split into two! This discovery, which caused great discussion and surprise presented a puzzle. However, in our derivation of allowed angular momentum eigenvalues we found that, although for any system the allowed values of m form a ladder with spacing !, we could not rule out half-integral values of m. The lowest such case, " = 1/2, would in fact have just two allowed m values: m = ±1/2. However, such an " value could not translate to an orbital angular momentum because the z-component of the orbital wavefunction, ψ has a factor e±iφ , and therefore acquires a factor −1 on rotating through 2π! This would imply that ψ is not single-valued, which doesn’t make sense for a Schr¨odinger-type wavefunction. Yet the experimental result was irrefutable. Therefore, this must be a new kind of non-orbital angular momentum – spin. Conceptually, just as the Earth has orbital angular momentum in its yearly circle around the sun, and also spin angular momentum from its daily turning, the electron has an analogous spin. But this analogy has obvious limitations: the Earth’s spin is after all made up of material orbiting around the axis through the poles. The electron spin cannot be imagined as arising from a rotating body, since orbital angular momenta always come in integral multiples of !. Fortunately, this lack of a simple quasi-mechanical picture underlying electron spin doesn’t Advanced Quantum Physics

Gerlach’s postcard, dated 8th February 1922, to Niels Bohr. It shows a photograph of the beam splitting, with the message, in translation: “Attached [is] the experimental proof of directional quantization. We congratulate [you] on the confirmation of your theory.” (Physics Today December 2003)

6.1. SPINORS, SPIN PPERATORS, PAULI MATRICES

54

prevent us from using the general angular momentum machinery developed ealier, which followed just from analyzing the effect of spatial rotation on a quantum mechanical system.

6.1

Spinors, spin pperators, Pauli matrices

The Hilbert space of angular momentum states for spin 1/2 is two-dimensional. Various notations are used: |", m# becomes |s, m# or, more graphically, |1/2, 1/2# = | ↑#,

|1/2, −1/2# = | ↓# .

A general state of spin can be written as the lienar combination, ! " α α| ↑# + β| ↓# = , β with the normalisation condition, |α|2 + |β|2 = 1, and this two-dimensional ket is called a spinor. Operators acting on spinors are necessarily of the form of 2 × 2 matrices. We shall adopt the usual practice of denoting the angular momentum components Li by Si for spins. (Once again, for clarity, we also drop the hats on the angular momentum operators!) From our definition of the spinor, it is evident that the z-component of the spin can be represented as the matrix, ! " ! 1 0 S z = σz , σz = . 0 −1 2 From the general formulae (4.5) for raising and lowering operators S± = Sx ± iSy , with s = 1/2, we have S+ |1/2, −1/2# = !|1/2, 1/2#, S− |1/2, 1/2# = !|1/2, −1/2#, or, in matrix form, ! " ! " 0 1 0 0 , Sx − iSy = S− = ! . Sx + iSy = S+ = ! 0 0 1 0 It therefore follows that an appropriate matrix representation for spin 1/2 is ggiven by the Pauli spin matrices, S = !2 σ where σx =

!

0 1 1 0

"

,

σy =

!

0 −i i 0

"

,

σz =

!

1 0 0 −1

"

.

(6.1)

These matrices are Hermitian, traceless, and obey the relations σi2 = I, σi σj = −σj σi , and σi σj = iσk for (i, j, k) a cyclic permutation of (1, 2, 3). These relations can be summarised by the identity, σi σj = Iδij + i)ijk σk . The total spin S2 =

!2 2 4 σ

= 34 !2 , i.e. s(s + 1)!2 for s = 1/2.

* Exercise. Explain why any 2 × 2 matrix can be written in the form α0 I + ˆ , and (b) α n · σ)2 = I for any unit vector n i i σi . Use your results to show that (a) (ˆ (σ · A)(σ · B) = (A · B)I + σ · (A × B).

#

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Wolfgang Pauli and Niels Bohr demonstrating ‘tippe top’ toy at the inauguration of the new Institute of Physics at Lund, Sweden 1954.

6.2. RELATING THE SPINOR TO THE SPIN DIRECTION

6.2

Relating the spinor to the spin direction

For a general state α| ↑# + β| ↓#, how do α, β relate to which way the spin is pointing? To find out, let us assume that it is pointing up along the unit ˆ = (sin θ cos φ, sin θ sin φ, cos θ), i.e. in the direction (θ, φ). In other vector n ˆ · σ having eigenvalue unity: words, the spin is an eigenstate of the operator n ! "! " ! " nz α α nx − iny = . β β nx + iny −nz From this expression, we find that α/β = (nx − iny )/(1 − nz ) = e−iφ cot(θ/2) (exercise). Then, making use of the normalisation, |α|2 + |β|2 = 1, we obtain (up to an arbitrary phase) !

α β

"

=

!

e−iφ/2 cos(θ/2) eiφ/2 sin(θ/2)

"

.

Since e−iφ cot(θ/2) can be used to specify any complex number with 0 ≤ θ ≤ π, 0 ≤ φ < 2π, so for any possible spinor, there is an associated direction along which the spin points up. * Info. The spin rotation operator: In general, the rotation operator for ˆ is given rotation through an angle θ about an axis in the direction of the unit vector n by eiθnˆ ·J/! where J denotes the angular momentum operator. For spin, J = S = 21 !σ, and the rotation operator takes the form1 eiθnˆ ·J/! = ei(θ/2)(ˆn·σ ) . Expanding the exponential, and making use of the Pauli matrix identities ((n · σ)2 = I), one can show that (exercise) ei(θ/2)(n·σ ) = I cos(θ/2) + in · σ sin(θ/2) . The rotation operator is a 2 × 2 matrix operating on the ket space. The 2 × 2 rotation matrices are unitary and form a group known as SU(2); the 2 refers to the dimensionality, the U to their being unitary, and the S signifying determinant +1. ˆ = (0, 0, 1), it is more natural to replace θ Note that for rotation about the z-axis, n with φ, and the rotation operator takes the form, " ! −iφ/2 0 e ei(θ/2)(n·σ ) = . 0 eiφ/2 In particular, the wavefunction is multiplied by −1 for a rotation of 2π. Since this is true for any initial wave function, it is clearly also true for rotation through 2π about any axis.

* Exercise. Construct the infinitesimal version of the rotation operator eiδθnˆ ·J/! for spin 1/2, and prove that eiδθnˆ ·J/! σe−iδθnˆ ·J/! = σ + δθˆ n × σ, i.e. σ is rotated in the same way as an ordinary three-vector - note particularly that the change depends on the angle rotated through (as opposed to the half-angle) so, reassuringly, there is no −1 for a complete rotation (as there cannot be - the direction of the spin is a physical observable, and cannot be changed on rotating the measuring frame through 2π). 1 Warning: do not confuse θ – the rotation angle - with the spherical polar angle used to ˆ. parameterise n

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6.3. SPIN PRECESSION IN A MAGNETIC FIELD

6.3

Spin precession in a magnetic field

Consider a magnetized classical object spinning about it’s centre of mass, with angular momentum L and parallel magnetic moment µ, µ = γL. The constant γ is called the gyromagnetic ratio. Now suppose that we impose a magnetic field B along, say, the z-direction. This will exert a torque T = µ × B = γL × B = dL dt . This equation is easily solved and shows that the angular momentum vector L precesses about the magnetic field direction with angular velocity of precession ω 0 = −γB.2 In the following, we will show that precisely the same result appears in the study of the quantum mechanics of an electron spin in a magnetic field. −e and the The electron has magnetic dipole moment µ = γS, where γ = g 2m e 3 gyromagnetic ratio, g, is very close to 2. The Hamiltonian for the interaction ˆ = of the electron’s dipole moment with the magnetic field is given by H −µ · B = −γS · B. Hence the time development is specified by the equation ˆ (t)|ψ(0)#, with the time-evolution operator (or propagator), U ˆ (t) = |ψ(t)# = U ˆ −i Ht/! iγσ·Bt/2 e = e . However, this is nothing but the rotation operator (as shown earlier) through an angle −γBt about the direction of B! For an arbitrary initial spin orientation " ! " ! −iφ/2 α e cos(θ/2) , = β eiφ/2 sin(θ/2) the propagator for a magnetic field in the z-direction is given by " ! −iω t/2 e 0 0 U (t) = eiγσ·Bt/2 = , 0 eiω0 t/2 so the time-dependent spinor is set by " ! " ! −i(φ+ω t)/2 0 α(t) e cos(θ/2) . = β(t) ei(φ+ω0 t)/2 sin(θ/2) The angle θ between the spin and the field stays constant while the azimuthal angle around the field increases as φ = φ0 +ω0 t, exactly as in the classical case. |e|B denotes the cyclotron frequency. For The frequency ω0 = gωc , where ωc = 2m e 11 a magnetic field of 1 T, ωc ( 10 rads/s. * Exercise. For a spin initially pointing along the x-axis, prove that )Sx (t)# =

(!/2) cos(ω0 t).

6.3.1

Paramagnetic Resonance

The analysis above shows that the spin precession frequency is independent of the angle of the spin with respect to the field direction. Consider then how this looks in a frame of reference which is itself rotating with angular ˆ, since velocity ω about the z-axis. Let us specify the magnetic field B0 = B0 z we’ll soon be adding another component. In the rotating frame, the observed precession frequency is ω r = −γ(B0 + ω/γ), so there is a different effective dL

Proof: From the equation of motion, with L+ = Lx + iLy , dt+ = −iγBL+ , L+ = z Of course, dL = 0, since dL = γL × B is perpendicular to B, which is in the dt dt z-direction. 3 This g-factor terminology is used more widely: the magnetic moment of an atom is e! written µ = gµB , where µB = 2m is the known as the Bohr magneton, and g depends e on the total orbital angular momentum and total spin of the particular atom. 2

L0+ e−iγBt .

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6.3. SPIN PRECESSION IN A MAGNETIC FIELD

57

field (B0 + ω/γ) in the rotating frame. Obviously, if the frame rotates exactly at the precession frequency, ω = ω 0 = −γB0 , spins pointing in any direction will remain at rest in that frame – there is no effective field at all. Suppose we now add a small rotating magnetic field with angular frequency ω in the xy plane, so the total magnetic field, ˆy sin(ωt)) . ˆ + B1 (ˆ ex cos(ωt) − e B = B0 z The effective magnetic field in the frame rotating with the same frequency ω as the small added field is then given by ˆx . z + B1 e Br = (B0 + ω/γ)ˆ Now, if we tune the angular frequency of the small rotating field so that it exactly matches the precession frequency in the original static magnetic field, ω = ω 0 = −γB0 , all the magnetic moment will see in the rotating frame is the small field in the x-direction! It will therefore precess about the x-direction at the slow angular speed γB1 . This matching of the small field rotation frequency with the large field spin precession frequency is the “resonance”. If the spins are lined up preferentially in the z-direction by the static field, and the small resonant oscillating field is switched on for a time such that γB1 t = π/2, the spins will be preferentially in the y-direction in the rotating frame, so in the lab they will be rotating in the xy plane, and a coil will pick up an a.c. signal from the induced e.m.f. * Info. Nuclear magnetic resonance is an important tool in chemical analysis. As the name implies, the method uses the spin magnetic moments of nuclei (particularly hydrogen) and resonant excitation. Magnetic resonance imaging uses the same basic principle to get an image (of the inside of a body for example). In basic NMR, a strong static B field is applied. A spin 1/2 proton in a hydrogen nucleus then has two energy eigenstates. After some time, most of the protons fall into the lower of the two states. We now use an electromagnetic wave (RF pulse) to excite some of the protons back into the higher energy state. The proton’s magnetic moment interacts with the oscillating B field of the EM wave through the Hamiltionian, ˆ = −µ · B = gp e S · B = gp e! σ · B = gp µN σ · B , H 2mp c 4mp c 2 where the gyromagnetic ratio of the proton is about +5.6. The magnetic moment is 2.79µN (nuclear magnetons). Different nuclei will have different gyromagnetic ratios giving more degrees of freedom with which to work. The strong static B field is chosen to lie in the z direction and the polarization of the oscillating EM wave is chosen so that the B field points in the x direction. The EM wave has (angular) frequency ω, ! " Bx cos(ωt) Bz ˆ = gp µN Bz σz + Bx cos(ωt)σx = gp µN H . −Bz Bx cos(ωt) 2 2 ˆ i.e. If we now apply the time-dependent Schr¨odinger equation, i!∂t χ = Hχ, ! " ! "! " a˙ ω0 ωI cos(ωt) a = −i , ωI cos(ωt) b −ω0 b˙ where ω0 = gp µN Bz /2! and ωI = gp µN Bx /2!, we obtain, % ωI $ i(ω−2ω0 )t ∂t (be−iω0 t ) = − e + e−i(ω+2ω0 )t . 2

The second term oscillates rapidly and can be neglected. The first term will only result in significant transitions if ω ≈ 2ω0 . Note that this is exactly the condition that ensures that the energy of the photons in the EM field E = !ω is equal to the energy difference between the two spin states ∆E = 2!ω0 . The conservation of energy

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A proton NMR spectrum of a solution containing a simple organic compound, ethyl benzene. Each group of signals corresponds to protons in a different part of the molecule.

6.4. ADDITION OF ANGULAR MOMENTA condition must be satisfied well enough to get a significant transition rate. In NMR, we observe the transitions back to the lower energy state. These emit EM radiation at the same frequency and we can detect it after the stronger input pulse ends (or by more complex methods). NMR is a powerful tool in chemical analysis because the molecular field adds to the external B field so that the resonant frequency depends on the molecule as well as the nucleus. We can learn about molecular fields or just use NMR to see what molecules are present in a sample. In MRI, we typically concentrate on the one nucleus like hydrogen. We can put a gradient in Bz so that only a thin slice of the material has ω tuned to the resonant frequency. Therefore we can excite transitions to the higher energy state in only a slice of the sample. If we vary (in the orthogonal direction!) the B field during the decay, we can recover 3d images.

6.4

Addition of angular momenta

In subsequent chapters, it will be necessary to add angular momentum, be it ˆ = L+S, ˆ the addition of orbital and spin angular momenta, J as with the study of spin-orbit coupling in atoms, or the addition of general angular momenta, ˆ 2 as occurs in the consideration of multi-electron atoms. In the ˆ = J ˆ1 + J J following section, we will explore three problems: The addition of two spin 1/2 degrees of freedom; the addition of a general orbital angular momentum and spin; and the addition of spin J = 1 angular momenta. However, before addressing these examples in turn, let us first make some general remarks. Without specifying any particular application, let us consider the total ˆ = J ˆ1 + J ˆ 2 where J ˆ 1 and J ˆ 2 correspond to distinct angular momentum J ˆ ˆ degrees of freedom, [J1 , J2 ] = 0, and the individual operators obey angular momentum commutation relations. As a result, the total angular momentum also obeys angular momentum commutation relations, [Jˆi , Jˆj ] = i!)ijk Jˆk . For each angular momentum component, the states |j1 , m1 # and |j2 , m2 # where mi = −ji , · · · ji , provide a basis of states of the total angular momentum ˆ 2 and the projection Jˆiz . Together, they form a complete basis operator, J i which can be used to span the states of the coupled spins,4 |j1 , m1 , j2 , m2 # ≡ |j1 , m1 # ⊗ |j2 , m2 # . These product states are also eigenstates of Jˆz with eigenvalue !(m1 + m2 ), ˆ2. but not of J * Exercise. Show that [Jˆ 2 , Jˆiz ] != 0. However, for practical application, we require a basis in which the total angular ˆ 2 is also diagonal. That is, we must find eigenstates momentum operator J ˆ 2 , Jˆz , J ˆ 2 , and J ˆ2. |j, mj , j1 , j2 # of the four mutually commuting operators J 2 1 In general, the relation between the two basis can be expressed as & |j, mj , j1 , j2 # = |j1 , m1 , j2 , m2 #)j1 , m1 , j2 , m2 |j, mj , j1 , j2 # , m1 ,m2

4 Here ⊗ denotes the “direct product” and shows that the two constituent spin states access their own independent Hilbert space.

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High resolution MRI scan of a brain!

6.4. ADDITION OF ANGULAR MOMENTA where the matrix elements are known as Clebsch-Gordon coefficients. In general, the determination of these coefficients from first principles is a somewhat soul destroying exercise and one that we do not intend to pursue in great detail.5 In any case, for practical purposes, such coefficients have been tabulated in the literature and can be readily obtained. However, in some simple cases, these matrix elements can be determined straightforwardly. Moreover, the algorithmic programme by which they are deduced offer some new conceptual insights. Operationally, the mechanism for finding the basis states of the total angular momentum operator follow the strategy: 1. As a unique entry, the basis state with maximal Jmax and mj = Jmax is easy to deduce from the original basis states since it involves the product of states of highest weight, |Jmax , mj = Jmax , j1 , j2 # = |j1 , m1 = j1 # ⊗ |j2 , m2 = j2 # , where Jmax = j1 + j2 . 2. From this state, we can use of the total spin lowering operator Jˆ− to find all states with J = Jmax and mj = −Jmax · · · Jmax . 3. From the state with J = Jmax and mj = Jmax − 1, one can then obtain the state with J = Jmax − 1 and mj = Jmax − 1 by orthogonality.6 Now one can return to the second step of the programme and repeat until J = |j1 − j2 | when all (2j1 + 1)(2j2 + 1) basis states have been obtained.

