Quantum Mechanics 3540431098, 9783540431091

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A.A. Sokolov T. M. Loskutov I. M. Ternov

Quantum meEhaniE5 •

A. A. SOKOLOV

•

Y. M. LOSKUTOV

•

I. M. TERNOV Moscow State University

Translated by Scripta Technica, Inc. Problems prepared by Graham Frye Physics Department The City College of The City University of New York

HOLT, RINEHART NEW YORK

-

CHICAGO -

AND

WINSTON, INC.

SAN FRANCISCO

-

TORONTO

-

LONDON

E n g li s h t r a n s l a t i o n c o p y rig h t (c) 1966 by H o lt, R in e h a r t and W in ston, Inc. All R ig h t s R e s e r v e d Library of C o n g r e s s C a t a l o g C ard Number: 6 6 —17275

28036-0116 P r i n te d in th e U n ite d S t a t e s of A m erica

O rig in a lly p u b li s h e d in R u s s i a as K v a n to v a y a M ekhanika

by A. A. S okolov, Y. M. L o s k u to v , and I. M. T e rn o v T e x tb o o k P u b li s h i n g H o u se of th e Ministry of E d u c a tio n of R S F S R , M oscow, 1962

Preface to the English Edition This book is specifically a “ textbook” for learning the physical content of quantum mechanics. There is a pleasing progression from the gross quantum effects (blackbody radiation, photoelectric effect, specific heats) to typical quantum mechanical behavior (spreading of wave packets, barrier penetration, stationary states, spin and angular momentum multiplets) to the more refined quantum phenomenology (fine structure, effect of the nucleus on atomic structure, quantum fluctuations of the electromagnetic field, coupling of angular momentums in multielectron atoms and molecules). At each stage of remoteness from everyday experi­ ence some of the conceptually and computationally abstruse parts of the theory are dealt with in explicit detail that emphasizes the real observability of the phenomenon. The mathematical form of the theory is thereby dictated by the necessity of having a notational apparatus that is sufficiently rich and flexible to embrace the scope of actual observable effects. While it is an exposition of the principles of quantum mechanics, the selection of material is unusual because the book includes much that is ordinarily regarded as atomic structure and omits any long excursion into the formal mathematical structure of the theory. The formalism is a part of the practice of quantum mechanics, however, so to be complete we ought to recognize these tools and provide some guide to their practical utilization. To accomplish this we added two appendices in which are collected many definitions and formal statements and a few examples to show how the notational apparatus is used. The material is pre­ sented in a way that is very abstract and condensed. It is not intended as an expository treatment of the subject but rather as an outline of useful reference material on the formal aspects of the theory. The justification for this mode of presentation is that textbook expositions are widely available and that once the material is grasped a concise summary of definitions and results is often all that is needed for reference. The active participation of the student in solving problems is an indispensable part of the discipline of quantum mechanics. We have therefore included a relatively large number of problems to supplement the text. The problems are coordinated with the organi­ zation of material in the text so that they serve to illustrate in context the applications and principal ramifications of the theory. GRAHAM F R Y E N E W YORK,

NOVEMBER

1965

Contents PREFACE TO THE ENGLISH EDITION PREFACE

xiii

INTRODUCTION PART 1.

Mi

NONRELATIVISTIC QUANTUM MECHANICS

1. The Quantum Theory of Light

xv 1 3

The Maxwell-Lorentz equations. Radiation of electromagnetic waves. Lorentz force. R elativistic Legrangian, Hamiltonian and equations of motion for a particle in an external field. Scalar and vector potentials. The Lorentz condition. Blackbody radi­ ation. Spectral density. Average energy of a harmonic oscillator. Equipartition of energy. The Rayleigh-Jeans formula. Wien’s thermodynamic law. The ultraviolet catastrophe. Planck’s equation. Planck’s co n sta n ts. Stephan-Boltzmann law. Wien’s displacement law. Einstein’s photon theory. Energy and mo­ mentum relations of a photon. Photoelectric effect. Transfer of energy and momentum. Compton effect. 2. The Bohr Quantum Theory

20

Emission of light by atoms. Spectral terms. The classical model of the atom. Continuous radiation of light. The Thompson model of the atom. The Rutherford experiment. The planetary model of the atom. The Coulomb potential. Differential scattering cross section. The Rutherford formula. Applicability of Coulomb’s law. The effect of nuclear size. The Bohr theory. The clas­ sical solution in terms of adiabatic invariants. Periodic and quasi-periodic motion. Frequency and rate of cla ssica l radia­ tion. The postulate of stationary states. The frequency postu­ late. Franck and Hertz experiment. Ionization of atoms. Radius of the first Bohr orbit. Balmer formula. The correspondence principle. E lliptic orbits. Selection rules. Somerfeld formula for relativistic Coulomb energy levels. Existence of stationary states. R elativistic Coulomb scattering. (Quantization, selec­ tion rules and lifetimes in terms of the correspondence principle. Universality of 7t. R elativistic harmonic oscillator.) 3. Wave Properties of Particles De Broglie waves. Single valuedness. Davisson-Germer and Tartakovskiy-Thompson diffraction experiment. Wave packets. Group and phase velocities. The uncertainty principle. Local­ ization of particles. Spreading time of wave packets. Born’s

41

CONTENTS

VI

statistical interpretation. The complimentarity principle. Heisen­ berg’s ‘‘ultramicroscope’’. Opinion of the Copenhagen school. Gaussian wave packet. Fourth uncertainty relation. (Universal­ ity of ~h. Uncertainty principle for angular momentum. Zero point energy. Use of the uncertainty principle.) 4. The Time-Independent Schrodinger Wave Equation

57

Monochromatic waves. Local wavelength of a de Broglie wave. Probability density. Normalization. Continuity. Single valued­ ness. Boundedness. Boundary conditions. Eigenvalues and eigenfunctions. Energy spectrum. Orthonormality. A particle in a potential well, discrete spectrum. Motion of free particles. Normalization of wave functions in the case of a continuous spectrum. Born’s periodic boundary condition. Separation of variables. Plane waves in three dimensions. The Dirac 5-function. 5-function normalization in the case of a continuous spectrum. Cdthplete set of orthonormal functions. Completeness relation. (The 5-function potential. Sturm-Liouville theorem. Comparison potential for the existence and number of bound states.) 5. Ihc Iime-Dependenl Schrodinger Wave Equation

