THE FOUNDATIONS OF QUANTUM THEORY SOL WIEDER Fairleigh Dickinson University ACADEMIC PRESS New York and London COPYRIGHT © 1973, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 LIBRARY OF CONGRESS CATALOG CARD NUMBER: 72-9422 PRINTED IN THE UNITED STATES OF AMERICA TO MY PARENTS MAYER AND TOBY Preface The undergraduate physics curriculum at many institutions has been revised to include a two-semester junior level course which surveys topics in modern physics and treats quantum mechanics on a semiquantitative basis. The student later takes a course which stresses formal quantum theory. This book provides a smooth transition from early undergraduate work to the more advanced material. The text evolved from lecture notes used in a two-semester course at Fairleigh Dickinson University. It draws on the stu­ dent's background in mechanics, electricity and magnetism, and modern physics. From my own experience, I have found two pedagogical approaches to quantum theory useful. The first relies rather heavily on the analogy between the wave equation in physical optics and the Schroedinger equation. This viewpoint is likely responsible for the somewhat restrictive and misleading label "wave mechanics." Since the de Broglie hypothesis is used as a corner­ stone in wave mechanics, this approach is understandably consistent with a historical development. In addition the classical limit can easily be explained in terms of an analogy with geometrical optics by illustrating the similarities between the Hamilton-Jacobi equation and the eikonal equation. However, xiii xiv PREFACE the formal correspondence between classical and quantum mechanics is not immediately clear. In this text, I have chosen the second alternative, sometimes referred to as the "canonical" approach. The importance of first setting up the classical Hamiltonian in terms of the canonical coordinates and their conjugate momenta and then studying the inherent symmetries of the problem is an underlying theme of this book. In this respect the book follows closely the elegant but formal text of Dirac* This book approaches the subject in a manner more palatable to the average senior. The correspondence between the classical and quantum theories is made via the Poisson bracket-commuta­ tor analogy. Ehrenfest's theorem is used as a postulate to connect the classical equations of motion with the Schroedinger equation. After presenting the general form of wave mechanics in bra-ket notation, it is shown that wave mechanics is only one of the many representations of quantum theory. There is general agreement that where wave mechanics is applicable, it is by far the simplest and most direct way of dealing with a problem. Most of the problems in the text deal with particles in prescribed potentials and so wave mechanics is used extensively and the techniques for solving Schroedinger's differential equation are discussed in detail. However, where feasible, solutions are also obtained using the more general methods of Dirac. The quantum theory is applied to selected problems in modern physics with the expressed purpose of teaching the former and not the latter. It has been my feeling that modern physics is best appreciated after a thorough exposition of quan­ tum mechanics. Since the electromagnetic interactions are understood more completely than are the other interactions, I have deliberately avoided most of the applications to nuclear physics and have limited most of the discussions to the atomic and molecular domain. The text is divided into three parts—One-Particle Systems (Chapters 1-8), Many-Particle Systems (Chapters 9 and 10), and Relativistic Quantum Mechanics and Field Theory (Chapters 11 and 12). In the first part, Chapters 3 and 7 are crucial and will probably be the most difficult for the student to master. Without proper attention to this material, what follows may be meaningless. In Part II, Chapter 9 deals with noninteracting indistinguishable particles and the material covered is fundamental to almost all branches of physics. Chapter 10 covers interacting particle systems and if necessary certain sections may be omitted. Part III contains somewhat more advanced material and if time is short this part may be neglected altogether. If all twelve chapters are to be covered, at least half of Chapter 7 should be completed by the end of the first semester. * P. A. M. Dirac, "The Principles of Quantum Mechanics," 4th ed., Oxford University Press, London, 1958. PREFACE xv I am very much indebted to my colleagues, students, and editors who have contributed their time and effort to make this text a better one. My thanks to my Honors student Mr. Alan Blumberg who scanned the original manuscript and to Professor W. Arthur whose careful scrutiny of the galleys has reduced the number of errors significantly. I am especially grateful to Professors D. Flory and R. Zeidler for the many interesting discussions and invaluable suggestions on quantum theory which have improved this book immeasurably. Any remaining errors or shortcomings are entirely my own fault. My wife Suzanne patiently typed and retyped the manuscript making many valuable comments while at the same time caring for our young sons Ari, Jonah, and Jeremy. For this and for her tolerance throughout the course of this work, she deserves my most special thanks. 