6.4.1

Addition of two spin 1/2 degrees of freedom

For two spin 1/2 degrees of freedom, we could simply construct and diagonalize the complete 4 × 4 matrix elements of the total spin. However, to gain some intuition for the general case, let us consider the programme above. Firstly, the maximal total spin state is given by |S = 1, mS = 1, s1 = 1/2, s2 = 1/2# = |s1 = 1/2, ms1 = 1/2# ⊗ |s2 = 1/2, ms2 = 1/2# . Now, since s1 = 1/2 and s = 1/2 is implicit, we can rewrite this equation in a more colloquial form as |S = 1, mS = 1# = | ↑1 # ⊗ | ↑2 # . We now follow step 2 of the programme and subject the maximal spin state to the total spin lowering operator, Sˆ− = Sˆ1− + Sˆ1+ . In doing so, making use of Eq. (4.5), we find √ Sˆ− |S = 1, mS = 1# = 2!|S = 1, mS = 0# = ! (| ↓1 # ⊗ | ↑2 # + | ↑1 # ⊗ | ↓2 #) , 5

In fact, one may show that the general matrix element is given by s (j1 + j2 − j)!(j + j1 − j2 )!(j + j2 − j1 )!(2j + 1) $j1 , m1 , j2 , m2 |j, mj , j1 , j2 % = δmj ,m1 +m2 (j + j1 + j2 + 1)! p X (−1)k (j1 + m1 )!(j1 − m1 )!(j2 + m2 )!(j2 − m2 )!(j + m)!(j − m)! × . k!(j1 + j2 − j − k)!(j1 − m1 − k)!(j2 + m2 − k)!(j − j2 + m1 + k)!(j − j1 − m2 + k)! k

6 Alternatively, as a maximal spin state, |J = Jmax − 1, mj = Jmax − 1, j1 , j2 % can be identified by the “killing” action of the raising operator, Jˆ+ .

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6.4. ADDITION OF ANGULAR MOMENTA i.e. |S = 1, mS = 0# =

√1 (| 2

Sˆ− |S = 1, mS = 0# =

↓1 # ⊗ | ↑2 # + | ↑1 # ⊗ | ↓2 #). Similarly,

√

2!|S = 1, mS = −1# =

√ 2!| ↓1 # ⊗ | ↓2 # ,

i.e. |S = 1, mS = −1# = | ↓1 # ⊗ | ↓2 #. This completes the construction of the manifold of spin S = 1 states – the spin triplet states. Following the programme, we must now consider the lower spin state. In this case, the next multiplet is the unique total spin singlet state |S = 0, mS = 0#. The latter must be orthogonal to the spin triplet state |S = 1, mS = 0#. As a result, we can deduce that 1 |S = 0, mS = 0# = √ (| ↓1 # ⊗ | ↑2 # − | ↑1 # ⊗ | ↓2 #) . 2

6.4.2

Addition of angular momentum and spin

We now turn to the problem of the addition of angular momentum and spin, ˆ =L ˆ + S. ˆ In the original basis, for a given angular momentum ", one can J identify 2 × (2" + 1) product states |", m% # ⊗ | ↑# and |", m% # ⊗ | ↓#, with ˆ 2, L ˆ 2 and Sˆz , but not J ˆ 2 . From ˆz, S m% = −", · · · ", involving eigenstates of L ˆ 2 and S ˆ 2 . To ˆ 2 , Jˆz , L these basis states, we are looking for eigenstates of J undertake this programme, it is helpful to recall the action of the angular momentum raising and lower operators, ˆ ± |", m% # = ((" ± m% + 1)(" ∓ m% ))1/2 !|", m% ± 1# , L as well as the identity ˆ ·S ˆ 2L () * ' 2 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ J = L + S + 2Lz Sz + L+ S− + S+ L− .

ˆ 2 and S ˆ 2 we will adopt the notation |j, mj , "# ˆ 2 , Jˆz , L For the eigenstates of J leaving the spin S = 1/2 implicit. The maximal spin state is given by7 |" + 1/2, " + 1/2, "# = |", "# ⊗ | ↑# . To obtain the remaining states in the multiplet, |j = " + 1/2, mj=%+1/2 , "#, we may simply apply the total spin lowering operator Jˆ− , Jˆ− |", "# ⊗ | ↑# = !(2")1/2 |", " − 1# ⊗ | ↑# + !|", "# ⊗ | ↓# . Normalising the right-hand side of this expression, one obtains the spin state, + + 2" 1 |", " − 1# ⊗ | ↑# + |", "# ⊗ | ↓# . |" + 1/2, " − 1/2, "# = 2" + 1 2" + 1 7

The proof runs as follows: ˆ z + Sˆz )|$, $% ⊗ | ↑% = ($ + 1/2)!|$, $% ⊗ | ↑% , Jˆz |$, $% ⊗ | ↑% = (L

and ˆ 2 |$, $% ⊗ | ↑% = !2 ($($ + 1) + 1/2(1/2 + 1) + 2$ 1 )|$, $% ⊗ | ↑% J 2 = !2 ($ + 1/2)($ + 3/2)|$, $% ⊗ | ↑% .

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6.4. ADDITION OF ANGULAR MOMENTA By repeating this programme, one can develop an expression for the full set of basis states, + " + mj + 1/2 |j = " + 1/2, mj , "# = |", mj − 1/2# ⊗ | ↑# 2" + 1 + " − mj + 1/2 + |", mj + 1/2# ⊗ | ↓# , 2" + 1 with mj = " + 1/2, · · · , −(" + 1/2). In order to obtain the remaining states with j = " − 1/2, we may look for states with mj = " − 1/2, · · · , −(" − 1/2) which are orthogonal to |" + 1/2, mj , "#. Doing so, we obtain + " − mj + 1/2 |", mj − 1/2# ⊗ | ↑# |" − 1/2, mj , "# = − 2" + 1 + " + mj + 1/2 + |", mj + 1/2# ⊗ | ↓# . 2" + 1 Finally, these states can be cast in a compact form by setting |j = " ± 1/2, mj , "# = α± |", mj − 1/2# ⊗ | ↑# + β± |", mj + 1/2# ⊗ | ↓# , (6.2) ,

where α± = ±

6.4.3

%±mj +1/2 2%+1

= ±β∓ .

Addition of two angular momenta J = 1

As mentioned above, for the general case the programme is algebraically technical and unrewarding. However, for completeness, we consider here the explicit example of the addition of two spin 1 degrees of freedom. Once again, the maximal spin state is given by |J = 2, mJ = 2, j1 = 1, j2 = 1# = |j1 = 1, m1 = 1# ⊗ |j2 = 1, m2 = 1# , or, more concisely, |2, 2# = |1# ⊗ |1#, where we leave j1 and j2 implicit. Once again, making use of Eq. (4.5) and an ecomony of notation, we find (exercise)  |2, 2# = |1# ⊗ |1#     √1   |2, 1# = 2 (|0# ⊗ |1# + |1# ⊗ |0#) |2, 0# = √16 (| − 1# ⊗ |1# + 2|0# ⊗ |0# + |1# ⊗ | − 1#) .    |2, −1# = √12 (|0# ⊗ | − 1# + | − 1# ⊗ |0#)    |2, 2# = | − 1# ⊗ | − 1#

Then, from the expression for |2, 1#, we can construct the next maximal spin state |1, 1# = √12 (|0# ⊗ |1# − |1# ⊗ |0#), from the orthogonality condition. Once again, acting on this state with the total spin lowering operator, we obtain the remaining members of the multiplet,  √1   |1, 1# = 2 (|0# ⊗ |1# − |1# ⊗ |0#) |1, 0# = √12 (| − 1# ⊗ |1# − |1# ⊗ | − 1#) .   |1, −1# = √1 (| − 1# ⊗ |0# − |0# ⊗ | − 1#) 2

Finally, finding the state orthogonal to |1, 0# and |2, 0#, we obtain the final state, 1 |0, 0# = √ (| − 1# ⊗ |1# − |0# ⊗ |0# + |1# ⊗ | − 1#) . 3 Advanced Quantum Physics

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Chapter 7

Approximation methods for stationary states 7.1

Time-independent perturbation theory

While we have succeeded in deriving formal analytical solutions for stationary states of the Schr¨odinger operator in a variety of settings, in the majority of practical applications, exact solutions are inaccessible.1 For example, if an atom is placed in an external electric field, the energy levels shift, and the wavefunctions become distorted — the Stark effect. The new energy levels and wavefunctions could in principle be obtained by writing down a complete Hamiltonian, including the external field. Indeed, such a programme may be achieved for the hydrogen atom. But even there, if the external field is small compared with the electric field inside the atom (which is billions of volts per meter) it is easier to compute the changes in the energy levels and wavefunctions within a scheme of successive corrections to the zero-field values. This method, termed perturbation theory, is the single most important method for solving problems in quantum mechanics, and is widely used in atomic physics, condensed matter and particle physics. ! Info. It should be acknowledged that there are – typically very interesting – problems which cannot be solved using perturbation theory, even when the perturbation is very weak; although such problems are the exception rather than the rule. One such case is the one-dimensional problem of free particles perturbed by a localized potential of strength λ. As we found earlier in chapter 2, switching on an arbitrarily weak attractive potential causes the k = 0 free particle wavefunction to drop below the continuum of plane wave energies and become a localized bound state with binding energy of order λ2 . However, on changing the sign of λ to give a repulsive potential, there is no bound state; the lowest energy plane wave state stays at energy zero. Therefore the energy shift on switching on the perturbation cannot be represented as a power series in λ, the strength of the perturbation. This particular difficulty does not typically occur in three dimensions, where arbitrarily weak potentials do not in general lead to bound states.

! Exercise. Focusing on the problem of bound state formation in one-dimension described above, explore the dependence of the ground state energy on λ. Consider why a perturbative expansion in λ is infeasible. 1 Indeed, even if such a solution is formally accessible, its complexity may render it of no practical benefit.

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7.1. TIME-INDEPENDENT PERTURBATION THEORY

7.1.1

63

The Perturbation Series

ˆ (0) , having known eigenLet us then consider an unperturbed Hamiltonian, H (0) states |n(0) ! and eigenvalues En , ˆ (0) |n(0) ! = En(0) |n(0) ! . H

(7.1)

In the following we will address the question of how the eigenstates and ˆ (1) eigenenergies are modified by the imposition of a small perturbation, H (such as that imposed by an external electric or magnetic field on a charged particle, or the deformation of some other external potential). In short, we are interested in the solution of the Schr¨odinger equation, ˆ (1) )|n! = En |n! . ˆ (0) + H (H

(7.2)

ˆ (1) |n(0) ! # En , it seems natural to If the perturbation is small, "n(0) |H (1) ˆ suppose that, on turning on H , the eigenfunctions and eigenvalues will change adiabatically from their unperturbed to their perturbed values, a situation described formally as “adiabatic continuity”, (0)

|n(0) ! $−→ |n!,

En(0) $−→ En .

However, note that this is not always the case. For example, as mentioned above, an infinitesimal perturbation has the capacity to develop a bound state not present in the unperturbed system. For now, let us proceed with the perturbative expansion and return later to discuss its potential range of validity. The basic assumption that underpins the perturbation theory is that, for ˆ (1) small, the leading corrections are of the same order of magnitude as H ˆ (1) itself. The perturbed eigenenergies and eigenvalues can then be obH tained to a greater accuracy by a successive series of corrections, each of order ˆ (1) !/"H ˆ (0) ! compared with the previous. To identify terms of the same "H ˆ (0) !, it is convenient to extract from H ˆ (1) a dimensionless ˆ (1) !/"H order in "H parameter λ, characterising the relative magnitude of the perturbation against ˆ (0) , and then expand |n! and En as a power series in λ, i.e. H |n! = |n(0) ! + λ|n(1) ! + λ2 |n(2) ! + · · · = En = En(0) + λEn(1) + λ2 En(2) + · · · =

∞ !

m=0

∞ !

λm |n(m) !,

λm En(m) .

m=0

One may think of the parameter λ as an artifical book-keeping device to organize the perturbative expansion, and which is eventually set to unity at the end of the calculation. Applied to the stationary form of the Schr¨odinger equation (7.2), an expansion of this sort leads to the relation ˆ (1) )(|n(0) ! + λ|n(1) ! + λ2 |n(2) ! + · · ·) ˆ (0) + λH (H

= (En(0) + λEn(1) + λ2 En(2) + · · ·)(|n(0) ! + λ|n(1) ! + λ2 |n(2) ! + · · ·) .(7.3) From this equation, we must relate terms of equal order in λ. At the lowest order, O(λ0 ), we simply recover the unperturbed equation (7.1). In practical applications, one is usually interested in determining the first non-zero perturbative correction. In the following, we will explore the form of the first and second order perturbative corrections. Advanced Quantum Physics

7.1. TIME-INDEPENDENT PERTURBATION THEORY

7.1.2

64

First order perturbation theory

Isolating terms from (7.3) which are first order in λ, ˆ (1) |n(0) ! = En(0) |n(1) ! + En(1) |n(0) ! . ˆ (0) |n(1) ! + H H

(7.4)

and taking the inner product with the unperturbed states "n(0) |, one obtains ˆ (0) |n(1) ! + "n(0) |H ˆ (1) |n(0) ! = "n(0) |E (0) |n(1) ! + "n(0) |E (1) |n(0) ! . "n(0) |H n n ˆ (0) = "n(0) |En , and exploiting the presumed normalizaNoting that "n(0) |H (0) (0) tion "n |n ! = 1, one finds that the first order shift in energy is given simply by the expectation value of the perturbation taken with respect to the unperturbed eigenfunctions, (0)

ˆ (1) |n(0) ! . En(1) = "n(0) |H

(7.5)

Turning to the wavefuntion, if we instead take the inner product of (7.4) with "m(0) | (with m '= n), we obtain ˆ (0) |n(1) ! + "m(0) |H ˆ (1) |n(0) ! = "m(0) |E (0) |n(1) ! + "m(0) |E (1) |n(0) ! . "m(0) |H n n ˆ (0) = "m(0) |Em and the orthogonality condition on Once again, with "m(0) |H (0) the wavefunctions, "m |n(0) ! = 0, one obtains an expression for the first order shift of the wavefunction expressed in the unperturbed basis, (0)

"m(0) |n(1) ! =

ˆ (1) |n(0) ! "m(0) |H (0)

(0)

E n − Em

(7.6)

.

In summary, setting λ = 1, to first order in perturbation theory, we have the eigenvalues and eigenfunctions, ˆ (1) |n(0) ! En ( En(0) + "n(0) |H ! ˆ (1) |n(0) ! "m(0) |H |n! ( |n(0) ! + |m(0) ! . (0) (0) E n − Em m"=n Before turning to the second order of perturbation theory, let us first consider a simple application of the method. ! Example: Ground state energy of the Helium atom: For the Helium atom, two electrons are bound to a nucleus of two protons and two neutrons. If one neglects altogether the Coulomb interaction between the electrons, in the ground state, both electrons would occupy the ground state hydrogenic wavefunction (scaled appropriately to accommodate the doubling of the nuclear charge) and have opposite spin. Treating the Coulomb interaction between electrons as a perturbation, one may then use the basis above to estimate the shift in the ground state energy with ˆ (1) = H

e2 1 . 4π$0 |r1 − r2 |

As we have seen, the hydrogenic wave functions are specified by three quantum numbers, n, %, and m. In the ground state, the corresponding wavefunction takes the spatially isotropic form, "r|n = 1, % = 0, m = 0! = ψ100 (r) =

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"

1 πa3

#1/2

e−r/a ,

(7.7)

7.1. TIME-INDEPENDENT PERTURBATION THEORY 2

a0 0 ! where a = 4π" Ze2 me = Z denotes the atomic Bohr radius for a nuclear charge Z. For the Helium atom (Z = 2), the symmetrized ground state of the unperturbed Hamiltonian is then given by the spin singlet (S = 0) electron wavefunction,

1 |g.s.(0) ! = √ (|100, ↑! ⊗ |100, ↓! − |100, ↓! ⊗ |100, ↑!) . 2 Here we have used the direct product ⊗ to discriminate between the two electrons. Then, applying the perturbation theory formula above (7.5), to first order in the Coulomb interaction, the energy shift is given by $ 2 e2 C0 1 e−2(r1 +r2 )/a ˆ (1) |g.s.(0) ! = e En(1) = "g.s.(0) |H dr1 dr2 = , 3 2 4π$0 (πa ) |r1 − r2 | 4π$0 2a % −(z1 +z2 ) 1 where we have defined the dimensionless constant C0 = (4π) dz1 dz1 e|z1 −z2 | . Then, 2 making use of the identity, $ 1 1 1 dΩ1 dΩ2 = , 2 (4π) |z1 − z2 | max(z1 , z2 ) where the integrations runs over the angular coordinates of the vectors z1 and z2 , %∞ %∞ and z1,2 = |z1,2 |, one finds that C0 = 2 0 dz1 z12 e−z1 z1 dz2 z2 e−z2 = 5/4. As a 2

e 1 result, noting that the Rydberg energy, Ry = 4π" , we obtain the first order 0 2a0 5 energy shift ∆E = 4 ZRy ( 34eV for Z = 2. This leads to a total ground state energy of (2Z 2 − 54 Z) Ry = −5.5Ry ( −74.8eV compared to the experimental value of −5.807Ry.