73

Time-dependent wave functions; expression in terms of energy eigenfunctions. The energy, momentum and kinetic energy oper­ ators. The Hamiltonian operator. Operator form of the wave equation. Charge density and current density. Equation of con­ tinuity. Conservation of charge. Probability amplitudes Cn. Quantum ensembles. Pure and mixed states. Interference of do Broglie waves. Connection with the cla ssica l HamiltonJacobi equation. Action function. Q uasi-classical approximation. The WKB method. Derivation of the Bohr quantization rule. Zero point energy. Symmetric and antisymmetric wave functions. Quasi-stalionary levels. (Motion of a wave packet. Dispersionfree approximation. Green’s function solution of initial value problem. Flux of plane and spherical waves.) (>.

Basic

I ’ r i n c i p l c s o l i h c Quant um T h e o r y o l C o n d u c t i v i t y

Transmission of a particle through a potential barrier. Trans­ mission and reflection coefficients. The tunnel effect as a manifestation of wave properties. Distribution of real momenta in the cla ssica lly forbidden region. Motion of electrons in a metal. Specific heat of free' electron gas. Fermi-Dirac quantum statistics, distribution function. Pauli exclusion principle. Density ol states. Electron density. Fermi energy. Average energy in thermal (equilibrium. Degeneracy temperature. Removal o( electrons from a metal. Cold emission. Contact potentials. Viol nmol electrons in a periodic potential. The one-dimensional Kronig-Peiiney model. Basic principles of the electron theory of conductivity of crystals. Allowed and forbidden energy bands.

97

CONTENTS

vii

(Gamow penetration factor. Pressure of an electron gas at 0°K. Block wave functions.) 7. Statistical Interpretation of Quantum Mechanics

119

Elements of the theory of linear operators. Principle of super­ position. Differential and integral operators. The Laplacian and inverse Laplacian operators. The coordinate variable and potential function as linear operators. Elements of represen­ tation theory. Cononical commutation relations. Momentum rep­ resentation. Average values of operators. Reality and self-conju­ gate or Hermitian operators. Integration by parts, “transferring a derivative.” Average in terms of eigenvalues and probabilities. (Translation operator. Time evolution operator. Inner product. Operator equations. Matrix elements of an operator. Hermitian conjugate or adjoint operator. Unitary operators.) 8. Average Values of Operators. Change of Dynamic Variables with Time

127

Derivation of the uncertainty principle. Schwartz inequality. Condition of simultaneous measurement of two dynamic quanti­ ties. Poisson brackets in cla ssica l and quantum theory. Com­ mutator bracket. Constant of motion. Ehrenfest’s theorem. Transition from quantum to cla ssica l equations of motion. Conditions for validity of the classica l approximation. Hydrogen atom. Motion in constant and homogeneous electric and magnetic fields. (Time dependence of off-diagonal matrix elements. Operator solution for time evolution operator. Schrbdinger picture and Heisenberg picture. Virial theorem. Sum rules.) 9. Elemenlary Theory of Radiation

141

Spontaneous and induced transitions. Einstein coefficients. Emission and absorption. Virtual photons. Vacuum fluctuations. Quantum electrodynamics. Matrix elements of the position op­ erator. Allowed and forbidden transitions. Selection rules. Electric dipole and quadrupole radiation. Charge to mass ratio. Gravitational radiation. Nuclear transitions. 10. The Linear Harmonic Oscillalor Description in the classical and Bohr theories. Energy eigen­ functions and eigenvalues. Asymptotic behavior. Hermite poly­ nomials. Normalization. Orthogonality. C lassical limit of spatial probability distribution. Zero-point energy and the uncertainty principle. Selection rules. Intensity of radiation. Matrix elements of the position operator. Energy eigenvalues by the WKB method. Theory in the momentum representation. Motion of a particle in a uniform magnetic field. Diamagnetism of an electron gas. Quadrupole radiation by a harmonic oscillator. Matrix elements of x 2. Motion of a wave packet in a harmonic potential. Clas-

149

v iii

CONTENTS

sical limit. (Gaussian wave packet. Generating function and properties of Tchebycheff-Hermite polynomials. Fourier-type transform. Heisenberg operators. The harmonic oscillator Green’s function. Operator solution. Creation and annihilation operators.) 11. General Theory of Motion of a Particle in a Centrally Symmetric Field

168

SchrOdinger’s equation in spherical coordinates. Separation of variables. Radial and angular Schrbdinger equations. Eigen­ functions for angular dependence. Eigenvalues. Associated Legendre polynomials. Normalization. Orthogonality. Spherical harmonics. Physical meaning of the quantum numbers I and m. Angular momentum operators. Commutation relations. Eigen­ functions of operators L z and L2. Raising and lowering operators. Connection with the Bohr theory. (Generating function for Le. gendre polynomials. Vector operators. Matrix representations of the angular momentum commutation relations.) 12. The Rotator

185

Eigenfunctions of the rotator. Effect of non-commutativity of angular momentum components. Energy lev els. Degeneracy. Angular distribution and orientation. Selection rules. Matrix elements of r. Spectra of diatomic molecules. Reduced mass. Vibrational-rotational spectra. Motion of a free particle in spherical coordinates. Quadrupole selection rules. Expansion of the plane wave in terms of spherical waves. (Isotropic har­ monic oscillator. D iscrete and continuous spectra in an isotropic potential. Phase shift. Spherical B essel functions.) 13. The Theory of the Hydrogen-like Atom (Kepler’s Problem)

203

Energy eigenfunctions and eigenvalues. Radial wave equation. Centrifugal barrier. Effective potential. Behavior at r = 0 and Associated Lagucrre polynomials. Normalization and expec­ tation values of r"y. Energy levels. Degeneracy with respect to angular momentum. Scm iclassical interpretation. E lliptic orbits. Radial probability density. Selection rules. Radial matrix ele­ ments. Emission spectra of hydrogen-like atoms. Continuous spectrum of a particle in a Coulomb potential. WKB method. Asymptotic form of the radial wave function. Phase shift. Scat­ tering problems. Ionization energy. Effects of motion of the nucleus. Experimental values of the Rydberg constant for //, I), T i//'"*, 2 //e 4 1. Average electrostatic potential of a hydrogen atom. Discrete s|>ectrum by the WKB-mcthod. Magnetic field at nucleus due to a 2p electron. i 1. I i mc - l nd r p r i i d c n l 1'rrlurhnlioii ' t h e o r y

Basic principles and fundamental equations of perturbation theory. Non-degenerate case. Degenerate case. Secular equation.