1 Historical Aspects The development of physics in the twentieth century has been marked by two great discoveries. The first, special relativity (Einstein, 1905), corrects the equations of classical dynamics when the characteristic speed of matter becomes comparable to that of light. The second, quantum mechanics (Schroedinger, Heisenberg, Born, Dirac, 1925-1928), provides us with a more accurate picture of the dynamics of microscopic systems than do Newton's laws. By the end of the nineteenth century, experimental evidence had gradually accumulated to suggest that the classical theories of Newton and Maxwell were not adequate to explain many phenomena associated with matter and radiation. As a first step in our study of quantum mechanics, we examine some of the problems that faced the physicist at the turn of the century. I Black-Body Radiation Matter is constantly emitting and absorbing radiation. A material emits radiation due to thermal agitation. For example, a metal may become "red hot " when heated to a few thousand degrees Kelvin. Thus, when any material 3 4 1 HISTORICAL ASPECTS at a temperature Γ is fashioned into a cavity to enclose a region in space, the cavity will contain electromagnetic radiation. In equilibrium, it is this radia­ tion which is known as "black-body" radiation and is found experimen­ tally to contain a characteristic mixture of frequencies (that is, color) which depends only on the Kelvin temperature Γ and not on the chemical composi­ tion, contents, or shape of the enclosure. We define the spectral density of the radiation ρ(ω, Τ) as the energy (per unit volume) of that radiation lying in the frequency (radians/sec) range between ω and ω + dœ. Figure 1-1 gives the spectral density at two different ρ (ω, Γ) ω Figure 1-1 The black-body spectral density at two different Kelvin temperatures temperatures. I t is the curves, of which these two are typical, that we shall try to deriv e from classical theory. Our failure in this regard will give us a better perspective o n the need for quantu m mechanics and at the same time will provide u s with an exercise in classical physics. Mathematically we have dê — ρ(ω, Τ) dœ. (1-1) The tota l energy density contained by all possible frequencies is (1-2) where Ε is the total energy and V is the volume of the cavity. Various attempts were made to explain the origin of the black-body spectrum. Wien suggested that a general form for the spectral density could I BLACK-BODY RADIATION 5 be derived by performing a thermodynamic process (Carnot cycle) on the radiation in the cavity. By taking the radiation as the working substance in the Carnot engine, he concluded that ρ(ω, Τ) must be of the general form 3 ρ(ω, Τ) = ω ^^ (1-3) where F is some function of the variable χ = ω/Τ. Thermodynamics alone could not determine the function F, but it would eliminate those theories that did not conform to Wien's law (1-3). Using Wien's law and (1-2), the total energy density within the cavity becomes Λ » /QJ\ -00 3 <r(T) = J Λ - |ώ = η x F(x)dx (1-4) where χ = ω/Τ. The integral (if it converg4es) implies that the total energy density within the cavity is proportional to T , that is, 4 S{T) = σ'Γ (l-4a) 3 where the constant is σ' = x F(x) dx. Equation (l-4a) relates the electro­ magnetic energy density to the Kelvin temperature. For isotropic radiation, the radiative flux J (energy crossing unit area per unit time) can in turn be related to the energy density using ia J = where c is the speed of light (see Problem 1-2). Equation (l-4a) becomes 4 J = \c&T* = σΤ . (1-5) This is the Stefan-Boltzmann law where the Stefan-Boltzmann constant is 00 3 c Λ σ = - xF(x) dx. 4 Jo 4 While σ was experimentally known to be σ = 0.567 χ 10" (cgs), F(x) remained to be determined theoretically. A second consequence of (1-3) is the "displacement" law which suggests that the dominant color (that is, the frequency at which ρ(ω, Τ) is a maxi­ mum) within the cavity is proportional to the temperature, that is, ω0 oc T. (1-6) This shift in frequency with temperature was also confirmed experimentally. Increasing the temperature of a substance produces a shift from "red hot" 6 1 HISTORICAL ASPECTS to "white hot," the whiteness indicating the presence of a bluish component. The frequency ω at which ρ is a maximum can be derived from (1-3) by 0 differentiation. Using the variable χ = ω/Τ we find 3 4 3 δωΡ(ω/Τ) = T (-^- xF(x)\ = 0 δω »0 4\U3X }x = x02 = T{x F(x) + 3x F(x)} = 0 0 0 0 0 or x F'(x )-3F(x ) = 0. (1-7) 0 0 0 For any "reasonable" function F(x) in Wien's law, (1-7) represents an ordinary equation which can be solved for x giving 0 x = — = const. 0 as the required displacement law (1-6). Of course, the value of the constant depends on the choice of F. In fact for certain functions, (1-7) has no solutions and the displacement law fails. Summarizing, we observe that Wien's law leads to a spectral density which agrees with both the Stefan-Boltzmann law and the displacement law. The constants associated with these laws depend on the particular function F, which cannot be determined using thermodynamics alone. It was in fact the search for the function F that led Planck to the discovery of quantum mechanics. We shall next apply the laws of mechanics and electromagnetism to various models in the hope of obtaining the function F(co/T). II Characteristic Modes within a Cavity In classical theory, electromagnetic radiation is composed of vibrating electric and magnetic fields Ε and B. Since black-body radiation within a cavity is independent of the shape and composition of the enclosure, no generality is lost and some mathematical simplification is gained by assuming the enclosure to be a large metallic cube. The laws of electromagnetism require that the fields satisfy the wave equations