7.1.3

Second order perturbation theory

With the first order of perturbation theory in place, we now turn to consider the influence of the second order terms in the perturbative expansion (7.3). Isolating terms of order λ2 , we have ˆ (1) |n(1) ! = En(0) |n(2) ! + En(1) |n(1) ! + En(2) |n(0) ! . ˆ (0) |n(2) ! + H H

As before, taking the inner product with "n(0) |, one obtains ˆ (0) |n(2) ! + "n(0) |H ˆ (1) |n(1) ! "n(0) |H

= "n(0) |En(0) |n(2) ! + "n(0) |En(1) |n(1) ! + "n(0) |En(2) |n(0) ! .

Noting that the first two terms on the left and right hand sides cancel, we are left with the result ˆ (1) |n(1) ! − En(1) "n(0) |n(1) ! . En(2) = "n(0) |H

Previously, we have made use of the normalization of the basis states, |n(0) !. We have said nothing so far about the normalization of the exact eigenstates, |n!. Of course, eventually, we would like to ensure normalization of these states too. However, to facilitate the perturbative expansion, it is operationally more convenient to impose a normalization on |n! through the condition "n(0) |n! = 1. Substituting the λ expansion for |n!, we thus have "n(0) |n! = 1 = "n(0) |n(0) ! + λ"n(0) |n(1) ! + λ2 "n(0) |n(2) ! + · · · .

From this relation, it follows that "n(0) |n(1) ! = "n(0) |n(2) ! = · · · = 0.2 We can (1) therefore drop the term En "n(0) |n(1) ! from consideration. As a result, we 2 Alternatively, would we suppose that |n! and |n(0) ! were both normalised to unity, to leading order, |n! = |n(0) ! + |n(1) !, and "n(0) |n(1) ! + "n(1) |n(0) ! = 0, i.e. "n(0) |n(1) ! is pure imaginary. This means that if, to first order, |n! has a component parallel to |n(0) !, that component has a small imaginary amplitude allowing us to define |n! = eiφ |n(0) !+orthog. components. However, the corresponding phase factor φ can be eliminated (1) by redefining the phase of |n!. Once again, we can conclude that the term En "n(0) |n(1) ! can be eliminated from consideration.

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obtain ˆ (1) |n(1) ! = "n(0) |H ˆ (1) En(2) = "n(0) |H

!

m"=n

i.e.

|m(0) !

ˆ (1) |n(0) ! "m(0) |H (0)

! |"m(0) |H ˆ (1) |n(0) !|2

En(2) =

(0)

m"=n

(0)

En − Em

(7.8)

.

(0)

En − Em

,

From this result, we can conclude that, ! for the ground state, the second order shift in energy is always negative; ˆ (1) are of comparable magnitude, neighbour! if the matrix elements of H ing levels make a larger contribution than distant levels; ! Levels that lie in close proximity tend to be repelled; ! If a fraction of the states belong to a continuum, the sum in Eq. (7.8) should be replaced by an intergral. Once again, to illustrate the utility of the perturbative expansion, let us consider a concrete physical example. ! Example: The Quadratic Stark Effect: Consider the influence of an external electric field on the ground state of the hydrogen atom. As the composite electron and proton are drawn in different directions by the field, the relative displacement of the electon cloud and nucleus results in the formation of a dipole which serves to lower the overall energy. In this case, the perturbation due to the external field takes the form ˆ (1) = qEz = qEr cos θ , H where q = −|e| denotes the electron charge, and the electric field, E = Eˆez is oriented (0) (0) along the z-axis. With the non-perturbed energy spectrum given by En#m ≡ En = (0) −Ry/n2 , the ground state energy is given by E (0) ≡ E100 = −Ry. At first order in the electric field strength, E, the shift in the ground state energy is given by E (1) = "100|qEz|100! where the ground state wavefunction was defined above (7.7). Since the potential perturbation is antisymmetric in z, it is easy to see that the energy shift vanishes at this order. We are therefore led to consider the contribution second order in the field strength. Making use of Eq. (7.8), and neglecting the contribution to the energy shift from the continuum of unbound positive energy states, we have E (2) =

!

n#=1,#,m

|"n%m|eEz|100!|2 (0)

(0)

E1 − En

,

where |n%m! denote the set of bound state hydrogenic wavefunctions. Although the expression for E (2) can be computed exactly, the programme is somewhat tedious. However, we can place a strong bound on the energy shift through the following (0) (0) (0) (0) argument: Since, for n > 2, |E1 − En | > |E1 − E2 |, we have ! 1 |E (2) | < (0) "100|eEz|n%m!"n%m|eEz|100! . (0) E2 − E1 n#=1,#,m & & Since n,#,m |n%m!"n%m| = I, we have n#=1,#,m |n%m!"n%m| = I−|100!"100|. Finally, since "100|z|100! = 0, we can conclude that |E (2) | < (0) 1 (0) "100|(eEz)2 |100!. With (0)

"100|z 2 |100! = a20 , E1

E2 −E1 (0) (0) = E1 /4,

2

e 1 = − 4π" = −Ry, and E2 0 2a0

|E (2) |
|$210|eEz|100%| . From this result, one (0) (0) E2 −E1

can show that 0.55 × 83 4π$0 E 2 a30 < |E (2) | < 83 4π$0 E 2 a30 (exercise).3

7.2

Degenerate perturbation theory

The perturbative analysis above is reliable providing that the successive terms in the expansion form a convergent series. A necessary condition is that the matrix elements of the perturbing Hamiltonian must be smaller than the corresponding energy level differences of the original Hamiltonian. If it has different states with the same energy (i.e. degeneracies), and the perturbation has non-zero matrix elements between these degenerate levels, then obviously the theory breaks down. However, the problem is easily fixed. To understand how, let us consider a particular example. ˆ = pˆ2 + 1 mω 2 x2 , the Recall that, for the simple harmonic oscillator, H 2m 2 1/4 e−ξ 2 /2 , where ξ = ) ground state wavefunction is given by "x|0! = ( mω π! ' 1/4 ξe−ξ 2 /2 . The wavex mω/! and the first excited state by "x|1! = ( 4mω π! ) functions for the two-dimensional harmonic oscillator, 1 ˆ (0) = 1 (ˆ p2 + pˆ2y ) + mω 2 (x2 + y 2 ) . H 2m x 2 are given simply by the product of two one-dimensional oscillators. So, setting ' ( )1/2 −(ξ2 +η2 )/2 e , η = y mω/!, the ground state is given by "x, y|0, 0! = mω π! and the two degenerate first excited states, an energy !ω above the ground state, are given by, * * + mω ,1/2 ξ "x, y|1, 0! −(ξ 2 +η 2 )/2 . e = η "x, y|0, 1! π! Suppose now we add to the Hamiltonian a perturbation, ˆ (1) = αmω 2 xy = α!ωξη , H controlled by a small parameter α. Notice that, by symmetry, the following ˆ (1) |0, 0! = "1, 0|H ˆ (1) |1, 0! = "0, 1|H ˆ (1) |0, 1! = matrix elements all vanish, "0, 0|H 0. Therefore, according to a na¨ıve perturbation theory, there is no first-order correction to the energies of these states. However, on proceeding to consider the second-order correction to the energy, the theory breaks down. The offˆ (1) |0, 1! = 0 is non-zero, but the two states diagonal matrix element, "1, 0|H |0, 1! and |1, 0! have the same energy! This gives an infinite term in the series (2) for En=1 . Yet we know that a small perturbation of this type will not wreck a twodimensional simple harmonic oscillator – so what is wrong with our approach? To understand the origin of the problem and its fix, it is helpful to plot the original harmonic oscillator potential 12 mω 2 (x2 + y 2 ) together with the perturbing potential αmω 2 xy. The first of course has circular symmetry, the second has two symmetry axes oriented in the directions x = ±y, climbing most steeply from the origin along x = y, falling most rapidly in the directions x = y. If we combine the two potentials into a single quadratic form, " " #2 #2 . 1 x + y 1 x − y √ √ mω 2 (x2 + y 2 ) + αmω 2 xy = mω 2 (1 + α) . + (1 − α) 2 2 2 2 3

Energetic readers might like to contemplate how the exact answer of |E (2) | = 94 E 2 a30 can be found exactly. The method can be found in the text by Shankar.

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7.2. DEGENERATE PERTURBATION THEORY the original circles of constant potential become ellipses, with their axes aligned along x = ±y. As soon as the perturbation is introduced, the eigenstates lie in the direction of the new elliptic axes. This is a large change from the original x and y bases, which is not proportional to the small parameter α. But the original unperturbed problem had circular symmetry, and there was no particular reason to choose the x and y axes as we did. If we had instead chosen as our original axes the lines x = ±y, the basis states would not have undergone large changes on switching on the perturbation. The resolution of the problem is now clear: Before switching on the perturbation, one must choose a set of basis states in a degenerate subspace in which the perturbation is diagonal. In fact, for the simple harmonic oscillator example above, the problem can of course be solved exactly √ by rearranging the coordinates to lie along the symmetry axes, (x ± y)/ 2. It is then clear that, despite the results of na¨ıve first order perturbation theory, there is indeed a first order shift in the energy √ levels, !ω → !ω 1 ± α ≈ !ω(1 ± α/2). ! Example: Linear Stark Effect: As with the two-dimensional harmonic oscillator discussed above, the hydrogen atom has a non-degenerate ground state, but degeneracy in its lowest excited states. Specifically, there are four n = 2 states, all having energy − 14 Ry. In spherical coordinates, these wavefunctions are given by ,  +   #1/2 " 2 − ar0  ψ200 (r)  1 r ψ210 (r) = e−r/2a0 . θ   ar0 cos 32πa30  ±iφ ψ21,±1 (r) sin θ a0 e

ˆ (1) = When perturbing this system with an electric field oriented in the z-direction, H qEr cos θ, a na¨ıve application of perturbation theory predicts no first-order shift in any of these energy levels. However, at second order in E, there is a non-zero matrix ˆ (1) |210!. All the other matrix element between two degenerate levels ∆ = "200|H elements between these basis states in the four-dimensional degenerate subspace are zero. So the only diagonalization necessary is within the two-dimensional degenerate subspace spanned by |200! and |210!, i.e. " # 0 ∆ (1) ˆ H = , ∆ 0 with ∆ = qE

+

1 32πa30

,% +

2−

r a0

,+

r cos θ a0

,2

e−r/a0 r2 dr sin θ dθ dφ = −3qEa0 .

ˆ (1) within this sub-space, the new basis states are given by the Diagonalizing H √ symmetric and antisymmetric combinations, (|200! ±| 210!)/ 2 with energy shifts ±∆, linear in the perturbing electric field. The states |2%, ±1! are not changed by the presence of the field to this level of approximation, so the complete energy map of the n = 2 states in the electric field has two states at the original energy of −Ry/4, one state moved up from that energy by ∆, and one down by ∆. Notice that the new √ eigenstates (|200! ±| 210!)/ 2 are not eigenstates of the parity operator -- a sketch of their wavefunctions reveals that, in fact, they have non-vanishing electric dipole moment µ. Indeed this is the reason for the energy shift, ±∆ = ∓2eEa0 = ∓µ · E.

! Example: As a second and important example of the degenerate perturbation theory, let us consider the problem of a particle moving in one dimension and subject to a weak periodic potential, V (x) = 2V cos(2πx/a) – the nearly free electron model. This problem provides a caricature of a simple crystalline solid in which (free) conduction electrons propagate in the presence of a periodic background lattice potential. Here we suppose that the strength of the potential V is small as compared to the typical energy scale of the particle so that it may be treated as a small perturAdvanced Quantum Physics

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7.3. VARIATIONAL METHOD bation. In the following, we will suppose that the total one-dimensional system is of length L = N a, with periodic boundary conditions. For the unperturbed free particle system, the eigenstates are simply plane waves ψk (x) = "x|k! = √1L eikx indexed by the wavenumber k = 2πn/L, n integer, and (0)

the unperturbed spectrum is given by Ek = !2 k 2 /2m. The matrix elements of the perturbation between states of different wavevector are given by $ " 1 L dxei(k −k)x 2V cos(2πx/a) "k|V |k ! = L 0 $ L+ , " " V = ei(k −k+2π/a)x + ei(k −k−2π/a)x = V δk" −k,±2π/a . L 0 '

Note that all diagonal matrix elements of the perturbation are identically zero. In general, for wavevectors k and k ' separated by G = 2π/a, the unperturbed states are non-degenerate. For these states one can compute the relative energy shift within the framework of second order perturbation theory. However, for states k = −k ' = G/2 ≡ π/a, the unperturbed free particle spectrum is degenerate. Here, and in the neighbourhood of these k values, we must implement a degenerate perturbation theory. For the sinusoidal potential considered here, only states separated by G = 2π/a are coupled by the perturbation. We may therefore consider matrix elements of the full Hamiltonian between pairs of coupled states, |k = G/2 + q! and |k = −G/2 + q! 3 (0) 4 EG/2+q V H= . (0) V E−G/2+q As a result, to leading order in V , we obtain the eigenvalues, Eq =

#1/2 " π 2 !4 q 2 !2 2 (q + (π/a)2 ) ± V 2 + . 2m 4m2 a2

In particular, this result shows that, for k = ±G/2, the degeneracy of the free particle system is lifted by the potential. In the vicinity, |q| # G, the spectrum of eigenvalues is separated by a gap of size 2V . The appearance of the gap mirrors the behaviour found in our study of the Kronig-Penney model of a crystal studied in section 2.2.3. The appearance of the gap has important consequences in theory of solids. Electrons are fermions and have to obey Pauli’s exclusion principle. In a metal, at low temperatures, electrons occupy the free particle-like states up to some (Fermi) energy which lies away from gap. Here, the accessibility of very low-energy excitations due to the continuum of nearby states allows current to flow when a small electric field is applied. However, when the Fermi energy lies in the gap created by the lattice potential, an electric field is unable to create excitations and induce current flow. Such systems are described as (band) insulators.