231

CONTENTS

ix

Removal of degeneracy. The Stark effect. Splitting of spectral terms. The Lorentz cla ssica l theory of dispersion. Index of of refraction. Polarization. Radiation damping. Quantum theory of dispersion. Oscillator strengths. Ramon effect. “Stokes" and “anti-Stokes" lines. Energy correction in second order per­ turbation theory. A.iharmonic oscillator. Harmonic oscillator. Matrix elements of x^. (Perturbation theory for the continuous spectrum. Ingoing and outgoing wave boundary conditions. Phase shift. Connection between partial wave and three di­ mensional formulations.) PART II. RELATIVISTIC QUANTUM MECHANICS 15. The Klein-Gordon Scalar Relativistic Wave Equation

257 259

R elativistic invariance of the de Broglie relations. R elativistic energy-momentum relation of a free particle. The Klein-Gordon equation. Charge and current density. Nonrelativistic limit. The initial data problem. Indefiniteness of the sign of charge. Inter­ action with an external electromagnetic field. R elativistic energy levels of a spinless particle in a Coulomb field. Fine structure constant. The case Za > V2 . Charge and current density in the presence of an electromagnetic field. 16. Motion of an Electron in a Magnetic Field. Electron Spin

268

C lassical theory of the Zeeman effect. Interaction energy of a magnetic dipole. Larmor precession. Magnetic moment of a moving electron. Zeeman effect in nonrelativistic SchrOdinger theory. Orbital magnetic moment. Bohr magneton. Normal and anomalous Zeeman splitting. Einstein-de Haas experiment. Landd g factor. Stern-Gerlach experiment. Uhlenbeck-Goudsmit hypothesis of intrinsic angular momentum. Half-integral quantum numbers for angular momentum. Electron spin. The Pauli equation. Two-component wave functions. The operator for intrinsic mag­ netic moment. Pauli matricies. Coupled SchrOdinger equations. Matrix elements. Spin operators. Commutation relations for spin operators. Vectorial character of spin operators. Separation of spin and space variables in a homogeneous magnetic field. Eigenvalues of the spin operator along an arbitrary direction. Probability distribution of spin directions. 17. The Dirac Wave Equation Linearization of the energy operator. Dirac matricies and their relation to Pauli matricies. The Dirac equation. Charge and current density. External electromagnetic field. Velocity oper­ ator. Statistics in second quantization. Transformation proper­ ties of the spinor wave function under Lorentz transformations and spatial rotations.

285

X

CONTENTS

18. The Dirac Theory of the Motion of an Electron in a Central Field of Force

293

Orbital, spin and total angular momenta. Conservation laws. Properties of the total angular momentum operators. Quantization of total angular momentum. Clebsch-Gordan coefficients. Spher­ ical spinors. The vector model of the addition of angular mo­ menta. Motion in a central field including spin effects. Theory of the rotator. Selection rules. Parity of a state. Conservation of parity. Solution of the Dirac equation for a free particle. Negative energy states. Nonrelativistic limit. Four-vector trans­ formation law of the energy-momentum operators under Lorentz transformations. R elativistic invariance of the scalar wave equation. Vector model. Charge conjugation. 19. The Dirac Equation in Approximate Form

308

•Two component Pauli form. “Small” and “large” components. Correction terms to order ( e /c ) 2. R elativistic increase of mass. Interaction of the intrinsic magnetic moment. Spin-orbit inter­ action. Contact interaction. The velocity operator and Ehrenfe st’s theorem in the Dirac theory. 20. The Fine Structure of the Spectra of Hydrogen-like Atoms

314

Advantages of the approximate method. R elativistic and spin effects. Contact interaction. Stable motion for Z < 137. Fine structure in the Dirac theory. Experimental verification of the fine structure theory. Lamb-Rutherford experiment. Anomalous Zeeman effect. Weak magnetic field. Landd g factor. Strong magnetic fields. Paschen-Back effect. “Breaking” of spin-orbit coupling. Paramagnetism and diamagnetism. Anomalous Zeeman effect in the vector model. (Stark effect. Quenching of metastable states. Intermediate field Paschen-Back effect.) 21. The Effect of Nuclear Si rue lure on Atomic Spectra

334

Reduced mass. Effect of finite nuclear size. Mcsic atoms. Ap­ proximate harmonic oscillator potential for large Z. Spin of the muon. Application of the Dirac equation to the neutron and pro­ ton. Anomalous (Pauli) magnetic moment. Experimental deter­ mination of the magnetic moments of the neutron and proton. Limitations on the measurement of angular momentum. Experi­ ments of Bloch and Alvarez and of Rabi. Nuclear magneton, llyperfine structure of the hydrogen spectrum. 22. The Electron-Positron \ncuum and the Electromagnetic \ aciitim A. Dirac theory of “h o les.” Negative energy stales. Discovery of the positron. Pair creation and annihilation. Anliparticles. Rigorous validity of conservation laws. Positronium. Interconvertibility of particles. B. Tile Lamb shift of energy levels of atomic electrons. Fluctuations of the electromagnetic vacuum.

