Quantum Mechanics

Quantum Mechanics

by John L. Powell, Bernd Crasemann

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Overview

Suitable for advanced undergraduates, this thorough text focuses on the role of symmetry operations and the essentially algebraic structure of quantum-mechanical theory. Based on courses in quantum mechanics taught by the authors, the treatment provides numerous problems that require applications of theory and serve to supplement the textual material.
Starting with a historical introduction to the origins of quantum theory, the book advances to discussions of the foundations of wave mechanics, wave packets and the uncertainty principle, and an examination of the Schrödinger equation that includes a selection of one-dimensional problems. Subsequent topics include operators and eigenfunctions, scattering theory, matrix mechanics, angular momentum and spin, and perturbation theory. The text concludes with a brief treatment of identical particles and a helpful Appendix.

Product Details

ISBN-13: 9780486794594
Publisher: Dover Publications
Publication date: 06/17/2015
Series: Dover Books on Physics Series
Pages: 512
Product dimensions: 8.90(w) x 6.00(h) x 1.20(d)

About the Author

John L. Powell and Bernd Crasemann were Professors of Physics at the University of Oregon.

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Quantum Mechanics


By John L. Powell, Bernd Crasemann

Dover Publications, Inc.

Copyright © 1989 John L. Powell and Bernd Crasemann
All rights reserved.
ISBN: 978-0-486-80478-1



CHAPTER 1

HISTORICAL ORIGINS OF THE QUANTUM THEORY


1–1 Difficulties with classical models. Classical physics deals primarily with macroscopic phenomena. Most of the effects with which classical theory is concerned are either directly observable or can be made observable with relatively simple instruments. There is a close link between the world of classical physics and the world of sensory perception.

During the first decades of the present century, physicists turned their attention to the study of atomic systems, which are inherently inaccessible to direct observation. It soon became clear that the concepts and methods of classical macroscopic physics could not be applied directly to atomic phenomena. If classical laws of physics are to be applied at all to systems of atomic size, they can be considered only in connection with models of such systems. Models are usually conceived as systems of particles which interact with one another and with electromagnetic radiation according to assumed simple laws. One attempts to construct a microscopic model which will reproduce, as nearly as possible, observed macroscopic effects.

Many of the observed characteristics of atomic systems, however, are such that they cannot be reproduced by any model which behaves according to classical laws. The early development of atomic theory consisted of efforts to overcome these difficulties by modifying the laws of classical physics and the properties of the models to which they were applied. These efforts reached their successful conclusion in the period from 1925 to 1930, when an entirely new theoretical discipline, quantum mechanics, was developed by Schrödinger, Heisenberg, Dirac, and others. The present book is an introduction to this theory. The first chapter is a review of the history of the subject and provides a background for later work. Some of the problems which must be faced in an attempt to understand atomic phenomena will be pointed out.

The earliest evidence of the need for revision of classical concepts came from the field of chemistry. It had long been realized that the molecules of which a pure substance is composed are chemically identical, and that they retain their identity over long periods of time. A molecule of nitrogen, for example, consists of a number of positively and negatively charged particles which are held together by electrostatic forces. However, it is stated by Earnshaw's theorem that a system of charged particles cannot remain at rest in stable equilibrium under the influence of purely electrostatic forces. If a classical picture is adopted, these particles must, therefore, be in relative motion. Yet, this motion must be such as not to destroy the identity of the molecules, and must persist indefinitely. If the particles are confined to a restricted region of space, they must frequently or continuously change their direction of motion, i.e., be accelerated. It is, however, a well-known fact that accelerated charged particles radiate energy in the form of electromagnetic waves. Hence, the particles in a molecule should progressively change their state of motion in accordance with this loss of energy — a conclusion which does not agree with observation. This fact alone shows that the stability of molecules cannot be understood on the basis of a classical model.

The enormous range of electrical conductivities of solid materials is an example of a property of matter which cannot be reasonably explained in terms of classical ideas. For instance, the conductivity of silver is more than 1024 times larger than that of fused quartz. Electrical conduction presumably consists of a relative motion of the negatively and positively charged components of the material under the influence of an applied electric field. It is not possible to comprehend, in terms of a classical picture, how such motion occurs readily in silver, but not at all in quartz. It seems to be necessary to recognize the existence of a principle of selection which prohibits the motion in quartz, but not in silver. A similar situation is found in ferromagnetism: the magnetic susceptibility of iron is observed to be of the order of 109 times larger than that of other metals. Quite generally, any attempts to understand the chemical behavior of matter, as summarized in the periodic table of the elements, must take into account the fact that not all the states of motion permitted by a classical model are accessible to the systems of particles which comprise the molecules of a chemical substance.


1–2 Optical spectra. Light is emitted by substances which are raised to a high temperature or subjected to an electrical discharge. This light can be separated into its spectral components by means of a diffraction grating, and the wavelengths of the various components can be measured with great precision. Characteristic emission spectra, consisting of discrete frequencies or lines, are obtained in this way for each element.

The emission of this electromagnetic radiation must be associated with the accelerated motion of the charged particles in the atoms of the emitting substance. An attempt to construct a classical model reproducing the observed frequencies naturally leads to the conclusion that these must be the same as the frequencies of the periodic motions of the particles. The spectral lines should therefore fall into groups within which the frequencies v are related by the harmonic law where v1, v2, v3, ... are the fundamental frequencies of the motions of the atomic system, and n1, n2, n3, ... are integers denoting harmonics of these frequencies. Such a relationship does not describe the experimental facts. Rather, spectral lines occur in series, such as the Balmer series in the visible and near ultraviolet region of the hydrogen spectrum, which is reproduced in Fig. 1–1. Within a series, the lines are most widely spaced on the long wavelength side and are crowded together near the short wavelength series limit.

v = n1v1 + n2v2 + n3v3 + ···,

The empirically observed regularities among spectral lines are expressed by the Ritz combination principle, first formulated in 1908. According to this principle, a small set of numbers or spectroscopic terms can be assigned to the atoms of a substance, such that the wave numbers of the numerous spectral lines are equal to differences between terms:

[??]12 = T1 - T2. (1–1)

Classical physics does not tell how the terms arise, nor does it explain the combination principle.