7.3

Variational method

So far, in devising approximation methods for quantum mechanics, we have focused on the development of a perturbative expansion scheme in which the states of the non-perturbed system provided a suitable platform. Here, by suitable, we refer to situations in which the states of the unperturbed system mirror those of the full system – adiabatic contunity. For example, the states of the harmonic oscillator potential with a small perturbation will mirror those of the unperturbed Hamiltonian: The ground state will be nodeless, the first excited state will be antisymmetric having one node, and so on. However, often we working with systems where the true eigenstates of the problem may not be adiabatically connected to some simple unperturbed reference state. This Advanced Quantum Physics

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7.3. VARIATIONAL METHOD situation is particularly significant in strongly interacting quantum systems where many-particle correlations can effect phase transitions to new states of matter – e.g. the development of superfluid condensates, or the fractional quantum Hall fluid. To address such systems if it is often extremely effective to “guess” and then optimize a trial wavefunction. The method of optimization relies upon a simple theoretical framework known as the variational approach. For reasons that will become clear, the variational method is particularly wellsuited to addressing the ground state. The variational method involves the optimization of some trial wavefunction on the basis of one or more adjustable parameters. The optimization is achieved by minimizing the expectation value of the energy on the trial function, and thereby finding the best approximation to the true ground state wave function. This seemingly crude approach can, in fact, give a surprisingly good approximation to the ground state energy but, it is usually not so good for the wavefunction, as will become clear. However, as mentioned above, the real strength of the variational method arises in the study of many-body quantum systems, where states are more strongly constrained by fundamental symmetries such as “exclusion statistics”. To develop the method, we’ll begin with the problem of a single quantum ˆ = pˆ2 + V (r). If the particle is restricted to particle confined to a potential, H 2m one dimension, and we’re looking for the ground state in any fairly localized potential well, it makes sense to start with a trial wavefunction which belongs 2 to the family of normalized Gaussians, |ψ(α)! = (α/π)1/4 e−αx /2 . Such a trial state fulfils the criterion of being nodeless, and is exponentially localized to the region of the binding potential. It also has the feature that it includes the exact ground states of the harmonic binding potential. The variation approach involves simply minimizing the expectaton value of ˆ the energy, E = "ψ(α)|H|ψ(α)!, with respect to variations of the variational parameter, α. (Of course, as with any minimization, one must check that the variation does not lead to a maximum of the energy!) Not surprisingly, this programme leads to the exact ground state for the simple harmonic oscillator potential, while it serves only as an approximation for other potentials. What is perhaps surprising is that the result is only off by only ca. 30% or so for the attractive δ-function potential, even though the wavefunction looks substantially different. Obviously, the Gaussian family cannot be used if there is an infinite wall anywhere: one must find a family of wavefunctions vanishing at such a boundary. ! Exercise. Using the Gaussian trial state, find the optimal value of the variational state energy, E, for an attractive δ-function potential and compare it with the exact result. To gain some further insight into the approach, suppose the Hamiltonian ˆ has a set of eigenstates, H|n! ˆ H = En |n!. Since the Hamiltonian is Hermitian, these states span the space of possible wave functions, including our & variational family of Gaussians, so we can write, |ψ(α)! = n an (α)|n!. From this expansion, we have ! ˆ "ψ(α)|H|ψ(α)! = |an |2 En ≥ E0 , "ψ(α)|ψ(α)! n for any |ψ(α)!. (We don’t need the denominator if we’ve chosen a family of normalized wavefunctions, as we did with the Gaussians above.) Evidently, Advanced Quantum Physics

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minimizing the left hand side of this equation as function of α provides an upper bound on the ground state energy. We can see immediately that this will probably be better for finding the ground state energy than for the wavefunction: Suppose the optimum state in our family is given by, say, |αmin ! = N (|0!+0.2|1!) with the normalization N ( 0.98, i.e. a 20% admixture of the first excited state. Then the wavefunction is off by ca. 20%, but the energy estimate will be too high by only 0.04(E1 −E0 ), usually a much smaller error. ! Example: To get some idea of of how well the variational approach works, consider its application to the to the ground state of the hydrogen atom. Taking into account the spherical symmetry of the ground state, we may focus on the onedimensional radial component of the wavefunction. Defining the trial radial wavefunction u(ρ) (presumed real), where ρ = r/a0 , the variational energy is given by , + 2 %∞ d 2 + dρ u(ρ) 2 dρ ρ u(ρ) 0 %∞ . E(u) = −Ry 2 dρ u (ρ) 0 For the three families of trial functions, u1 (ρ) = ρe−αρ ,

u2 (ρ) =

ρ , α 2 + ρ2

u3 (ρ) = ρ2 e−αρ ,

and finds αmin = 1, π/4, and 3/2 respectively (exercise). The first family, u1 , includes the exact result, and the minimization procedure correctly finds it. For the three families, the predicted energy of the optimal state is off by 0, 25%, and 21% respectively. The corresponding error in the wavefunction is defined by how far the square of the overlap with the true ground state wavefunction falls short of unity. For the three families, ε = 1 − |"ψ0 |ψvar |2 = 0, 0.21, and 0.05. Notice here that our handwaving argument that the energies would be found much more accurately than the wavefunctions seems to come unstuck. The third family has far better wavefunction overlap than the second, but only a slightly better energy estimate. Why? A key point is that the potential is singular at the origin; there is a big contribution to the potential energy from a rather small region, and the third family of trial states is the least accurate of the three there. The second family of functions are very inaccurate at large distances: the expectation value "r! = 1.5a0 , ∞, 1.66a0 for the three families. But at large distances, both kinetic and potential energies are small, so the result can still look reasonable. These examples reinforce the point that the variational method should be implemented with some caution.

In some cases, one can exploit symmetry to address the properties of higher-lying states. For example, if the one-dimensional attractive potential is symmetric about the origin, and has more than one bound state, the ground state will be even, the first excited state odd. Therefore, we can estimate the energy of the first √ excited state by minimizing a family of odd functions, such π 1/2 −αx2 /2 as ψ(x, α) = ( 2α3/2 ) xe . ! Example: Helium atom addressed by the variational approach: For the hydrogen atom, we know that the ground state energy is 1 Ry, or 13.6 eV. The He+ ion (with just a single electron) has a nuclear charge of Z = 2, so the ground state energy of the electron, being proportional to Z 2 , will now be equal to 4 Ry. Therefore, for the He atom, if we neglect their mutual interaction, the electrons will occupy the ground state wavefunction having opposite spin, leading to a total ground state energy of 8 Ry or 109 eV. In practice, as we have seen earlier, the repulsion between the electrons lowers ground state energy to 79 eV (see page 64). To get a better estimate for the ground state energy, one can retain the form of Z 3 1/2 −Zr/a0 the ionic wavefunction, ( πa e , but rather than setting the nuclear charge 3) 0

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Z = 2, leave it as a variational parameter. In other words, let us accommodate the effects of electron-electron repulsion, which must “push” the wavefunctions to larger radii, by keeping exactly the same wavefunction profile but lessening the effective nuclear charge as reflected in the spread of the wavefunction from Z = 2 to Z < 2. The precise value will be set by varying it to find the minimum total energy, including the term from electron-electron repulsion. To find the potential energy from the nuclear-electron interactions, we of course must use the actual nuclear charge Z = 2, but impose a variable Z for the wavefunction, so the nuclear potential energy for the two electrons is given by, $ ∞ e2 Z 3 e−2Zr/a0 2e2 = −Z 4πr2 dr 3 = −8Z Ry. p.e. = −2 × 4π$0 0 πa0 r π$0 a0 This could have been inferred from the formula for the one electron ion, where the potential energy for the one electron is −2Z 2 Ry, one factor of Z being from the nuclear charge, the other from the consequent shrinking of the orbit. The kinetic energy is even easier to determine: it depends entirely on the form of the wavefunction, and not on the actual nuclear charge. So for our trial wavefunction it has to be Z 2 Ry per electron. Finally, making use of our calculation on page 64, we can immediately write down the positive contribution to the energy expectation value from the electronelectron interaction, $ 5 e2 Z 5 e2 Z3 e−2Z(r1 +r2 )/a0 dr1 dr2 = = Z Ry . 3 2 4π$0 (πa0 ) |r1 − r2 | 4 4π$0 2a0 4 Collecting all of the terms, the total variational state energy is given by: " # 5 2 E = −2 4Z − Z − Z Ry . 8 5 , Minimization of this energy with respect to Z obtains the minimum at Z = 2 − 16 leading to an energy of 77.5 eV. This result departs from the true value by about 1 eV. So, indeed, the presence of the other electron leads effectively to a shielding of the nuclear charge by an amount of ca. (5/16)e.

This completes our discussion of the principles of the variational approach. However, later in the course, we will find the variational methods appearing in several important applications. Finally, to close this section on approximation methods for stationary states, we turn now to consider a framework which makes explicit the connection between the quantum and classical theory in the limit ! → 0.

7.4

Wentzel, Kramers and Brillouin (WKB) method

The WKB (or Wentzel, Kramers and Brillouin) approximation describes a “quasi-classical” method for solving the one-dimensional time-independent Schr¨odinger equation. Note that the consideration of one-dimensional systems is less restrictive that it may sound as many symmetrical higher-dimensional problems are rendered effectively one-dimensional (e.g. the radial equation for the hydrogen atom). The WKB method is named after physicists Wentzel, Kramers and Brillouin, who all developed the approach independently in 1926.4 Earlier, in 1923, the mathematician Harold Jeffreys had developed a general method of approximating the general class of linear, second-order 4 L. Brillouin, (1926). “La mcanique ondulatoire de Schr¨ odinger: une m´thode g´en´erale de resolution par approximations successives”, Comptes Rendus de l’Academie des Sciences 183: 2426; H. A. Kramers, (1926). “Wellenmechanik und halbz¨ ahlige Quantisierung”, Z. Phys. 39: 828840; G. Wentzel (1926). “Eine Verallgemeinerung der Quantenbedingungen f¨ ur die Zwecke der Wellenmechanik”. Z. Phys. 38: 518529.

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L´ eon Nicolas Brillouin 18891969 A French physicis, his father, Marcel Brillouin, grandfather, ´ Eleuth` ere Mascart, and greatgrandfather, Charles Briot, were physicists as well. He made contributions to quantum mechanics, radio wave propagation in the atmosphere, solid state physics, and information theory.

Hendrik Anthony Kramers 1894-1952

“Hans”

A Dutch physicist who conducted early and important work in quantum theory and electromagnetic dispersion relations, solid-state physics, and statistical mechanics. He was a long-time assistant and friend to Niels Bohr, and collaborated with him on a 1924 paper contending that light consists of probability waves, which became a foundation of quantum mechanics. He introduced the idea of renormalization, a cornerstone of modern field theory, and determined the dispersion formulae that led to Werner Heisenberg’s matrix mechanics. He is not as well known as some of his contemporaries (primarily because his work was not widely translated into English), but his name is still invoked by physicists as they discuss Kramers dispersion theory, Kramers-Heisenberg dispersion formulae, Kramers opacity formula, Kramers degeneracy, or Kramers-Kronig relations.

7.4. WENTZEL, KRAMERS AND BRILLOUIN (WKB) METHOD differential equations, which of course includes the Schr¨odinger equation.5 But since the Schr¨odinger equation was developed two years later, and Wentzel, Kramers, and Brillouin were apparently unaware of this earlier work, the contribution of Jeffreys is often neglected. The WKB method is important both as a practical means of approximating solutions to the Schr¨odinger equation, and also as a conceptual framework for understanding the classical limit of quantum mechanics. The WKB approximation is valid whenever the wavelength, λ, is small in comparison to other relevant length scales in the problem. This condition is not restricted to quantum mechanics, but rather can be applied to any wave-like system (such as fluids, electromagnetic waves, etc.), where it leads to approximation schemes which are mathematically very similar to the WKB method in quantum mechanics. For example, in optics the approach is called the eikonal method, and in general the method is referred to as short wavelength asymptotics. Whatever the name, the method is an old one, which predates quantum mechanics – indeed, it was apparently first used by Liouville and Green in the first half of the nineteenth century. In quantum mechanics, λ is interpreted as the de Broglie wavelength, and L is normally the length scale of the potential. Thus, the WKB method is valid if the wavefunction oscillates many times before the potential energy changes significantly.

7.4.1

Semi-classical approximation to leading order

Consider then the propagation of a quantum particle in a slowly-varying onedimensional potential, V (x). Here, by “slowly-varying” we mean that, in any small region the wavefunction is well-approximated by a plane wave, and that the wavelength only changes over distances that are long compared with the local value of the wavelength. We’re also assuming for the moment that the particle has positive kinetic energy in the region. Under these conditions, we can anticipate that the solution to the time-independent Schr¨odinger equation −

!2 2 ∂ ψ(x) + V (x)ψ(x) = Eψ(x) , 2m x

will take the form A(x)e±ip(x)x/! where p(x) is the “local” value of the momentum set by the classical value, p2 /2m + V (x) = E, and the amplitude, A(x), is slowly-varying compared with the phase factor. Clearly this is a semi-classical limit: ! has to be sufficiently small that there are many oscillations in the typical distance over which the potential varies.6 To develop this idea a little more rigorously, and to emphasize the rapid phase variation in the semi-classical limit, we can parameterize the wavefunction as ψ(x) = eiσ(x)/! . 5

H. Jeffreys, (1924). “On certain approximate solutions of linear differential equations of the second order”, Proc. Lon. Math. Soc. 23: 428436. 6 To avoid any point of confusion, it is of course true that ! is a fundamental constant – not easily adjusted! So what do we mean when we say that the semi-classical limit translates to ! → 0? The validity of the semi-classical approximation relies upon λ/L $ 1. Following the de Broglie relation, we may write this inequality as h/pL $ 1, where p denotes the particle momentum. Now, in this correspondence, both p and L can be considered as “classical” scales. So, formally, we can think of think of accessing the semi-classical limit by adjusting ! so that it is small enough to fulfil this inequality. Alternatively, at fixed !, we can access the semi-classical regime by reaching to higher and higher energy scales (larger and larger p) so that the inequality becomes valid.

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7.4. WENTZEL, KRAMERS AND BRILLOUIN (WKB) METHOD Here the complex function σ(x) encompasses both the amplitude and phase. Then, with −!2 ∂x2 ψ(x) = −i!eiσ(x)/!∂x2 σ(x) + eiσ(x)/!(∂x σ)2 , the Schr¨odinger equation may be rewritten in terms of the phase function as, −i!∂x2 σ(x) + (∂x σ)2 = p2 (x) .

(7.9)

Now, since we’re assuming the system is semi-classical, it makes sense to expand σ(x) as a power series in ! setting, σ = σ0 + (!/i)σ1 + (!/i)2 σ2 + · · · . At the leading (zeroth) order of the expansion, we can drop the first term in'(7.9), leading to (∂x σ0 )2 = p2 (x). Fixing the sign of p(x) by p(x) = + 2m(E − V (x)), we conclude that $ σ0 (x) = ± p(x)dx . For free particle systems – those for which the kinetic energy is proportional to p2 – this expression coincides with the classical action. From the form of the Schr¨odinger equation (7.9), it is evident that this approximate solution is only valid if we can ignore the first term. More precisely, we must have 5 5 5 !∂x2 σ(x) 5 5 5 5 (∂x σ(x))2 5 ≡ |∂x (!/∂x σ)| # 1 . But, in the leading approximation, ∂x σ ( p(x) and p(x) = 2π!/λ(x), so the condition translates to the relation 1 |∂x λ(x)| # 1 . 2π This means that the change in wavelength over a distance of one wavelength must be small. Obviously, this condition can not always be met: In particular, if the particle is confined by an attractive potential, at the edge of the classically allowed region, where E = V (x), p(x) must is zero and the corresponding wavelength infinite. The approximation is only valid well away from these classical turning points, a matter to which we will return shortly.

7.4.2

Next to leading order correction

Let us now turn to the next term in the expansion in !. Retaining terms from Eq. (7.9) which are of order !, we have −i!∂x2 σ0 + 2∂x σ0 (!/i)∂x σ1 = 0 . Rearranging this equation, and integrating, we find ∂x σ 1 = −

∂x2 σ0 ∂x p , =− 2∂x σ0 2p

1 σ1 (x) = − ln p(x) . 2

So, to this order of approximation, the wavefunction takes the form, C2 −(i/!) R p dx C1 (i/!) R p dx +' , e e ψ(x) = ' p(x) p(x)

(7.10)

where C1 and C2 denote constants of integration. To interpret the factors of ' p(x), consider the first term: a wave moving to the right. Since p(x) is real Advanced Quantum Physics

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7.4. WENTZEL, KRAMERS AND BRILLOUIN (WKB) METHOD (remember we are currently considering the classically allowed region where E > V (x)), the exponential has modulus unity, and the local probability density is proportional to 1/p(x), i.e. to 1/v(x), where v(x) denotes the velocity of the particle. This dependence has a simple physical interpretation: The probability of finding the particle in any given small interval is proportional to the time it spends there. Hence it is inversely proportional to its speed. We turn now to consider the wavefunction in the classically forbidden region where p2 (x) = E − V (x) < 0 . 2m Here p(x) is of course pure imaginary, but the same formal phase solution of the Schr¨odinger equation applies provided, again, that the particle is remote from the classical turning points where E = V (x). In this region, the wavefunction takes the general form, ψ(x) = '

C1% |p(x)|

e−(1/!)

R

|p| dx

R C% + ' 2 e(1/!) |p| dx . |p(x)|

(7.11)

This completes our study of the wavefunction in the regions in which the semi-classical approach can be formally justified. However, to make use of this approximation, we have to understand how to deal with the regions close to the classical turning points. Remember in our treatment of the Schr¨odinger equation, energy quantization derived from the implementation of boundary conditions.

7.4.3

Connection formulae, boundary conditions and quantization rules

Let us assume that we are dealing with a one-dimensional confining potential where the classically allowed region is unique and spans the interval b ≤ x ≤ a. Clearly, in the classically forbidden region to the right, x > a, only the first term in Eq. (7.11) remains convergent and can contribute while, for x < b it is only the second term that contributes. Moreover, in the classically allowed region, b ≤ x ≤ a, the wavefunction has the oscillating form (7.10). But how do we connect the three regions together? To answer this question, it is necessary to make the assumption that the potential varies sufficiently smoothly that it is a good approximation to take it to be linear in the vicinity of the classical turning points. That is to say, we assume that a linear potential is a sufficiently good approximation out to the point where the short wavelength (or decay length for tunneling regions) description is adequate. Therefore, near the classical turning at x = a, we take the potential to be E − V (x) ( F0 (x − a) , where F0 denotes the (constant) force. For a strictly linear potential, the wavefunction can be determined analytically, and takes the form of an Airy function.7 In particular, it is known that the Airy function to the right of the classical turning point has the asymptotic form Rx C lim ψ(x) = ' e−(1/!) a |p| dx , x&a 2 |p(x)|

7 For a detailed discussion in the present context, we refer to the text by Landau and Lifshitz.

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7.4. WENTZEL, KRAMERS AND BRILLOUIN (WKB) METHOD

76

translating to a decay into the classically forbidden region while, to the left, it has the asymptotic oscillatory solution, 7 6 $ a 6 7 $ C C 1 π 1 a π ≡' − cos cos p dx − p dx . lim ψ(x) = ' b'x 0, we obtain " ! " ! n 1 2 Zα 4 3 ˆ ˆ %H1 + H2 &n,j=!±1/2,mj ,! = mc − , 2 n 4 j + 1/2 while for ' = 0, we retain just the kinetic energy term (9.2). 4

For details see, e.g., Ref [1].