347

CONTENTS

XI

Virtual particles. “Smearing out” of a point electron. C. Elec­ tron-Positron vacuum. Vacuum polarization. Anomalous mag­ netic moments of electron, proton, and neutron. D. Renormal­ ization. Quantum electrodynamics. Quantum theory of fields. Cherenkov radiation. 23. Theory of llie Helium Atom Neglecting Spin Slates

358

Basic principles of the theory of multielectron atoms. Indistinguishability of electrons. Exchange forces. Perturbation theory solution of the helium atom. Permutation of electrons. Exchange degeneracy. Exchange energy. Symmetric and anti­ symmetric wave functions. Coulomb interaction between elec­ trons. Ionization energy. The variational method. Derivation of the SchrOdinger equation by the variational method. HartreeFock method of self-consistent fields. Investigation of the exchange energy. Exchange time. 24. Elementary Theory of Millticlcctron Atoms Including Spin States

378

Symmetric and antisymmetric states. Permutation operator. Eermi-Dirac and Bose-Einstein statistics. The Pauli exclusion principle. Fermions. Bosons. Determinental wave function. Addition of angular momentum. Russell-Saunders coupling. Clebsch-Gordan coefficients. LS coupling, jj coupling. Wave function of the helium atom including spins. Triplet and singlet states. Parahelium and orthohelium. Energy spectrum of the helium atom. Variational wave function for a Yukawa potential. Diamagnetic susceptibility of parahelium. 25. Optical Spectra of Alkali Metals

397

The structure of complex atoms. The Thomas-Fermi statistical method. Boundary conditions for neutral and ionized atoms. Solution of the Thomas-Fermi problem by the Ritz variational method. Total ionization energies. Charge distribution in argon. Energy levels of alkali atoms. Atomic core. “Penetrating” orbits. Polarization of the atomic core. “Effective principle quantum number.” Smearing of the atomic core. Fundamental series. Multiplet structure of spectral lines. Spectral terms of sodium. Sharp, principle and diffuse series. 26. Mendeleyev’s Periodic System of Elements X-ray spectra of atoms. Continuous spectra. Bremsstrahlung. Characteristic spectra of atoms and the structure of their inner sh ells. Moseley’s law. Multiplet structure of x-ray spectra. R elativistic and spin effects. Regular and irregular doublets. The discovery of Mendeleyev’s periodic law. Filling of the electron sh ells. Application of the Thomas-Fermi method. Peri­ odicity properties of the elements.

420

xii

CONTENTS

27. Tin1 Theory of Simple Molecules

437

Chemical bond. Heteropolar molecules. Affinity. Valence. Kossel. Molecular hydrogen ion. Exchange forces. Evaluation of some integrals by Fourier transforms. Homopolar atomic mole­ cules. Hcitler-London theory. Spin and syinmetiy. Orthohydrogen and parahydrogen. The valence theory. Spin valence. Masers and lasers. PART III. SOME APPLICATIONS TO NUCLEAR PHYSICS 28. Elastic Seatiering of Particles

465

Time-dependent perturbation theory. Golden rule. Cross section for elastic scattering. Uncertainty of energy. Scattering ampli­ tude. Bom approximation. Scattering by a Yukawa center of force. Range of nuclear force. Fast-electron scattering by neu­ tral atoms. Validity of Born approximation. Partial-wave cross sections. Phase shift. Scattering from a spherical barrier and spherical well. Resonant scattering. (Golden Rule #2, Density of final states.) 29. Second Quantization

480

Second quantization of the SchrOdinger equation. The Heisen­ berg equation of motion, q numbers and c numbers. Commutation relations for Boson field amplitudes. Creation and destruction operators. Anti commutation relations describing particles obey­ ing Fermi sta tistics. Quantization of Maxwell’s field equations. Spontaneous em ission. Dipole approximation. Beta decay. Pauli’s hypothesis of the neutrino. The Fermi theory. Weak and strong interactions. Fermi and Gamow-Teller selection rules. Feynman and Gell-Mann theory. /3-decay spectrum. Nonconservation of parity in weak interactions. Lee and Yang. Ilelicity of the neutrino. Pion decay. APPENDIX A. Hilbert Space and Transformation Theory

497

APPENDIX II. ’I lie Statistical Assertions of Quantum Mechanics

505

PROBLEMS

511

Preface

This textbook is based on my lectures to students at the Mos­ cow Regional Pedagogical Institute (1945 to 1948) and Moscow University from 1945 on. In writing this book we set ourselves the difficult task of treating in a single volume the fundamentals of atomic theory, that is, Schrodinger’s nonrelativistic theory, Dirac’s relativistic theory, the theory of multielectron atoms, and the basic applications of quantum mechanics to solid state physics. Our aim was to combine the exposition of general the­ oretical principles with examples of the application of quantum mechanics to specific problems connected with atomic structure. To avoid overloading this book, we have abridged the treatment of certain specialized topics, but in such cases we have endeavored to supply references to standard works on the subject. In most textbooks the solution of specific problems with the help of Schrodinger’s equation is handled in fairly elegant form. The basic mathematical tools required for this purpose are a knowledge of second-order differential equations and various spe­ cial functions (including the Hermite, Legendre and Laguerre polynomials). However, applications of Dirac’s theory to specific problems (such as the hydrogen atom) are on the whole handled le ss satisfactorily. In some cases the calculations are so long and cumbersome that it is difficult to perceive the physical mean­ ing of the solutions. In others there is no actual derivation of the results or only a rough proof is given. In an attempt to avoid these pitfalls, we have used an approximate form of Dirac’s equa­ tion for our treatment of the hydrogen atom (Chapter 19). This approximation still enables us to obtain the formula for the fine structure of the energy levels and the selection rules (Chapter 18 and 20). Our analysis of the Lamb shift due to the electronpositron vacuum is also somewhat simplified (Chapter 22). Several good problem books in quantum mechanics are avail­ able, and therefore we shall consider only a few problems chosen with the aim of elucidating and supplementing the general discus­ sion. The first part of this book was written jointly by me and Yu. M. Loskutov, and the second part jointly by me and I. M. Ternov. Great assistance was rendered by M. M. Kolesnikova in condens­ ing notes based on my lectures on quantum mechanics and in preparing the manuscript for the press. Chapter 25 was carefully

xiv

PREFACE

read by N. N. Kolesnikov, who made a number of valuable com­ ments. I would like to mention the great pains taken by S. I. Larin in editing the whole manuscript. A. A. Sokolov

Introduction Quantum mechanics dates only from the 1920’s. This important branch of theoretical physics deals with the fundamental problem of the behavior of microparticles (for instance, the behavior of electrons in an atom). As a theory, quantum mechanics represents an extension of classical mechanics, electrodynamics (including the theory of the electron and the theory of relativity), the kinetic theory of matter, and other branches of theoretical physics. Historically, the development of every branch of theoretical physics involves two main stages. First comes the accumulation of experimental facts, the discovery of semiempirical laws, and the development of preliminary hypotheses and theories. This is fol­ lowed by the discovery of general laws, which provide a basis for interpreting a large number of phenomena. For example, the first or pre-Newtonian stage of mechanics consisted of the discovery of a number of seemingly unrelated laws: the law of inertia, the law of free fall under the action of a gravitational field and Kepler’s laws of planetary motion. Most of these laws were discovered only after years of painstaking work by many scientists. Thus, many astronomical observations preceded the discovery of Kepler’s laws. We may recall the great efforts of Copernicus, Bruno, Gal­ ileo, and others to establish that the Sun is the center of our planetary system and that the Earth is only a planet like Mars, Venus, or Jupiter. It was only after working for fifteen years on Tycho Brahe’s extremely valuable observational data that Kepler found the semiempirical laws describing planetary motion. After these preliminary, seemingly independent laws had been estab­ lished, Newton was able to show that they all rested on the same theoretical foundation. Newton’s three laws of motion and the law of universal gravitation opened a new stage in the development of theoretical mechanics. One of the great triumphs of Newtonian mechanics was L everrier’s prediction of the existence of a new planet, Neptune, from perturbations in the motion of Jupiter. In a sim ilar fashion, Maxwell’s formulation of the laws of elec­ trodynamics was preceded by the discovery of empirical laws de­ scribing various electric and magnetic phenomena. Coulomb’s law of interaction between electric charges and magnetic poles1 and the Biot-Savart law of interaction between an electric current and a magnetic pole were found by analogy with Newton’s law of ' A s m a g n e t i c m o n o p o l e s do n o t e x i s t in n a t u r e , C o u l o m b ’s l a w in m a g n e t o s t a t i c s i s v e r i f i e d b y m e a n s of m a g n e t i c d i p o l e s .

xvi

INTRODUCTION

gravitation. All of these phenomena were explained on the basis of the principle of “ action at a distance,” according to which one charge acts directly on another through the intervening space. After Newton, and independently of investigations of electric and magnetic phenomena, considerable attention was devoted to optics. At a relatively early stage, it was established that light consists of transverse waves, propagating with a finite velocity of c«=1QI

4£2 + 2

ua

1

2

r2 io2

3 w0c!

■*■'’ 0 1

(1.27)

On the other hand, the energy density u, which is related to the electromagnetic field of the radiation by Eq. (1.16), can also be expressed in terms of | £

|2. Since the radia­

tion is isotropic we have, on the basis of Eq. (1.16), “ = L (£2 + "*) = 4 “

+ &y + Ei),

(1.28)

^ T h e r e a d e r s h o u l d n o t f e e l u n e a s y a b o u t o ur e q u a t i n g t h e d i f f e r e n t i a l d.

(1.41)

We then obtain Planck’s equation na) 3 fnu

Pm

>(1.42)

7i’>c, (eW — I)

which was a brilliant achievement of quantum theory. The quantity h = 1.05 • 1O' 27 erg.sec, which has the dimensions of action. Is called Planck’s constant.3 At low frequencies

1 :, the exponential e ^ ,kTmay be expanded

in the form of a power series in too kT. Restricting ouselves to the linear terms of the expansion, we have 1

kV

In Ihe 1i l e n i t ’iro, P l a n c k ’s c o n s t a n t is m ore o f t e n t a k e n a s t h e q u a n t i t y h 6 . 6 2 4 9 • 10 erf» • s e c , w h i c h r e l a t e s th e e n e r g y ( to th e f r e q u e n c y v:

f

his.

2 nfi =

THE QUANTUM THEORY OF

13

LIGHT

Thus, Planck’s equation (1.42) reduces to the Rayleigh-Jeans formula (1.35). In the case of high frequencies l), we may neglect the 1 in the denominator of Eq. (1.42) and write pm in the form a - 4 3 )

Planck’s equation (1.42), which describes the dependence of the spectral density of thermal radiation on the frequency w, is in excellent agreement with experiment (see Fig. 1.1). From Eqs. (1.42) and (1.18), we can find the total radiation density oo

u=

0)3dm

(1.44)

e 1lw/kT __J

Introducing the variable £= / ) i o a n d considering that r e3 dk ] 7 -1

7t4 15’

we obtain the well-known Stefan-Boltzmann law 4 u

15

cW

(1.45)

= a7’4,

where a = i h w ==7-56 • 10'" e rg -cm ' 3 -deg'4.

(1.46)

From Eq. (1.42), it can be seen that the spectral density of black-body radiation has a maximum at some value of i» and that the position of this maximum changes with temperature. The equation governing the position of this maximum is called Wien’s displacement law. More often, Wien’s displacement law is expressed

^ U s u a l l y it i s n o t t h e d e n s i t y u w h i c h i s m e a s u r e d , b u t t h e e n e r g y a T 4 w h i c h i s r a d i a t e d p e r s e c o n d p e r s q u a r e c e n t i m e t e r o f t h e b l a c k b o d y ’s s u r f a c e w i th in a s o l i d a n g l e o f 277 (U = 277).

c

C

In t h i s c a s e , t h e Stefa n -B o ltzm a n n c onstant i s O ~ 277 ---- a = — a = 5.67 • 10 8 77 4

e r g •cm ^ •d e g

4

c

NONRELATIVISTIC

14

QUANTUM

MECHANICS

in term s of the spectral distribution with respect to the wavelengths X. To determine px, we can use the expression for u w = J P\dk Since X— 2 kc / w , transforming to the spectral distribution over the frequencies, we have CO

00

«=

jpo.du), a

o

(1.47)

from which we find Px--

2Ttc

16it3cft

X2 Pm--

(1.48)

2 r.cH

X5(eftrx._i)

To determine the wavelength Xmax at which px has its maximum, we set ^ (M =

0

2Itch

—5

*rxr 2

e

h T l

rcrfl

=

0.

m ax .