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For ' > 0, $

1 r3

%

= n!

!

mcαZ !n

"3

1 . '(' + 12 )(' + 1)

9.1. THE “REAL” HYDROGEN ATOM

9.1.3

93

Darwin term

The final contribution to the Hamiltonian from relativistic effects is known as the Darwin term and arises from the “Zitterbewegung” of the electron – trembling motion – which smears the effective potential felt by the electron. Such effects lead to a perturbation of the form, ! " 2 2 !2 e Ze2 ˆ 3 = ! ∇2 V = ! Q (r) = 4πδ (3) (r) , H nuclear 2 2 2 2 8m c 8m c "0 4π"0 8(mc)2 where Qnuclear (r) = Zeδ (3) (r) denotes the nuclear charge density. Since the perturbation acts only at the origin, it effects only states with ' = 0. As a result, one finds that 2 !2 1 ˆ 3 &njm ! = Ze %H 4π|ψ!n (0)|2 = mc2 j 2 4π"0 8(mc) 2

!

Zα n

"4

nδ!,0 .

Intriguingly, this term is formally identical to that which would be obtained ˆ 2 & at ' = 0. As a result, combining all three contributions, the total from %H energy shift is given simply by 1 ∆En,j=!±1/2,mj ,! = mc2 2

!

Zα n

"4 !

n 3 − 4 j + 1/2

"

,

(9.3)

a result that is independent of ' and mj . To discuss the predicted energy shifts for particular states, it is helpful to introduce some nomenclature from atomic physics. For a state with principle quantum number n, total spin s, orbital angular momentum ', and total angular momentum j, one may use spectroscopic notation n2s+1 Lj to define the state. For a hydrogen-like atom, with just a single electron, 2s + 1 = 2. In this case, the factor 2s + 1 is often just dropped for brevity. If we apply our perturbative expression for the relativistic corrections (9.3), how do we expect the levels to shift for hydrogen-like atoms? As we have seen, for the non-relativistic Hamiltonian, each state of given n exhibits a 2n2 -fold degeneracy. For a given multiplet specified by n, the relativistic corrections depend only on j and n. For n = 1, we have ' = 0 and j = 1/2: Both 1S1/2 states, with mj = 1/2 and −1/2, experience a negative energy shift by an amount ∆E1,1/2,mj ,0 = − 41 Z 4 α2 Ry. For n = 2, ' can take the values of 0 or 1. With j = 1/2, both the former 2S1/2 state, and the latter 2P1/2 states share 5 4 2 the same negative shift in energy, ∆E2,1/2,mj ,0 = ∆E2,1/2,mj ,1 = − 64 Z α Ry, 1 4 2 while the 2P3/2 experiences a shift of ∆E2,3/2,mj ,1 = − 64 Z α Ry. Finally, for n = 3, ' can take values of 0, 1 or 2. Here, the pairs of states 3S1/2 and 3P1/2 , and 3P3/2 and 2D3/2 each remain degenerate while the state 3D5/2 is unique. These predicted shifts are summarized in Figure 9.1. This completes our discussion of the relativistic corrections which develop from the treatment of the Dirac theory for the hydrogen atom. However, this does not complete our discription of the “real” hydrogen atom. Indeed, there are further corrections which derive from quantum electrodynamics and nuclear effects which we now turn to address.

9.1.4

Lamb shift

According to the perturbation theory above, the relativistic corrections which follow from the Dirac theory for hydrogen leave the 2S1/2 and 2P1/2 states degenerate. However, in 1947, a careful experimental study by Willis Lamb Advanced Quantum Physics

Willis Eugene Lamb, 1913-2008 A physicist who won the Nobel Prize in Physics in 1955 “for his discoveries concerning the fine structure of the hydrogen spectrum”. Lamb and Polykarp Kusch were able to precisely determine certain electromagnetic properties of the electron.

9.1. THE “REAL” HYDROGEN ATOM

94

Figure 9.1:

Figure showing the heirarchy of energy shifts of the spectra of hydrogen-like atoms as a result of relativistic corrections. The first column shows the energy spectrum predicted by the (non-relativistic) Bohr theory. The second column shows the predicted energy shifts from relativistic corrections arising from the Dirac theory. The third column includes corrections due quantum electrodynamics and the fourth column includes terms for coupling to the nuclear spin degrees of freedom. The H-α line, particularly important in the astronomy, corresponds to the transition between the levels with n = 2 and n = 3.

and Robert Retherford discovered that this was not in fact the case:5 2P1/2 state is slightly lower in energy than the 2S1/2 state resulting in a small shift of the corresponding spectral line – the Lamb shift. It might seem that such a tiny effect would be deemed insignificant, but in this case, the observed shift (which was explained by Hans Bethe in the same year) provided considerable insight into quantum electrodynamics. In quantum electrodynamics, a quantized radiation field has a zero-point energy equivalent to the mean-square electric field so that even in a vacuum there are fluctuations. These fluctuations cause an electron to execute an oscillatory motion and its charge is therefore smeared. If the electron is bound by a non-uniform electric field (as in hydrogen), it experiences a different potential from that appropriate to its mean position. Hence the atomic levels are shifted. In hydrogen-like atoms, the smearing occurs over a length scale, %(δr)2 & $

2α π

!

! mc

"2

ln

1 , αZ

some five orders of magnitude smaller than the Bohr radius. This causes the electron spin g-factor to be slightly different from 2, ! " α2 α − 0.328 2 + · · · . gs = 2 1 + 2π π

Hans Albrecht Bethe 1906-2005 A GermanAmerican physicist, and Nobel laureate in physics “for his work on the theory of stellar nucleosynthesis.” A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and particle astrophysics. During World War II, he was head of the Theoretical Division at the secret Los Alamos laboratory developing the first atomic bombs. There he played a key role in calculating the critical mass of the weapons, and did theoretical work on the implosion method used in both the Trinity test and the “Fat Man” weapon dropped on Nagasaki.

There is also a slight weakening of the force on the electron when it is very close to the nucleus, causing the 2S1/2 electron (which has penetrated all the way to the nucleus) to be slightly higher in energy than the 2P1/2 electron. Taking into account these corrections, one obtains a positive energy shift ∆ELamb

" ! "4 ! 1 Z 8 2 α ln δ!,0 , $ nα Ry × n 3π αZ

for states with ' = 0. 5

W. E. Lamb and R. C. Retherfod, Fine Structure of the Hydrogen Atom by a Microwave Method, Phys. Rev. 72, 241 (1947).

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Hydrogen fine structure and hyperfine structure for the n = 3 to n = 2 transition (see Fig. 9.1).

9.1. THE “REAL” HYDROGEN ATOM

9.1.5

Hyperfine structure

So far, we have considered the nucleus as simply a massive point charge responsible for the large electrostatic interaction with the charged electrons which surround it. However, the nucleus has a spin angular momentum which is associated with a further set of hyperfine corrections to the atomic spectra of atoms. As with electrons, the protons and neutrons that make up a nucleus are fermions, each with intrinsic spin 1/2. This means that a nucleus will have some total nuclear spin which is labelled by the quantum number, I. The latter leads to a nuclear magnetic moment, µN = gN

Ze I. 2MN

where MN denotes the mass of the nucleus, and gN denotes the gyromagnetic ratio. Since the nucleus has internal structure, the nuclear gyromagnetic ratio is not simply 2 as it (nearly) is for the electron. For the proton, the sole nuclear constituent of atomic hydrogen, gP ≈ 5.56. Even though the neutron is charge neutral, its gyromagnetic ratio is about −3.83. (The consitituent quarks have gyromagnetic ratios of 2 (plus corrections) like the electron but the problem is complicated by the strong interactions which make it hard to define a quark’s mass.) We can compute (to some accuracy) the gyromagnetic ratio of nuclei from that of protons and neutrons as we can compute the proton’s gyromagnetic ratio from its quark constituents. Since the nuclear mass is several orders of magnitude higher than that of the electron, the nuclear magnetic moment provides only a small perturbation. According to classical electromagnetism, the magnetic moment generates a magnetic field B=

2µ0 µ0 µ δ (3) (r) . (3(µN · er )er − µN ) + 4πr3 3 N

To explore the effect of this field, let us consider just the s-electrons, i.e. ' = 0, for simplicity.6 In this case, the interaction of the magnetic moment of the electrons with the field generated by the nucleus, gives rise to the hyperfine interaction, ˆ · B. ˆ hyp = −µe · B = e S H mc For the ' = 0 state, the first contribution to the first order correction, ! "4 Z nα2 Ry × %Hhyp &n,1/2,0 = n

B vanishes while second leads to m 1 8 gN S · I. 3 MN ! 2

Once again, to evaluate the expectation values on the spin degrees of freedom, it is convenient to define the total spin F = I + S. We then have 1 1 1 S · I = 2 (F2 − S2 − I2 ) = (F (F + 1) − 3/4 − I(I + 1)) 2 ! 2 *2! 1 I F = I + 1/2 = 2 −I − 1 F = I − 1/2 Therefore, the 1s state of Hydrogen is split into two, corresponding to the two possible values F = 0 and 1. The transition between these two levels has frequency 1420 Hz, or wavelength 21 cm, so lies in the radio waveband. It 6

For a full discussion of the influence of the orbital angular momentum, we refer to Ref. [6].

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9.2. MULTI-ELECTRON ATOMS

96

is an important transition for radio astronomy. A further contribution to the hyperfine structure arises if the nuclear shape is not spherical thus distorting the Coulomb potential; this occurs for deuterium and for many other nuclei. Finally, before leaving this section, we should note that the nucleus is not point-like but has a small size. The effect of finite nuclear size can be estimated perturbatively. In doing so, one finds that the s (' = 0) levels are those most effected, because these have the largest probability of finding the electron close to the nucleus; but the effect is still very small in hydrogen. It can be significant, however, in atoms of high nuclear charge Z, or for muonic atoms. This completes our discussion of the “one-electron” theory. We now turn to consider the properties of multi-electron atoms.

9.2

Multi-electron atoms

To address the electronic structure of a multi-electron atom, we might begin with the hydrogenic energy levels for an atom of nuclear charge Z, and start filling the lowest levels with electrons, accounting for the exclusion principle. The degeneracy for quantum numbers (n, ') is 2 × (2' + 1), where (2' + 1) is the number of available m! values, and the factor of 2 accounts for the spin degeneracy. Hence, the number of electrons accommodated in shell, n, would be 2 × n2 , n 1 2 3 4

' 0 0, 1 0, 1, 2 0, 1, 2, 3

Degeneracy in shell 2 (1 + 3) × 2 = 8 (1 + 3 + 5) × 2 = 18 (1 + 3 + 5 + 7) × 2 = 32

Cumulative total 2 10 28 60

We would therefore expect that atoms containing 2, 10, 28 or 60 electrons would be especially stable, and that in atoms containing one more electron than this, the outermost electron would be less tightly bound. In fact, if we look at data (Fig. 9.2) recording the first ionization energy of atoms, i.e. the minimum energy needed to remove one electron, we find that the noble gases, having Z = 2, 10, 18, 36 · · · are especially tightly bound, and the elements containing one more electron, the alkali metals, are significantly less tightly bound. The reason for the failure of this simple-minded approach is fairly obvious – we have neglected the repulsion between electrons. In fact, the first ionization energies of atoms show a relatively weak dependence on Z; this tells us that the outermost electrons are almost completely shielded from the nuclear charge.7 Indeed, when we treated the Helium atom as an example of the variational method in chapter 7, we found that the effect of electron-electron repulsion was sizeable, and really too large to be treated accurately by perturbation theory. 7

In fact, the shielding is not completely perfect. For a given energy shell, the effective nuclear charge varies for an atomic number Z as Zeff ∼ (1 + α)Z where α > 0 characterizes 2 the ineffectiveness of screening. This implies that the ionization energy IZ = −EZ ∼ Zeff ∼ (1 + 2αZ). The near-linear dependence of IZ on Z is reflected in Fig. 9.2.

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A muon is a particle somewhat like an electron, but about 200 times heavier. If a muon is captured by an atom, the corresponding Bohr radius is 200 times smaller, thus enhancing the nuclear size effect.

9.2. MULTI-ELECTRON ATOMS

97

Figure 9.2: Ionization energies of the elements.

9.2.1

Central field approximation

Leaving aside for now the influence of spin or relativistic effects, the Hamiltonian for a multi-electron atom can be written as - + + , !2 1 e2 1 Ze2 2 ˆ ∇i − + − , H= 2m 4π"0 ri 4π"0 rij i 2mc2 . In that case, the total change in energy of producing an electron-positron pair, subsequently binding the electron in the lowest state and letting the positron escape to infinity (it is repelled by the nucleus), is negative. There is an instability! The attractive electrostatic energy of binding the electron pays the price of producing the pair. Nuclei with very high atomic mass spontaneously “screen” themselves by polarising the vacuum via electron-positron production until the they lower their charge below a critical value Zc . This implies that objects with a charge greater than Zc are unobservable due to screening. ! Info. An estimate based on the non-relativistic formula above gives Zc $

270. Taking into account relativistic effects, the result is renormalised downwards to 137, while taking into account the finite size of the nucleus one finally obtains Zc ∼ 165. Of course, no such nuclei exist in nature, but they can be manufactured, fleetingly, in uranium ion collisions where Z = 2 × 92 = 184. Indeed, the production rate of positrons escaping from the nucleus is seen to increase dramatically as the total Z of the pair of ions passes 160.

! Spin: Finally, while the phenomenon of electron spin has to be grafted Advanced Quantum Physics

169 artificially onto the non-relativistic Schr¨odinger equation, it emerges naturally from a relativistic treatment of quantum mechanics. When do we expect relativity to intrude into quantum mechanics? According to the uncertainty relation, ∆x∆p ≥ !/2, the length scale at which the kinetic energy is comparable to the rest mass energy is set by the Compton wavelength h ≡ λc . mc We may expect relativistic effects to be important if we examine the motion of particles on length scales which are less than λc . Note that for particles of zero mass, λc = ∞! Thus for photons, and neutrinos, relativity intrudes at any length scale. What is the relativistic analogue of the Schr¨odinger equation? Non-relativistic quantum mechanics is based on the time-dependent Schr¨odinger equation ˆ = i!∂t ψ, where the wavefunction ψ contains all information about a given Hψ system. In particular, |ψ(x, t)|2 represents the probability density to observe a particle at position x and time t. Our aim will be to seek a relativistic version of this equation which has an analogous form. The first goal, therefore, is to find the relativistic Hamiltonian. To do so, we first need to revise results from Einstein’s theory of special relativity: ∆x ≥

! Info. Lorentz Transformations and the Lorentz Group: In the special theory of relativity, a coordinate in space-time is specified by a 4-vector. A contravariant 4-vector x = (xµ ) ≡ (x0 , x1 , x2 , x3 ) ≡ (ct, x) is transformed into the covariant 4-vector xµ = gµν xν by the Minkowskii metric   1 −1   (gµν ) =  gµν gνλ = δµλ , , −1 −1

Here, by convention, summation is assumed over repeated indicies. Indeed, summation covention will be assumed throughout this chapter. The scalar product of 4-vectors is defined by x · y = xµ y µ = xµ y ν gµν = xµ yµ .

The Lorentz group consists of linear Lorentz transformations, Λ, preserving x·y, µ i.e. for xµ )→ x! = Λµν xν , we have the condition gµν Λµα Λνβ = gαβ .

(15.1)

Specifically, a Lorentz transformation along the x1 direction can be expressed in the form   γ −γv/c γ  −γv/c  Λµν =   1 0 0 1

where γ = (1 − v 2 /c2 )−1/2 .1 With this definition, the Lorentz group splits up into four components. Every Lorentz transformation maps time-like vectors (x2 > 0) into 1

Equivalently the Lorentz transformation can be represented in the form 1 0 0 −1 C B −1 0 C, Λ = exp[ωK1 ], [K1 ]µν = B @ 0 0A 0 0

where ω = tanh−1 (v/c) is known as the rapidity, and K1 is the generator of velocity transformations along the x1 -axis.