Setting 2nchlkT\max = y, we obtain the equation £/ = 5(1 — e~v),

whose solution can be given with good accuracy in the form i / ^ 5 ( l — e-») = 4.965.

Thus

is related to the temperature T by the equation Xmax7' = 4?9 6 M = A= 0-29 cm -deg,

(1.49)

which expresses Wien’s displacement law, and where b is the Wien’s constant. According to this law, as the temperature of an ideal black body increases, the maximum of the radiation intensity is shifted towards shorter wavelengths (see Fig. 1.2).

TH E QUANTUM THEORY OF

LIGHT

15

Equations (1.46) and (1.49) relate Planck’s constant /2 and Boltz­ mann’s constant k to the constants a and b. Knowing the numerical values of a and b , we can determine tl and k. This is the way in which a numerical value was first obtained for U and a better value found for k.

F i g . 1.2. C u r v e s of t h e a function of body:

W i e n 's d i s p l a c e m e n t law. sp e ctral energy distribution as t e m p e r a t u r e for an i d e a l b l a c k Xm a x T = 0 . 2 9 cm . deg.

Recapitulating, it follows from Planck’s hypothesis that proc­ esses such as em ission and absorption involve discrete quanta. In other words, the energy change of particles involved in these proc­ esses is discontinuous and not smooth as would follow from the laws of classical physics. D. EINSTEIN’S PHOTON THEORY In deriving his equation, Planck assumed that the energy of the oscillators is quantized. The original version of the theory, however, does not provide any physical justification for this property. Indeed, Planck himself chose to attribute the “ special properties” to the heated body rather than to the electromagnetic radiation. The second important step towards the development of quantum theory consisted of Einstein’s hypothesis that oscillators absorb and emit radiation in discrete amounts because electromagnetic radiation itself consists of discrete particles, calledphotons, which carry an amount of energy hw. In effect, Einstein interpreted Planck’s equation as a description of the corpuscular properties of light. We shall now attempt to develop an elementary theory of photons.

16

N O N R E L A T I V I S T 1C

QUANTUM

MECHANICS

According to classical theory, the energy of a light wave is s=

( £ 2

+ m d*x =

4= 0. From these considerations, Einstein concluded that an electro­ magnetic field canbe considered as a set of particles called photons, with zero rest mass and the energy £= /ko).

(1.53)

For the photon momentum the following equation is obtained: n = k'>= k'>-K = flk, C

(1.54)

2~b° / 2 it where h — 2-h, and k = — is the wave vector! k = — is the wave number). On the basis of these concepts, in 1905, Einstein constructed a qunnfitative theory of the photoelectric effect, which had been dis­ covered by Hertz in 1887. What is observed in the photoelectric effect is the following: the potential difference required for a spark to jump between two small charged spheres is reduced if the

THE QUANTUM THEORY OF LIGHT

17

cathode is illuminated. To explain this phenomenon, Einstein postulated the simple equation =

(1.55)

This is essentially the law of conservation of energy and indicates that the kinetic energy

of the ejected electron is equal to the

difference between the energy of the absorbed photon /Zoo and the work function W of an electron in the metal. It is obvious that if fiu> exceeds a certain limit (so-called threshold frequency)

The implications of the photon theory were brought out and verified in 1923 by experiments on the scattering of x-rays by free electrons (the Compton effect). The Compton effect was particularly interesting because it confirmed not only the law of conservation of energy (which was already verified by the photoelectric effect) but also the law F i g . 1.3. S c a t t e r i n g o f l i g h t by a free of conservation of momentum. e l e c t r o n ( t h e C o m p to n e f fe c t ). It is well known that, in classical theory, the frequency of light does not change when it is scattered by a free electron (u/ = u)). By contrast, in quantum theory, part of the photon’s energy e = /Zu> is transferred to the electron (see Fig. 1.3). Consequently, the energy and frequency of a scattered photon should generally be somewhat sm aller (s' — h w ’ — c i ( m — m 0),

Uk — hk' — mv.

(1.56)

Here m0 and m = mnj Y 1 — represent the mass of the electron before and after collision; v is its velocity; (3= u/c; Hk = Hu>lc and

18

NONRELATIVISTIC

QUANTUM

MECHANICS

Hk' — hm'lc represent the momentum of the photon before and after scattering. We rewrite Eq. (1.56) in the form c8

u> —

— m 0),

=

(1.56a) * -* = " • Taking the square of these equations and subtracting the first equation from the second, we obtain u)d/ (1 — cos 0)

(^^ ■— —rwf).

(1.57)

Substituting X= 2rc/cu, \’— 2 tcc/u>', and dividing (1.57) by mu/, we find an expression for the increase in wavelength of the scattered light AX= X' — X= 2X0 sina y ,

(1.58)

where X0 is the Compton wavelength of the electron lo

2

rn

t l ’ oC

-IthC - - = 2 .4 - 10"‘“ cm.

We therefore see that, according to quantum concepts, the wave­ length of the scattered light X'must be greater than the initial wave­ length X(X'^>X) since «/. This difference increases with the scattering angle < ‘t. Since the Compton wavelength X0 is relatively small, Compton scattering is observed at relatively short wave­ lengths ( x-rays and gamma rays ). Indeed, for visible light (X~ 10 5 cm ) I0~B== i ° whereas for x-rays (X ~

10

8— 1 0 -y- ~

10

9

1

' 3

7*.

cm) = lO°/0.