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15.1. KLEIN-GORDON EQUATION

170

time-like vectors. Time-like vectors can be divided into those pointing forwards in time (x0 > 0) and those pointing backwards (x0 < 0). Lorentz transformations do not always map forward time-like vectors into forward time-like vectors; indeed Λ does so if and only if Λ00 > 0. Such transformations are called orthochronous. (Since Λµ0 Λµ0 = 1, (Λ00 )2 − (Λj0 )2 = 1, and so Λ00 += 0.) Thus the group splits into two according to whether Λ00 > 0 or Λ00 < 0. Each of these two components may be subdivided into two by considering those Λ for which det Λ = ±1. Those transformations Λ for which det Λ = 1 are called proper. Thus the subgroup of the Lorentz group for which det Λ = 1 and Λ00 > 0 is called the proper orthochronous Lorentz group, sometimes denoted by L↑+ . It contains neither the time-reversal nor parity transformation,     −1 1 1 −1     T = P = (15.2) , . 1 −1 1 −1

We shall call it the Lorentz group for short and specify when we are including T or P . In particular, L↑+ , L↑ = L↑+ ∪ L↑− (the orthochronous Lorentz group), L+ = L↑+ ∪ L↓+ (the proper Lorentz group), and L0 = L↑+ ∪ L↓− are subgroups, while L↓− = P L↑+ , L↑− = T L↑+ and L↓+ = T P L↑+ are not. Special relativity requires that theories should be invariant under Lorentz transe transformations xµ → formations xµ )−→ Λµν xν , and, more generally, Poincar´ Λµν xν + aµ . The proper orthochronous Lorentz transformations can be reached continuously from identity.2 Loosely speaking, we can form them by putting together infinitesimal Lorentz transformations Λµν = δ µν + ω µν , where the elements of ω µν - 1. Applying the identity gαβ = Λµα Λµβ = gαβ + ωαβ + ωβα + O(ω 2 ), we obtain the relation ωαβ = −ωβα . ωαβ has six independent components: L↑+ is a six-dimensional (Lie) group, i.e. it has six independent generators: three rotations and three boosts. Finally, according to the definition of the 4-vectors, the covariant and contravari∂ , ∇), ∂ µ = ∂x∂ µ = ant derivative are respectively defined by ∂µ = ∂x∂ µ = ( 1c ∂t

∂ , −∇). Applying the scalar product to the derivative we obtain the d’Alembertian ( 1c ∂t ∂2 2 operator (sometimes denoted as !), ∂ 2 = ∂µ ∂ µ = c12 ∂t 2 − ∇ .

15.1

Klein-Gordon equation

Historically, the first attempt to construct a relativistic version of the Schr¨odinger equation began by applying the familiar quantization rules to the relativistic Oskar Benjamin Klein 1894energy-momentum invariant. In non-relativistic quantum mechanics the cor1977 A Swedish theorespondence principle dictates that the momentum operator is associated with retical physicist, ˆ = −i!∇, and the energy operator with the time derivathe spatial gradient, p Klein is credited ˆ = i!∂t . Since (pµ ≡ (E/c, p) transforms like a 4-vector under Lorentz for inventing tive, E the idea, part transformations, the operator pˆµ = i!∂ µ is relativistically covariant. of Kaluza-Klein Non-relativistically, the Schr¨odinger equation is obtained by quantizing theory, that extra dimensions may the classical Hamiltonian. To obtain a relativistic version of this equation, be physically real one might apply the quantization relation to the dispersion relation obtained but curled up and very small, an idea essential to from the energy-momentum invariant p2 = (E/c)2 − p2 = (mc)2 , i.e. string theory/M-theory. ) 2 4 * , + 2 4 2 2 1/2 2 2 2 1/2 E(p) = + m c + p c ⇒ i!∂t ψ = m c − ! c ∇ ψ

where m denotes the rest mass of the particle. However, this proposal poses a dilemma: how can one make sense of the square root of an operator? Interpreting the square root as the Taylor expansion, i!∂t = mc2 ψ − 2

!4 (∇2 )2 !2 ∇2 ψ− ψ + ··· 2m 8m3 c2

They are said to form the path component of the identity.

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15.1. KLEIN-GORDON EQUATION

171

we find that an infinite number of boundary conditions are required to specify the time evolution of ψ.3 It is this effective “non-locality” together with the asymmetry (with respect to space and time) that suggests this equation may be a poor starting point. A second approach, and one which circumvents these difficulties, is to apply the quantization procedure directly to the energy-momentum invariant: ) * E 2 = p2 c2 + m2 c4 , −!2 ∂t2 ψ = −!2 c2 ∇2 + m2 c4 ψ.

Recast in the Lorentz invariant form of the d’Alembertian operator, we obtain the Klein-Gordon equation * ∂ 2 + kc2 ψ = 0 ,

)

(15.3)

where kc = 2π/λc = mc/!. Thus, at the expense of keeping terms of second order in the time derivative, we have obtained a local and manifestly covariant equation. However, invariance of ψ under global spatial rotations implies that, if applicable at all, the Klein-Gordon equation is limited to the consideration of spin-zero particles. Moreover, if ψ is the wavefunction, can |ψ|2 be interpreted as a probability density? To associate |ψ|2 with the probability density, we can draw intuition from the consideration of the non-relativistic Schr¨odinger equation. Applying the 2 ∇2 ψ) = 0, together with the complex conjugate of this identity ψ ∗ (i!∂t ψ + !2m equation, we obtain ∂t |ψ|2 − i

! ∇ · (ψ ∗ ∇ψ − ψ∇ψ ∗ ) = 0 . 2m

Conservation of probability means that density ρ and current j must satisfy the continuity relation, ∂t ρ + ∇ · j = 0, which states simply that the rate of decrease of density in any volume element is equal to the net current flowing out of that element. Thus, for the Schr¨odinger equation, we can consistently ! define ρ = |ψ|2 , and j = −i 2m (ψ ∗ ∇ψ − ψ∇ψ ∗ ). Applied to the Klein-Gordon equation (15.3), the same consideration implies !2 ∂t (ψ ∗ ∂t ψ − ψ∂t ψ ∗ ) − !2 c2 ∇ · (ψ ∗ ∇ψ − ψ∇ψ ∗ ) = 0 , from which we deduce the correspondence, ρ=i

! (ψ ∗ ∂t ψ − ψ∂t ψ ∗ ) , 2mc2

j = −i

! (ψ ∗ ∇ψ − ψ∇ψ ∗ ) . 2m

The continuity equation associated with the conservation of probability can be expressed covariantly in the form ∂µ j µ = 0 ,

(15.4)

where j µ = (ρc, j) is the 4-current. Thus, the Klein-Gordon density is the time-like component of a 4-vector. From this association it is possible to identify three aspects which (at least initially) eliminate the Klein-Gordon equation as a wholey suitable candidate for the relativistic version of the wave equation: 3

You may recognize that the leading correction to the free particle Schr¨ odinger equation is precisely the relativistic correction to the kinetic energy that we considered in chapter 9.

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15.2. DIRAC EQUATION

172

! The first disturbing feature of the Klein-Gordon equation is that the density ρ is not a positive definite quantity, so it can not represent a probability. Indeed, this led to the rejection of the equation in the early years of relativistic quantum mechanics, 1926 to 1934. ! Secondly, the Klein-Gordon equation is not first order in time; it is necessary to specify ψ and ∂t ψ everywhere at t = 0 to solve for later times. Thus, there is an extra constraint absent in the Schr¨odinger formulation. ! Finally, the equation on which the Klein-Gordon equation is based, E 2 = m2 c4 + p2 c2 , has both positive and negative solutions. In fact the apparently unphysical negative energy solutions are the origin of the preceding two problems. To circumvent these difficulties one might consider dropping the negative energy solutions altogether. For a free particle, whose energy is thereby constant, we can simply supplement the Klein-Gordon equation with the condition p0 > 0. However, such a definition becomes inconsistent in the presence of local interactions, e.g. * ) 2 self − interaction ∂ + kc2 ψ = F (ψ) . 2 2 (∂ + iqA/!c) + kc ψ = 0 interaction with EM field.

The latter generate transitions between positive and negative energy states. Thus, merely excluding the negative energy states does not solve the problem. Later we will see that the interpretation of ψ as a quantum field leads to a resolution of the problems raised above. Historically, the intrinsic problems confronting the Klein-Gordon equation led Dirac to introduce another equation.4 However, as we will see, although the new formulation implied a positive norm, it did not circumvent the need to interpret negative energy solutions.

15.2

Dirac Equation

Dirac attached great significance to the fact that Schr¨odinger’s equation of motion was first order in the time derivative. If this holds true in relativistic quantum mechanics, it must also be linear in ∂. On the other hand, for free particles, the equation must imply pˆ2 = (mc)2 , i.e. the wave equation must be consistent with the Klein-Gordon equation (15.3). At the expense of introducing vector wavefunctions, Dirac’s approach was to try to factorise this equation: (γ µ pˆµ − m) ψ = 0 .

(15.5)

(Following the usual convention we have, and will henceforth, adopt the shorthand convention and set ! = c = 1.) For this equation to be admissible, the following conditions must be enforced: ! The components of ψ must satisfy the Klein-Gordon equation. 4

The original references are P. A. M. Dirac, The Quantum theory of the electron, Proc. R. Soc. A117, 610 (1928); Quantum theory of the electron, Part II, Proc. R. Soc. A118, 351 (1928). Further historical insights can be obtained from Dirac’s book on Principles of Quantum mechanics, 4th edition, Oxford University Press, 1982.

Advanced Quantum Physics

Paul A. M. Dirac 1902-1984 Dirac was born on 8th August, 1902, at Bristol, England, his father being Swiss and his mother English. He was educated at the Merchant Venturer’s Secondary School, Bristol, then went on to Bristol University. Here, he studied electrical engineering, obtaining the B.Sc. (Engineering) degree in 1921. He then studied mathematics for two years at Bristol University, later going on to St. John’s College, Cambridge, as a research student in mathematics. He received his Ph.D. degree in 1926. The following year he became a Fellow of St.John’s College and, in 1932, Lucasian Professor of Mathematics at Cambridge. Dirac’s work was concerned with the mathematical and theoretical aspects of quantum mechanics. He began work on the new quantum mechanics as soon as it was introduced by Heisenberg in 1928 – independently producing a mathematical equivalent which consisted essentially of a noncommutative algebra for calculating atomic properties – and wrote a series of papers on the subject, leading up to his relativistic theory of the electron (1928) and the theory of holes (1930). This latter theory required the existence of a positive particle having the same mass and charge as the known (negative) electron. This, the positron was discovered experimentally at a later date (1932) by C. D. Anderson, while its existence was likewise proved by Blackett and Occhialini (1933) in the phenomena of “pair production” and “annihilation”. Dirac was made the 1933 Nobel Laureate in Physics (with Erwin Schr¨ odinger) for the discovery of new productive forms of atomic theory.

15.2. DIRAC EQUATION

173

! There must exist a 4-vector current density which is conserved and whose time-like component is a positive density. ! The components of ψ do not have to satisfy any auxiliary condition. At any given time they are independent functions of x. Beginning with the first of these requirements, by imposing the condition [γ µ , pˆν ] = γ µ pˆν − pˆν γ µ = 0, (and symmetrizing) " ! 1 ν µ ν µ 2 (γ pˆν + m) (γ pˆµ − m) ψ = {γ , γ } pˆν pˆµ − m ψ = 0 , 2

the latter recovers the Klein-Gordon equation if we define the elements γ µ such that they obey the anticommutation relation,5 {γ ν , γ µ } ≡ γ ν γ µ + γ µ γ ν = 2g µν – thus γ µ , and therefore ψ, can not be scalar. Then, from the expansion of ˆ − m)ψ = i∂t ψ − γ 0 γ · p ˆ ψ − mγ 0 ψ = 0, the Dirac Eq. (15.5), γ 0 (γ 0 pˆ0 − γ · p equation can be brought to the form ˆ i∂t ψ = Hψ,

ˆ =α·p ˆ + βm , H

(15.6)

where the elements of the vector α = γ 0 γ and β = γ 0 obey the commutation relations, {αi , αj } = 2δij ,

β 2 = 1,

{αi , β} = 0 .

(15.7)

ˆ is Hermitian if, and only if, α† = α, and β † = β. Expressed in terms of H † γ, this requirement translates to the condition (γ 0 γ)† ≡ γ † γ 0 = γ 0 γ, and † γ 0 = γ 0 . Altogether, we thus obtain the defining properties of Dirac’s γ matrices, γ µ† = γ 0 γ µ γ 0 ,

{γ µ , γ ν } = 2g µν .

(15.8)

Given that space-time is four-dimensional, the matrices γ must have dimension of at least 4 × 4, which means that ψ has at least four components. It is not, however, a 4-vector; it does not transform like xµ under Lorentz transformations. It is called a spinor, or more correctly, a bispinor with special Lorentz transformations which we will shall discuss presently. ! Info. An explicit representation of the γ matrices which most easily captures the non-relativistic limit is the following, ! " ! " I2 0 σ 0 γ0 = , γ= , (15.9) −σ 0 0 −I2

where σ denote the familiar 2 × 2 Pauli spin matrices which satisfy the relations, σi σj = δij + i$ijk σk , σ † = σ. The latter is known in the literature as the DiracPauli representation. We will adopt the particular representation, ! " ! " ! " 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = . 1 0 i 0 0 −1 Note that with this definition, the matrices α and β take the form, ! " ! " 0 σ I2 0 α= , β= . σ 0 0 −I2

5

Note that, in some of the literature, you will see the convention [ , ]+ for the anticommutator.

Advanced Quantum Physics

15.2. DIRAC EQUATION

15.2.1

174

Density and Current

Turning to the second of the requirements placed on the Dirac equation, we now seek the probability density ρ = j 0 . Since ψ is a complex spinor, ρ has to be of the form ψ † M ψ in order to be real and positive. Applying hermitian conjugation to the Dirac equation, we obtain µ← − [(γ µ pˆµ − m)ψ]† = ψ † (−iγ † ∂ µ − m) = 0 ,

← − where ψ † ∂ µ ≡ (∂µ ψ)† . Making use of (15.8), and defining ψ¯ ≡ ψ † γ 0 , the − ¯ ← Dirac equation takes the form ψ(i + ∂ + m) = 0, where we have introduced the Feynman ‘slash’ notation + a ≡ aµ γ µ . Combined with Eq. (15.5) (i.e. − → (i + ∂ − m)ψ = 0), we obtain / 0 ) µ * ← − − → ¯ ψ = 0. ψ¯ + ∂ + + ∂ ψ = ∂µ ψγ From this result and the continuity relation (15.4) we can identify ¯ µψ , j µ = ψγ

(15.10)

(or, equivalently, (ρ, j) = (ψ † ψ, ψ † αψ)) as the 4-current. In particular, the density ρ = j 0 = ψ † ψ is, as required, positive definite.

15.2.2

Relativistic Covariance

To complete our derivation, we must verify that the Dirac equation remains invariant under Lorentz transformations. More precisely, if a wavefunction ψ(x) obeys the Dirac equation in one frame, its counterpart ψ $ (x$ ) in a Lorentz transformed frame x$ = Λx, must obey the Dirac equation, * ) µ $ (15.11) iγ ∂µ − m ψ $ (x$ ) = 0 . In order that an observer in the second frame can reconstruct ψ $ from ψ there must exist a local transformation between the wavefunctions. Taking this relation to be linear, we therefore must have, ψ $ (x$ ) = S(Λ)ψ(x) , where S(Λ) represents a non-singular 4 × 4 matrix. Now, using the identity, ∂xν ∂ −1 )ν ∂ = (Λ−1 )ν ∂ , the Dirac equation (15.11) ∂µ$ ≡ ∂x∂" µ = ∂x " µ ∂xν = (Λ µ ∂xν µ ν in the transformed frame takes the form, * ) µ −1 ν iγ (Λ ) µ ∂ν − m S(Λ)ψ(x) = 0 .

The latter is compatible with the Dirac equation in the original frame if S(Λ)γ ν S −1 (Λ) = γ µ (Λ−1 )νµ .

(15.12)

To define an explicit form for S(Λ) we must now draw upon some of the defining properties of the Lorentz group discussed earlier. For an infinitesimal proper Lorentz transformation we have Λνµ = g νµ + ω νµ and (Λ−1 )νµ = g νµ − ω νµ + · · ·, where the matrix ωµν is antisymmetric and g νµ ≡ δ νµ . Correspondingly, by Taylor expansion in ω, we can define i S(Λ) = I − Σµν ω µν + · · · , 4 Advanced Quantum Physics

i S −1 (Λ) = I + Σµν ω µν + · · · , 4

15.2. DIRAC EQUATION

175

where the matrices Σµν are also antisymmetric in µν. To first order in ω, Eq. (15.12) yields (a somewhat unrewarding exercise!) ) * (15.13) [γ ν , Σαβ ] = 2i g να γβ − g νβ γα . The latter is satisfied by the set of matrices (another exercise!)6 Σαβ =

i [γα , γβ ] . 2

(15.14)

In summary, if ψ(x) obeys the Dirac equation in one frame, the wavefunction can be obtained in the Lorentz transformed frame by applying the transformation ψ $ (x$ ) = S(Λ)ψ(Λ−1 x$ ). Let us now consider the physical consequences of this Lorentz covariance.

15.2.3

Angular momentum and spin

To explore the physical manifestations of Lorentz covariance, it is instructive to consider the class of spatial rotations. For an anticlockwise spatial rotation by an infinitesimal angle θ about a fixed axis n, x )→ x$ = x − θx × n. In terms of the “Lorentz transformation”, Λ, one has x$i = [Λx]i ≡ xi − ωij xj where ωij = $ijk nk θ, and the remaining elements Λµ0 = Λ0µ = 0. Applied to the argument of the wavefunction we obtain a familiar result,7 ψ(x) = ψ(Λ−1 x$ ) = ψ(x$0 , x$ + x$ × nθ) = (1 − θn · x$ × ∇ + · · ·)ψ(x$ ) ˆ + · · ·)ψ(x$ ), = (1 − iθn · L

ˆ =x ˆ×p ˆ represents the non-relativistic angular momentum operator. where L Formally, the angular momentum operators represent the generators of spatial rotations.8 However, we have seen above that Lorentz covariance demands that the transformed wavefunction be multiplied by S(Λ). Using the definition of ωij above, one finds that i S(Λ) ≡ S(I + ω) = I − $ijk nk Σij θ + · · · 4 Then drawing on the Dirac/Pauli representation, 1! " ! "2 i i i 0 σi 0 σj Σij = [γi , γj ] = , = − [σi , σj ] ⊗ I2 = $ijk σk ⊗ I2 , −σ 0 −σ 0 2 2 2 i j one obtains S(Λ) = I − in · Sθ + · · · ,

1 S= 2

!

σ 0

0 σ

"

.

Combining both contributions, we thus obtain ˆ + · · ·)ψ(x$ ) , ψ $ (x$ ) = S(Λ)ψ(Λ−1 x$ ) = (1 − iθn · J 6 Since finite transformations are of the form S(Λ) = exp[−(i/4)Σαβ ω αβ ], one may show that S(Λ) is unitary for spatial rotations, while it is Hermitian for Lorentz boosts. 7 ˆ (θ) = exp(−iθn · Recall that spatial rotataions are generated by the unitary operator, U ˆ L). 8 ˆ For finite transformations, the generator takes the form exp[−iθn · L].

Advanced Quantum Physics

15.3. FREE PARTICLE SOLUTION OF THE DIRAC EQUATION ˆ=L ˆ + S can be identified as a total effective angular momentum of where J the particle being made up of the orbital component, together with an intrinsic contribution known as spin. The latter is characterised by the defining condition: [Si , Sj ] = i$ijk Sk ,

(Si )2 =

1 4

for each i .

(15.15)

Therefore, in contrast to non-relativistic quantum mechanics, the concept of spin does not need to be grafted onto the Schr¨odinger equation, but emerges naturally from the fundamental invariance of the Dirac equation under Lorentz transformations. As a corollary, we can say that the Dirac equation is a relativistic wave equation for particles of spin 1/2.

15.2.4

Parity

So far, our discussion of the covariance properties of the Dirac equation have only dealt with the subgroup of proper orthochronous Lorentz transformations, L↑+ – i.e. those that can be reached from Λ = I by a sequence of infinitesimal transformations. Taking the parity operation into account, relativistic covariance demands S −1 (P )γ 0 S(P ) = γ 0 ,

S −1 (P )γ i S(P ) = −γ i .

This is achieved if S(P ) = γ 0 eiφ , where φ denotes some arbitrary phase. Taking into account the fact that P 2 = I, φ = 0 or π, and we find ψ $ (x$ ) = S(P )ψ(x) = ηγ 0 ψ(P −1 x$ ) = ηγ 0 ψ(ct$ , −x$ ) ,

(15.16)

where η = ±1 represents the intrinsic parity of the particle.

15.3

Free Particle Solution of the Dirac Equation

Having laid the foundation we will now apply the Dirac equation to the problem of a free relativistic quantum particle. For a free particle, the plane wave ψ(x) = exp[−ip · x]u(p) , 3 with energy E ≡ p0 = ± p2 + m2 will be a solution of the Dirac equation if the components of the spinor u(p) are chosen to satisfy the equation (+ p − m)u(p) = 0. Evidently, as with the Klein-Gordon equation, we see that the Dirac equation therefore admits negative as well as positive energy solutions! Soon, having attached a physical significance to the former, we will see that it is convenient to reverse the sign of p for the negative energy solutions. However, for now, let us continue without worrying about the dilemma posed by the negative energy states. In the Dirac-Pauli block representation, " ! 0 p − m −σ · p µ γ pµ − m = . σ · p −p0 − m Thus, defining the spin elements u(p) = (ξ, η), where ξ and η represent twocomponent spinors, we find the conditions, (p0 − m)ξ = σ · p η and σ · p ξ =

Advanced Quantum Physics

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15.3. FREE PARTICLE SOLUTION OF THE DIRAC EQUATION ·p (p0 + m)η. With (p0 )2 = p2 + m2 , these equations are consistent if η = pσ 0 +m ξ. We therefore obtain the bispinor solution   χ(r) u(r) (p) = N (p)  σ · p (r)  , χ p0 + m

where χ(r) represents any pair of orthogonal two-component vectors, and N (p) is the normalisation. Concerning the choice of χ(r) , in many situations, the most convenient basis is the eigenbasis of helicity – eigenstates of the component of spin resolved in the direction of motion, S·

p (±) σ p (±) 1 ≡ · = ± χ(±) , χ χ |p| 2 |p| 2

ˆ3 , χ(+) = (1, 0) and χ(−) = (0, 1). Then, for the positive e.g., for p = p3 e energy states, the two spinor plane wave solutions can be written in the form   χ(±)  ψp(±) (x) = N (p)e−ip·x  |p| (±) χ ± 0 p +m

Thus, according to the discussion above, the Dirac equation for a free particle admits four solutions, two states with positive energy, and two with negative.

15.3.1

Klein paradox: anti-particles

While the Dirac equation has been shown to have positive definite density, as with the Klein-Gordon equation, it still exhibits negative energy states! To make sense of these states it is illuminating to consider the scattering of a plane wave from a potential step. To be precise, consider a beam of relativistic particles with unit amplitude, energy E, momentum pˆ e3 , and spin ↑ (i.e. χ = (1, 0)), incident upon a potential V (x) = V θ(x3 ) (see figure). At the potential barrier, spin is conserved, while a component of the beam with amplitude r is reflected (with energy E and momentum −pˆ e3 ), and a ˆ3 . component t is transmitted with energy E $ = E − V and momentum p$ e According to the energy-momentum invariant, conservation of energy across the interface dictates that E 2 = p2 + m2 and E $ 2 = p$2 + m2 . Being first order, the boundary conditions on the Dirac equation require only continuity of ψ (cf. the Schr¨odinger equation). Therefore, matching ψ at the step, we obtain the relations       1 1 1  0       p  + r  0p  = t  p0"  ,   −   "  E+m E+m E +m 0 0 0

from which we find 1 +r = t, and these equations lead to t=

2 , 1+ζ

p (1 − r) p0 +m

1+r 1 = , 1−r ζ

=

p" t. p"0 +m

r=

Setting ζ =

p" (E+m) p (E " +m) ,

1−ζ . 1+ζ

To interpret these solutions, let us consider the current associated with the reflected and transmitted components. Making use of the equation for the current density, j = ψ † αψ, and using the Dirac/Pauli representation wherein "! " ! " ! σ3 σ3 I2 = , α3 = γ0 γ3 = −I2 −σ3 σ3 Advanced Quantum Physics

177

15.3. FREE PARTICLE SOLUTION OF THE DIRAC EQUATION ˆ3 -direction is given by the current along e ! " σ3 j3 = ψ † ψ, σ3

j1 = j2 = 0 .

Therefore, up to an overall constant of normalisation, the current densities are given by (i)

j3 =

p0

2p , +m

(t)

j3 =

2(p$ + p$∗ ) 2 |t| , p$0 + m

(r)

j3 = −

2p |r|2 . p0 + m

From these relations we obtain (t)

(p$ + p$∗ ) p0 + m 1 4 = (ζ + ζ ∗ ) 2p p$0 + m |1 + ζ|2 2 4 4 4 1 − ζ 42 2 4 = −|r| = − 44 1+ζ4

j3

= |t|2

(i)

j3

(r)

j3

(i)

j3

from which current conservation can be confirmed: (r)

1+

j3

(i)

j3

(t)

=

|1 + ζ|2 − |1 − ζ|2 2(ζ + ζ ∗ ) j = = 3(i) . 2 2 |1 + ζ| |1 + ζ| j3

Interpreting these results, it is convenient to separate our consideration into three distinct regimes of energy: ! E $ ≡ (E − V ) > m: In this case, from the Klein-Gordon condition (the energy-momentum invariant) p$2 ≡ E $2 − m2 > 0, and (taking p$ > 0 – i.e. beam propagates to the right) ζ > 0 and real. From this result (r) (i) we find |j3 | < |j3 | – as expected, within this interval of energy, a component of the beam is transmitted and the remainder is reflected (cf. non-relativistic quantum mechanics). ! −m < E $ < m: In this case p$2 ≡ E $2 − m2 < 0 and p$ is purely imaginary. From this result it follows that ζ is also pure imaginary and (r) (i) |j3 | = |j3 |. In this regime the under barrier solutions are evanescent and quickly decay to the right of the barrier. All of the beam is reflected (cf. non-relativistic quantum mechanics). ! E $ < −m: Finally, in this case p$2 ≡ E $2 −m2 > 0 and, depending on the (r) (i) sign of p$ , j3 can be greater or less than j3 . But the solution has the " " form e−i(p x−E t) . Since we presume the beam to be propagating to the right, we require E $ < 0 and p$ > 0. From this result it follows that ζ < 0 (r) (i) and we are drawn to the surprising conclusion that |j3 | > |j3 | – more current is reflected that is incident! Since we have already confirmed (t) current conservation, we can deduce that j3 < 0. It is as if a beam of particles were incident from the right. The resolution of this last seeming unphysical result, known as the Klein paradox,9 in fact gives a natural interpretation of the negative energy solutions that plague both the Dirac and Klein-Gordon equations: Dirac particles are fermionic in nature. If we regard the vacuum as comprised of a filled Fermi sea of negative energy states or antiparticles (of negative charge), the Klein Paradox can be resolved as the stimulated emission of particle/antiparticle 9

Indeed one would reach the same conclusion were one to examine the Klein-Gordon equation.

Advanced Quantum Physics

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15.3. FREE PARTICLE SOLUTION OF THE DIRAC EQUATION Figure 15.2: The photograph shows a

small part of a complicated high energy neutrino event produced in the Fermilab bubble chamber filled with a neon hydrogen mixture. A positron (red) emerging from an electron-positron pair, produced by a gamma ray, curves round through about 180 degrees. Then it seems to change charge: it begins to curve in the opposite direction (blue). What has happened is that the positron has run head-on into a (moreor-less from the point of view of particle physics) stationary electron – transferring all its momentum. This tells us that the mass of the positron equals the mass of the electron.

pairs, the particles moving off towards x3 = −∞ and the antiparticles towards x3 = ∞. What about energy conservation? One might worry that the energy for these pairs is coming from nowhere. However, the electrostatic energy recovered by the antiparticle when its created is sufficient to outweigh the rest mass energy of the particle and antiparticle pair (remember that a repulsive potential for particles is attractive for antiparticles). Taking into account the fact that the minimum energy to create a particle/antiparticle pair is twice the rest mass energy 2 × m, the regime where stimulated emission is seen to occur can be understood. Negative energy states: With this conclusion, it is appropriate to revisit the definition of the free particle plane wave 3 state. In particular, for energies E < 0, it is more sensible to set p0 = + (p2 + m2 ), and redefine the plane wave solution as ψ(x) = v(p)eip·x , where the spinor satisfies the condition (+ p + m)v(p) = 0. Accordingly we find, 5 σ · p (r) 6 χ (r) v (p) = N (p) p0 + m . χ(r) So, to conclude, two relativistic wave equations have been proposed. The first of these, the Klein-Gordon equation was dismissed on the grounds that it exhibited negative probability densities and negative energy states. By contrast, the states of the Dirac equation were found to exhibit a positive definite probability density, and the negative energy states were argued to have a natural interpretation in terms of antiparticles: the vacuum state does not correspond to all states unoccupied but to a state in which all the negative energy states are occupied – the negative energy states are filled up by a Fermi sea of negative energy Fermi particles. For electron degrees of freedom, if a positive energy state is occupied we observe it as a (positive energy) electron of charge q = −e. If a negative energy state is unoccupied we observe it as a (positive energy) antiparticle of charge q = +e, a positron, the antiparticle of the electron. If a very energetic electron interacts with the sea causing a transition from a negative energy state to positive one (by communicating an energy of at least 2m) this is observed as the production of a pair of particles, an electron and a positron from the vacuum (pair production) (see Fig. 15.2). However, the interpretation attached to the negative energy states provides grounds to reconsider the status of the Klein-Gordon equation. Evidently, the Advanced Quantum Physics

179

15.4. QUANTIZATION OF RELATIVISTIC FIELDS Dirac equation is not a relativistic wave equation for a single particle. If it were, pair production would not appear. Instead, the interpretation above forces us to consider the wavefunction of the Dirac equation as a quantum field able to host any number of particles – cf. the continuum theory of the quantum harmonic chain. In the next section, we will find that the consideration of the wavefunction as a field revives the Klein-Gordon equation as a theory of scalar (interger spin) particles.

15.4

Quantization of relativistic fields

15.4.1

Info: Scalar field: Klein-Gordon equation revisited

Previously, the Klein-Gordon equation was abandoned as a candidate for a relativistic theory on the basis that (i) it admitted negative energy solutions, and (ii) that the probability density associated with the wavefunction was not positive definite. Yet, having associated the negative energy solutions of the Dirac equation with antiparticles, and identified ψ as a quantum field, it is appropriate that we revisit the Klein-Gordon equation and attempt to revive it as a theory of relativistic particles of spin zero. If φ were a classical field, the Klein-Gordon equation would represent the equation of motion associated with the Lagrangian density (exercise) L=

1 1 ∂µ φ ∂ µ φ − m2 φ2 , 2 2

(cf. our discussion of the low energy modes of the classical harmonic chain and the Maxwell field of the waveguide in chapter 11). Defining the canonical momentum ˙ π(x) = ∂φ˙ L(x) = φ(x) ≡ ∂0 φ(x), the corresponding Hamiltonian density takes the form , 1+ 2 π + (∇φ)2 + m2 φ2 . H = π φ˙ − L = 2

Evidently, the Hamiltonian density is explicitly positive definite. Thus, the scalar field is not plagued by the negative energy problem which beset the single-particle theory. Similarly, the quantization of the classical field will lead to a theory in which the states have positive energy. Following on from our discussion of the harmonic chain in chapter 11, we are already equipped to quantise the classical field theory. However, there we worked explicitly in the Schr¨odinger representation, in which the dynamics was contained within the time-dependent wavefunction ψ(t), and the operators were time-independent. Alternatively, one may implement quantum mechanics in a representation where the time dependence is transferred to the operators instead of the wavefunction — the Heisenberg representation. In this representation, the Schr¨odinger state vector ψS (t) is related to the Heisenberg state vector ψH by the relation, ˆ

ψS (t) = e−iHt ψH ,

ψH = ψS (0) .

ˆ S are related to the Heisenberg operators O ˆ H (t) by Similarly, Schr¨odinger operators O ˆ ˆ ˆ ˆ H (t) = eiHt O OS e−iHt .

ˆ S |ψS 4 and 3ψ ! |O ˆ H |ψH 4 are One can easily check that the matrix elements 3ψS! |O H equivalent in the two representations, and which to use in non-relativistic quantum mechanics is largely a matter of taste and convenience. However, in relativistic quantum field theory, the Heisenberg representation is often preferable to the Schr¨odinger representation. The main reason for this is that by using the former, the Lorentz covariance of the field operators is made manifest.