Therefore, the Compton shift can be observed experimentally only in the second case. In his experiments, Compton studied the scattering of radiation from an x-ray tube by graphite and other substances (lithium, beryllium, sodium, potassium, iron, nickel, copper, and so on) at different angles S). The spectral distribution of the intensity of the scattered radiation at different scattering angles was measured by means of an ionization chamber.

TH E QUANTUM THEORY OF

LIGHT

19

Figure 1.4 shows the spectral distribution of incident and scattered waves. If the incident wave (upper curve) has one maxi­ mum, the scattered wave (lower curve) will have, in addition to this maximum, a second maximum at a longer wavelength. The

Wavelength

F i g . 1.4. S p e c t r a l d i s t r i b u t i o n o f x - r a y s in t h e C o m p to n e f f e c t b e f o r e (u p p e r c u r v e ) a n d a f t e r ( lo w e r c u r v e ) s c a t t e r i n g .

distance between the wavelengths of the two maxima must corre­ spond to the Compton shift; this is because the distance increases with the scattering angle, and, in addition, it does not depend on the type of scattering material [both these facts are in accord with Eq. (1.58)5]. The unshifted maximum corresponds to scatter­ ing by electrons which are strongly bound to the nucleus (or more precisely, electrons whose binding energy is greater than the energy of the x-ray quanta). The shifted maximum corresponds to scattering by electrons which are so weakly bound to the nucleus that, in practice, they can be regarded as free. Thus the results of Compton’s experiments completely confirm the quantum nature of light (that is, the photon theory).

Only t h e i n t e n s i t y o f t h e m a x im a d e p e n d s on th e t y p e o f s c a t t e r i n g s u b s t a n c e . A s the a to m ic w e i g h t o f t h e s c a t t e r i n g s u b s t a n c e i n c r e a s e s , t h e i n t e n s i t y o f t h e u n s h i f t e d m a x i ­ mum i n c r e a s e s , a n d t h a t of t h e s h i f t e d maxim um d e c r e a s e s .

C hapter 2

The Bohr Quantum Theory A. BASIC INFORMATION ON PROPERTIES OF ATOMS A theory of the atom was developed only after reliable experi­ mental data had been obtained from studies of the effects described below. 1) Emission of light by atoms. From careful studies of the radiation of atoms, it was established that they have bright-line spectra and that the lines are arranged in certain definite series. For example, all the lines of hydrogen are described by Balm er’s formula 1 (2 . 1 )

where R is the Rydberg constant, and n'and n are integers. Setting n' = 1 and n — 2, 3, 4, . . . , we obtain the Lyman series, which lies in the ultraviolet part of the spectrum. F o rn '= 2 and n = 3, 4, 5 ....... we have the Balmer series, which is located in the visible part of the spectrum and is, therefore, easiest to study. Formula (2.1) can also be written in the form of a difference between two quantities (2 . 1 a) In s p e c t r o s c o p y , i t is c u s t o m a r y to w r i t e B a l m e r ' s fo rm u la in t h e form

A w h e r e t h e R y d b e rg c o n s t a n t for h y d r o g e n is R sp 1 0 9 , 6 7 7 . 6 c m c o n s t a n t in Kq. (2 1) is r e l a t e d to / f sp by t h e e q u a t i o n

It

2ncl< llp

277 ■ 3 29 ■ 1 0 1 5 s e c ' 1

1

A

2

nc

T h e v a l u e o f th e R y d b e r g

2 0 .6 6 • 1 0 15 s e c 1 -

THE

BOHR

QUANTUM

THEORY

21

which are called spectral terms. For the hydrogen atom, these terms are given by

This possibility of representing the radiation frequencies « as a difference between two terms is a consequence of the Ritz combina­ tion principle, which has important spectroscopic applications in regard to the hydrogen atom, as well as more complex atoms. For example, hydrogen was initially found to have two series, corre­ sponding to n '= 1 (the Lyman series) and to n’= 2 (the Balmer series). On the basis of the Ritz combination principle , 2 a third series was predicted with n’= 3 and n = 4, 5, 6 , . . . . This series was later discovered by Paschen in the infrared region of the spectrum. 2) The behavior of an atom in external electric and magnetic fields and, in particular, the interaction of the atoms of a substance with fast particles passing through it. The most important experi­ ments in this area were conducted by Rutherford, who succeeded in finding the distribution of positive charges inside the atom from the analysis of fast-a-particle scattering. 3) Finally, investigation of various properties of molecules provided important data pertaining to the properties of atoms. For example, the formation of simple homopolar molecules and the valence theory found their explanation only on the basis of the modern quantum theory of the atom. B. THE CLASSICAL MODEL OF THE ATOM Once it had been established that an atom consists of a positively charged part associated with most of the m ass, and of light, negatively charged electrons, attempts were made to construct a static model. The reason this approach to the problem was adopted is that, in classical electrodynamics, an accelerated electron emits radiation, the amount of energy emitted per unit time being dE_ 2 e2ws dt ¥ •

(2 . 2 )

T h e R i t z c o m b i n a t i o n p r i n c i p l e w a s f i rs t f o r m u l a t e d a s f o llo w s: if t h ere a r e tw o d i f fe r e n t f r e q u e n c i e s b e l o n g i n g to th e s a m e s e r i e s , t h e d i f f e r e n c e b e t w e e n t h e s e fre ­ q u e n c i e s i s a l s o a f r e q u e n c y w h i c h c a n a l s o b e e m i t t e d by t h e atom , but b e l o n g s to a n o t h e r s e r i e s . T h e c o n c e p t of “t e r m s '’, p e r m i t s a r e l a t i v e l y s i m p l e e x p l a n a t i o n o f t h i s . I n d e e d

Tv - T_ Hence COn n

- COn " n

T " - T

a n d t h u s the R i t z c o m b i n a t i o n p r i n c i p l e l e a d s d i r e c t l y to E q . (2 .1 a).

22

N O N R EL A TIV ISTIC

QUANTUM

M E CH A N IC S

where e = — ea is the electron charge ( e0 = 4.80 • 10~ 10 esu is the elementary charge), &yis the acceleration of the electron, and c is the velocity of light in vacuum. The minus sign in front of dE/dt shows that the energy of the electron decreases as a result of the em ission of radiant energy. Since an atom does not radiate in the ground state, it follows from the classical theory that the charges in the atom should be at rest. The most interesting classical model was that of Thomson, according to which the positive charge uniformly filled the entire atomic volume, and the electronic, that is, negative point charges were lo­ F i g . 2. 1. T h o m s o n ’s m o d el of cated inside the atom. t h e h y d r o g e n ato m ( 2 = 1). For example, in the hydrogen atom, the T h e p o s i t i v e c h a r g e Z cq is positive charge was supposed to fill uni­ un ifo rm ly distributed over formly a sphere of radius R0 (see Fig. t h e v o lu m e of a s p h e r e of 2.1). The charge density inside the sphere r a d i u s /?q. T h e e l e c t r o n (with c h a r g e - c 0) i s l o c a t e d a t a was (for Z— I) d i s t a n c e x from t h e c e n t e r o f

__

P= 4 ^ ' In the ground state, the electron was supposed to be located at the center of the sphere, where the electric field is zero. At a distance r = x

__Ze o

max~ Ro • To explain these results, Rutherford proposed a planetary model of the atom in which the structure of the atom resem bles a planetary system. A positively charged nucleus constituting almost the entire mass of the atom is concentrated at the center in a very small volume of radius 1 0 ~ 13 — 1 0 1 \ and charged electrons move about this nucleus in closed orbits like planets around the Sun. We note that the potential energy of the Newtonian attraction between a planet of m ass m and the Sun (of m ass M) y

_ Newt

V.mM r

’

where 7. is the gravitational constant and has the same form as the potential energy of the Coulomb attraction between an electron and a nucleus

From this model, Rutherford developed a quantitative theory of scattering. His calculations were based on the assumption of a Coulomb interaction between the alpha particles and the nucleus. The influence of the atomic electrons was neglected in the first approximation, since their energy is considerably lower than the energy of the bombarding particles. Let us find, following Rutherford, the trajectory of an alpha particle moving in the field of an infinitely heavy 5 point nucleus having a charge Ze0. Our calculations will be carried out in a 5If t h e f i n i t e n e s s o f th e n u c l e a r m a s s Mnuc *s t a k e n i n t o a c c o u n t , th e n u c l e u s h a s a c e r t a i n r e c o i l ( l ik e t h a t of th e a l p h a p a r t i c l e ) a s a r e s u l t of th e i n t e r a c t i o n . In t h i s c a s e , all th e c a l c u l a t i o n s m u s t b e p er fo r m ed in th e c e n t e r - o f - m a s s s y s t e m and, in t h e r e s u l t s o b t a i n e d for t h e c a s e ,\7nuc it i s n e c e s s a r y to r e p l a c e th e m a s s of t h e a l p h a p a r t i c l e Mg by t h e r e d u c e d m a s s

M0 »'nUc re d « 0 + Mnuc ( s e e C h a p t e r 12, S e c t i o n C for a d i s c u s s i o n o f the r e d u c e d mass).

26

NON R E L A T I V I S T 1C Q U ANT UM

M E CH A N IC S

coordinate system whose origin coincides with the nucleus (see Fig. 2.2). Since the field produced by the nucleus is centrally symmetric, in determining the trajectory of the alpha particles we can use both the law of conservation of energy E = const,

(2.7)

and the law of conservation of angular momentum (r

x v ) = const,

(2 . 8 )

where M0 is the m ass of the alpha particle, r is its coordinate, and v is its velocity.

F i g . 2.2 . D i a g r a m for t h e d e r i v a t i o n o f R u t h e r ­ f o r d ’s fo rm u la for t h e c r o s s s e c t i o n of e l a s t i c s c a t t e r i n g o f a l p h a p a r t i c l e s by n u c l e i .

Let us introduce the polar coordinates r and 9 . The velocity of the particle is given by & = v\ +w*J. = ^ + rY ,

where

(2.9)

and v i = r§ are the components of velocity parallel

and perpendicular to the radius vector r, respectively, andr=^T and 'f = ‘-Jt . We then obtain, instead of Eqs. (2.7) and (2.8), E ■=

-| 1/ =

(f2 + r'p) +

—

-5

= const,

Lz = Mn(r >v)z = M0r't§ = const.

(2.10) (2 . 1 1 )

In the absence of interaction, the alpha particle would pass the nucleus at a distance b (this distance b is called the impact para­ meter). Setting the initial velocity equal to u„ (that is, the velocity

THE

BOHR

QUANTUM

THEORY

27

r -* — oo and as follows from Fig. 2.2), then (2.10) and (2 . 1 1 ) can be reduced to the form Y ('* + r y ) +

T

Vo

8

mot2'

Since the wave function of the antisymmetric solution vanishes at .v=0, the antisymmetric solution for .v > 0 is also the solution for a particle in a potential field described by for A r>a (region III) Here As = ^,”“—,

z- =

(F0— E), A le‘kx and B\e~ikx characterize the incident and

reflected waves, respectively; A3e, klx~a> characterizes the transmitted wave; and Ih? ,k v , characterizes the reflected wave coming from infinity. Since we have no reflected wave from infinity in our case, we must set Z1„ = 0, To determine the transmission coefficient we shall use the boundary conditions at a = a and x = 0. We first express An and Bo in terms of Aa, making use of the fact that

' a ^ I:

A o = l 2 ‘n- A ae*a, Do = —

■A,e~

0,

and then e x p re s s Xl, in te rm s of /13

/l.

e*aAa

Hie transmission (diffusion) coefficient D is then found to be

,

i A,!: ^ “ M. r “-

(1

1 0 /t2 + n - f e~ x0 =

exp ^— 2xa -j- In

16n‘

\

(1+«2)2J’

THE

TIM E-DEPENDENT

SCHRODINGER

WAVE

EQUATION

95

where

k n= — x

E'

Neglecting the second term in the exponential for D (this is possible because the quantity 16n* Is only slightly different from unity), we finally obtain - 2± V

2m 0(Va - E )

D =