Advanced Quantum Physics

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15.4. QUANTIZATION OF RELATIVISTIC FIELDS In the Heisenberg representation, the quantisation of the fields is still enforced by ˆ but in this case, we promoting the classical fields to operators, π )→ π ˆ and φ )→ φ, impose the equal time commutation relations, -

. ˆ t), π φ(x, ˆ (x! , t) = iδ 3 (x − x! ),

-

. ˆ t), φ(x ˆ ! , t) = [ˆ φ(x, π (x, t), π ˆ (x! , t)] = 0 ,

ˆ In doing so, the Hamiltonian density takes the form with π ˆ = ∂0 φ. . ˆ 2 + m2 φˆ2 . ˆ=1 π ˆ 2 + (∇φ) H 2

To see the connection between the quantized field and particles we need to Fourier transform the field operators to obtain the normal modes of the Hamiltonian, 7 d4 k ˆ ˆ φ(x) = φ(k)e−ik·x . (2π)4

ˆ However the form of the Fourier elements φ(k) is constrained by the following conˆ ditions. Firstly to maintain Hermiticity of the field operator φ(x) we must choose † ˆ ˆ Fourier coefficients such that φ (k) = φ(−k). Secondly, to ensure that the field opˆ ˆ erator φ(x) obeys the Klein-Gordon equation,10 we require φ(k) ∼ 2πδ(k 2 − m2 ). Taking these conditions together, we require ) * ˆ φ(k) = 2πδ(k 2 − m2 ) θ(k 0 )a(k) + θ(−k 0 )a† (−k) ,

√ where k 0 ≡ ωk ≡ + k2 + m2 , and a(k) represent the operator valued Fourier coefficients. Rearranging the momentum integration, we obtain the Lorentz covariant expansion 7 , + d4 k ˆ φ(x) = 2πδ(k 2 − m2 )θ(k 0 ) a(k)e−ik·x + a† (k)eik·x . 4 (2π)

Integrating over k 0 , and making use of the identity 7 7 d4 k d4 k 2 2 0 2πδ(k − m )θ(k ) = δ(k02 − ωk2 )θ(k 0 ) 4 (2π) (2π)3 7 7 d4 k d4 k 1 0 = δ [(k − ω )(k + ω )] θ(k ) = [δ(k0 − ωk ) + δ(k0 + ωk )] θ(k 0 ) 0 k 0 k 3 (2π) (2π)3 2k0 7 7 7 d3 k dk0 d3 k 0 = δ(k − ω )θ(k ) = , 0 k 3 (2π) 2k0 (2π)3 2ωk one obtains ˆ φ(x) =

7

* d3 k ) a(k)e−ik·x + a† (k)eik·x . (2π)3 2ωk

More compactly, making use of the orthonormality of the basis 7 ↔ 1 e−ik·x , fk∗ (x)i ∂ 0 fk! (x)d3 x = δ 3 (k − k! ), fk = 3 3 (2π) 2ωk ↔

where A ∂0 B ≡ A∂t B − (∂t A)B, we obtain 7 + , d3 k ˆ 3 a(k)fk (x) + a† (k)fk∗ (x) . φ(x) = (2π)3 2ωk 10

Note that the field operators obey the equation of motion, π(x, ˙ t) = −

∂H = ∇ 2 φ − m2 φ . ∂φ(x, t)

˙ one finds (∂ 2 + m2 )φ = 0. Together with the relation π = φ,

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15.4. QUANTIZATION OF RELATIVISTIC FIELDS Similarly, ˆ π ˆ (x) ≡ ∂0 φ(x) =

7

3

d3 k (2π)3 2ω

k

, + iωk −a(k)fk (x) + a† (k)fk∗ (x) .

Making use of the orthogonality relations, the latter can be inverted to give 7 7 3 3 ↔ ↔ ˆ ˆ a(k) = (2π)3 2ωk d3 xfk∗ (x)i ∂0 φ(x), ∂0 fk (x) , a† (k) = (2π)3 2ωk d3 xφ(x)i or, equivalently, 7 / 0 ˆ − iˆ π (x) e−ik·x , a(k) = d3 x ωk φ(x)

a (k) = †

7

With these definitions, it is left as an exercise to show +

, a(k), a† (k! ) = (2π)3 2ωk δ 3 (k − k! ),

/ 0 ˆ d3 x ωk φ(x) + iˆ π (x) eik·x .

+ , [a(k), a(k! )] = a† (k), a† (k! ) = 0 .

The field operators a† and a can therefore be identified as operators that create and annihilate bosonic particles. Although it would be tempting to adopt a different normalisation wherein [a, a† ] = 1 (as is done in many texts), we chose to adopt the convention above where the covariance of the normalisation is manifest. Using this representation, the Hamiltonian is brought to the diagonal form 7 , d3 k ωk + † ˆ = a (k)a(k) + a(k)a† (k) , H 3 (2π) 2ωk 2 a result which can be confirmed by direct substitution. Defining the vacuum state |Ω4 as the state which is annhiliated by a(k), a single particle state is obtained by operating the creation operator on the vacuum, |k4 = a† (k)|Ω4 . Then 3k! |k4 = 3Ω|a(k! )a† (k)|Ω4 = 3Ω|[a(k! ), a† (k)]|Ω4 = (2π)3 2ωk δ 3 (k! − k). Manyparticle states are defined by |k1 · · · kn 4 = a† (k1 ) · · · a† (kn )|Ω4 where the bosonic statistics of the particles is assured by the commutation relations. Associated with these field operators, one can define the total particle number operator 7 d3 k ˆ= 3 N a† (k)a(k) . (2π)3 2ωk Similarly, the total energy-momentum operator for the system is given by 7 d3 k µ ˆ 3 P = k µ a† (k)a(k) . (2π)3 2ωk

The time component Pˆ 0 of this result can be compared with the Hamiltonian above. In fact, commuting the field operators, the latter is seen to differ from Pˆ 0 by an infinite 8 constant, d3 kωk /2. Yet, had we simply normal ordered11 the operators from the outset, this problem would not have arisen. We therefore discard this infinite constant.

15.4.2

Info: Charged Scalar Field

A generalization of the analysis above to the complex scalar field leads to the Lagrangian, L= 11

1 1 ∂µ φ∂ µ φ¯ − m2 |φ|2 . 2 2

Recall that normal ordering entails the construction of an operator with all the annihilation operators moved to the right and creation operators moved to the left.

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15.4. QUANTIZATION OF RELATIVISTIC FIELDS The latter can √be interpreted as the superposition of two independent scalar fields φ = (φ1 +iφ2 )/ 2, where, for each (real) component φ†r (x) = φr (x). (In fact, we could as easily consider a field with n components.) In this case, the canonical quantisation of the classical fields is achieved by defining (exercise) 7 + , d3 k ˆ 3 φ(x) = a(k)fk (x) + b† (k)fk∗ (x) . 3 (2π) 2ωk

(similarly φ† (x)) where both a and b obey bosonic commutation relations, + , + , a(k), a† (k! ) = b(k), b† (k! ) = (2π)3 2ωk δ 3 (k − k! ), + , [a(k), a(k! )] = [b(k), b(k! )] = a(k), b† (k! ) = [a(k), b(k! )] = 0 . With this definition, the total number operator is given by 7 + † , d3 k ˆa + N ˆb , ˆ = 3 a (k)a(k) + b† (k)b(k) ≡ N N 3 (2π) 2ωk

while the energy-momentum operator is defined by 7 + , d3 k 3 k µ a† (k)a(k) + b† (k)b(k) . Pˆ µ = 3 (2π) 2ωk

Thus the complex scalar field has the interpretation of creating different sorts of particles, corresponding to operators a† and b† . To understand the physical interpretation of this difference, let us consider the corresponding charge density operator, 8 ↔ ˆj0 = φˆ† (x)i ∂ 0 φ(x). Once normal ordered, the total charge Q = d3 xj0 (x) is given by 7 + † , d3 k ˆa − N ˆb . ˆ= 3 a (k)a(k) − b† (k)b(k) = N Q 3 (2π) 2ωk From this result we can interpret the particles as carrying an electric charge, equal in magnitude, and opposite in sign. The complex scalar field is a theory of charged particles. The negative density that plagued the Klein-Gordon field is simply a manifestation of particles with negative charge.

15.4.3

Info: Dirac Field

The quantisation of the Klein-Gordon field leads to a theory of relativistic spin zero particles which obey boson statistics. From the quantisation of the Dirac field, we expect a theory of Fermionic spin 1/2 particles. Following on from our consideration of the Klein-Gordon theory, we introduce the Lagrangian density associated with the Dirac equation (exercise) L = ψ¯ (iγ µ ∂µ − m) ψ , ↔

¯ 1 iγ µ ∂ µ −m)ψ). With this definition, the corresponding (or, equivalently, L = ψ( 2 ¯ 0 = iψ † . From the Lagrangian density, canonical momentum is given by ∂ψ˙ L = iψγ we thus obtain the Hamiltonian density, H = ψ¯ (−iγ · ∇ + m) ψ ,

¯ 0 ∂0 ψ = ψ † i∂t ψ. which, making use of the Dirac equation, is equivalent to H = ψiγ For the Dirac theory, we postulate the equal time anticommutation relations : ψα (x, t), iψβ† (x! , t) = iδ 3 (x − x! )δαβ ,

9

0 (or, equivalently {ψα (x, t), iψ¯β (x! , t)} = γαβ δ 3 (x − x! )), together with

; < {ψα (x, t), ψβ (x! , t)} = ψ¯α (x, t), ψ¯β (x! , t) = 0 .

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15.5. THE LOW ENERGY LIMIT OF THE DIRAC EQUATION Using the general solution of the Dirac equation for a free particle as a basis set, together with the intuition drawn from the study of the complex scalar field, we may with no more ado, introduce the field operators which diagonalise the Hamiltonian density 2 7 . = d3 k ψ(x) = ar (k)u(r) (k)e−ik·x + b†r (k)v (r) (k)eik·x 3 (2π) 2ωk r=1 7 2 . = d3 k - † (r) ik·x (r) −ik·x ¯ (k)¯ u (k)e + b (k)¯ v (k)e a , ψ(x) = r r (2π)3 2ωk r=1 where the annihilation and creation operators also obey the anticommutation relations, < ; < ; ar (k), a†s (k! ) = br (k), b†s (k! ) = (2π)3 2ωk δrs δ 3 (k − k! ) ; < ; < {ar (k), as (k! )} = a†r (k), a†s (k! ) = {br (k), bs (k! )} = b†r (k), b†s (k! ) = 0 .

The latter condition implies the Pauli exclusion principle a† (k)2 = 0. With this definition, a(k)u(k)e−ik·x annilihates a postive energy electron, and b† (k)v(k)eik·x creates a positive energy positron. From these results, making use of the expression for the Hamiltonian density operator, one obtains 2 7 = , + d3 k ˆ = ωk a†r (k)ar (k) − br (k)b†r (k) . H 3 (2π) 2ωk r=1

Were the commutation relations chosen as bosonic, one would conclude the existence of negative energy solutions. However, making use of the anticommutation relations, and dropping the infinite constant (or, rather, normal ordering) one obtains a positive definite result. Expressed as one element of the total energy-momentum operator, one finds Pˆ µ =

2 7 = r=1

+ , d3 k k µ a†r (k)ar (k) + b†r (k)br (k) . 3 (2π) 2ωk

Finally, the total charge is given by 7 7 ˆa − N ˆb . ˆ = ˆj 0 d3 x = d3 xψ † ψ = N Q

8 ˆ represents the total number operator. Na = d3 k a† (k)a(k) is the number where N 8 of the particles and Nb = d3 k b† (k)b(k) is the number of antiparticles with opposite charge.

15.5

The low energy limit of the Dirac equation

To conclude our abridged exploration of the foundations of relativistic quantum mechanics, we turn to the interaction of a relativistic spin 1/2 particle with an electromagnetic field. Suppose that ψ represents a particle of charge q (q = −e for the electron). From non-relativistic quantum mechanics, we expect to obtain the equation describing its interaction with an EM field given by the potential Aµ by the minimal substitution pµ )−→ pµ − qAµ , where A0 ≡ ϕ. Applied to the Dirac equation, we obtain for the interaction of a particle with a given (non-quantized) EM field, [γµ (pµ − qAµ ) − m]ψ = 0, or compactly (+ p − q+A − m)ψ = 0 . Advanced Quantum Physics

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15.5. THE LOW ENERGY LIMIT OF THE DIRAC EQUATION Previously, in chapter 9, we explored the relativistic (fine-structure) corrections to the hydrogen atom. At the time, we alluded to these as the leading relativistic contributions to the Dirac theory. In the following section, we will explore how these corrections are derived. In the Dirac-Pauli representation, ! " ! " 0 σ I2 0 α= , β= . σ 0 0 −I2 we have seen that the plane-wave solution to the Dirac equation for particles can be written in the form ! " χ ψp (x) = N ei(px−Et)/! , cσ ·ˆ p χ mc2 +E where we have restored the parameters ! and c. From this expression, we can see that, at low energies, where |E − mc2 | - mc2 , the second component of the bispinor is smaller than the first by a factor of order v/c. To obtain the non-relavistic limit, we can exploit this asymmetry to develop a perturbative expansion of the coefficients in v/c. Consider then the Dirac equation for a particle moving in a potential (φ, A). Expressed in matrix form, the Dirac equation H = cα · (−i!∇ − e 2 c A) + mc β + eφ is expressed as " ! cσ · (−i!∇ − ec A) mc2 + eφ . H= −mc2 + qφ cσ · (ˆ p − qc A) Defining the bispinor ψ T (x) = (ψa (x), ψb (x)), the Dirac equation translates to the coupled equations, q (mc2 + eφ)ψa + cσ · (ˆ p − A)ψb = Eψa c q cσ · (ˆ p − A)ψa − (mc2 − qφ)ψb = Eψb . c Then, if we define W = E − mc2 , a rearrangement of the second equation obtains ψb =

2mc2

q 1 cσ · (ˆ p − A)ψa . + W − qφ c

1 p − qc A)ψa . SubstiThen, at zeroth order in v/c, we have ψb $ 2mc 2 cσ ·(ˆ tuted into the first equation, we thus obtain the Pauli equation Hnon−rel ψa = W ψa , where

Hnonr el =

q .2 1 σ · (ˆ p − A) + qφ . 2m c

Making use of the Pauli matrix identity σi σj = δij + i$ijk σk , we thus obtain the familiar non-relativistic Schr¨odinger Hamiltonian, Hnon

rel

=

1 q! q (ˆ p − A)2 − σ · (∇ × A) + qφ . 2m c 2mc

As a result, we can identify the spin magnetic moment µS =

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q! ˆ q! σ= S, 2mc mc

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15.5. THE LOW ENERGY LIMIT OF THE DIRAC EQUATION with the gyromagnetic ratio, g = 2. This compares to the measured value of g = 2 × (1.0011596567 ± 0.0000000035), the descrepency form 2 being attributed to small radiative corrections. Let us now consider the expansion to first order in v/c. Here, for simplicity, let us suppose that A = 0. In this case, taking into account the next order term, we obtain ! " V −W 1 ˆ ψa ψb $ 1 + cσ · p 2mc2 2mc2 where V = qφ. Then substituted into the second equation, we obtain 1 2 1 1 2 ˆ) + ˆ )(V − W )(σ · p ˆ ) + V ψa = W ψ a . (σ · p (σ · p 2m 4m2 c2 At this stage, we must be cautious in interpreting ψa as a complete nonrelavistic wavefunction with leading relativistic corrections. To find the true wavefunction, we have to consider the normalization. If we suppose that the original wavefunction is normalized, we can conclude that, 7 7 / 0 d3 xψ † (x, t)ψ(x, t) = d3 x ψa† (x, t)ψa (x, t) + ψb† (x, t)ψb (x, t) 7 7 1 3 † $ d xψa (x, t)ψa (x, t) + d3 xψa† (x, t)ˆ p2 ψa (x, t) . (2mc)2 Therefore, at this order, the normalized wavefunction is set by, ψs = (1 + 1 ˆ 2 )ψa or, inverted, p 8m2 c2 " ! 1 2 ˆ ψs . p ψa = 1 − 8m2 c2 Substituting, then rearranging the equation for ψs , and retaining terms of ˆ non−rel ψs = W ψs , where order (v/c)2 , one ontains (exercise) H ˆ4 ˆ2 p 1 1 ˆ non−rel = p ˆ )V (σ · p ˆ) + V − ˆ2 + p ˆ 2V ) . − + (σ · p (V p H 2m 8m3 c2 4m2 c2 8m2 c2 Then, making use of the identities, ˆ 2 ] = !2 (∇2 V ) + 2i!(∇V ) · p ˆ [V, p

ˆ )V = V (σ · p ˆ ) + σ · [ˆ (σ · p p, V ]

ˆ + !σ · (∇V ) × p ˆ, ˆ )V (σ · p ˆ) = V p ˆ 2 − i!(∇V ) · p (σ · p

we obtain the final expression (exercise), ˆ4 ˆ2 p ! !2 ˆ non−rel = p ˆ − + σ · (∇V ) × p + (∇2 V ) . H 2 c2 2m 8m3 c2 >4m2 c2 ?@ 8m A > ?@ A spin−orbit coupling

Darwin term

The second term on the right hand side represents the relativistic correction to the kinetic energy, the third term denotes the spin-orbit interaction and the final term is the Darwin term. For atoms, with a central potential, the spin-orbit term can be recast as ˆ S.O. = H

!2 1 !2 1 ˆ. ˆ= (∂r V )σ · L σ · (∂r V )r × p 2 2 4m c r 4m2 c2 r

To address the effects of these relativistic contributions, we refer back to chapter 9. Advanced Quantum Physics

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