1–3 Blackbody radiation. The concept that atomic systems exchange radiant energy in discrete amounts, or quanta, rather than in a continuous way, was first enunciated by Max Planck in 1901, in his theoretical development of the law that describes the frequency distribution of heat radiation. The spectrum of the radiation emitted by a hot object is continuous; experimentally, it is known to cover a wide frequency range, with maximum intensity at a wavelength which depends upon the temperature of the radiating body.

It can be shown by thermodynamic reasoning that the spectral energy distribution of the radiation within an enclosure at constant temperature is independent of the shape and material of the enclosure (provided it is large enough), and is a function of temperature and frequency alone. The radiation within such an enclosure can be observed through a small hole in its wall, which is a perfect absorber for radiation incident from without and therefore acts as a black body.

The electromagnetic radiation can be considered to be a superposition of harmonic waves that correspond to the various normal modes of oscillation or degrees of freedom within the enclosed space. The frequency distribution of the radiation can be obtained if it is possible to enumerate the normal modes and determine the energy associated with each.

Consider the latter problem first. It is shown in statistical mechanics that, at temperature T, the probability of excitation of a mode whose energy lies in the range between E and E + dE is proportional to the Boltzmann factor exp (–E/kT) dE. Accordingly, the energy associated with each mode is, on the average,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–2)

This is an expression of the classical law of equipartition of energy.

The number of degrees of freedom can be determined by considering the enclosure as a large cubical cell with sides of length L. A closer examination of the problem shows that the shape of the enclosure is immaterial to the result. It can also be shown that the particular form of the boundary conditions at the walls does not matter. Assume, therefore, that the field is periodic with period L in each direction parallel to an edge of the cube, i.e., that an integral number of wavelengths must fit into the length L. There are two modes of oscillation (one for each independent direction of polarization) corresponding to each plane wave exp (ik·r) for which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–3)

Here, k is the propagation vector for the wave, with components kx, ky, and kz and of magnitude k = 2π/λ, where λ is the wavelength. Hence, there are two degrees of freedom for each triplet of integers (nx, ny, nz). These integers may conveniently be exhibited as a set of lattice points in (nx, ny, nz)-space, as shown in Fig. 1–2. There is one such point per unit volume of this space.

Now let n2 = n2x + n2y + n2z Then n is the radius of a sphere in (nx, ny, nz)-space, and the number of lattice points between n and n + dn is equal to the volume of a shell of radius n and thickness dn, that is, 4πn2dn. It follows from the definition of n and from the conditions (1–3) that k = (2π/L)n; hence the total number of modes for which k is in the range (k, k + dk) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–4)

where the factor 2 has been introduced to take account of the two possible directions of polarization of the transverse electromagnetic waves. This result may be rewritten in terms of the frequency v = kc/2π and divided by the volume L3, showing that the number of degrees of freedom per unit volume with frequency between v and v + dv is 8πv2dv/c3.

Now energy kT is to be associated with each degree of freedom, and the energy per unit volume within the frequency range dv is therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–5)

This is the Rayleigh-Jeans law, which results from the application of the classical equipartition theorem. The law agrees well with experimental results at low frequency (Fig. 1–3), but predicts a monotonic increase of the energy density uv with increasing frequency. The integral that represents the total energy density diverges:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–6)

This divergence at high frequencies has been called the ultraviolet catastrophe. Attempts to secure a more satisfactory law for the frequency distribution of blackbody radiation on a classical basis have failed.

The equipartition theorem applies only if the modes of oscillation in the cavity are continuously distributed as to energy. Planck's quantum hypothesis consisted in the assumption that the radiation oscillators are not excited to a continuum of energy states, but that these oscillators exchange energy with their surroundings in discrete units , so that the only available energy states are those for which E = 0, ε, 2ε, ..., etc. On this hypothesis, the average energy of an oscillator is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–7)

Using the formulae

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

we obtain for the average oscillator energy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–8)

In the limit ε [right arrow] 0, or at very high temperature, this result reduces to kT, in agreement with the value obtained from the equipartition theorem.

At the time of Planck's work, Wien had already shown on thermodynamic grounds that the energy density uv must have the functional form v3f(v/T). This requirement is satisfied if one sets

ε = hv, (1–9)

where h is a universal constant. If the expression 8πv2dv/c3 is introduced for the number of modes in the interval (v, v + dv), Planck's law results:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–10)

This law is in agreement with experiment. The avoidance of the ultraviolet catastrophe on the basis of the quantum partition law is easily understood, since the available states are now widely separated in energy when ? is large, and can be reached only by the absorption of very high-energy quanta, a relatively rare occurrence.

The constant h (Planck's constant), as well as the Boltzmann constant k, can be evaluated by comparison with experiment. The total energy density, obtained by integrating Planck's distribution function over the whole spectrum, must depend on the temperature in accordance with Stefan's law:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–11)

If the integral is evaluated, using Eq. (1–10) for the energy density, one obtains for the value of Stefan's constant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–12)

Furthermore, Planck's law must yield a maximum energy density at the wavelength predicted by Wien's law:

λmaxT = b. (1–13)

The experimental values for the Stefan-Boltzmann constant σ and for Wien's constant b then determine the value of k and of Planck's constant,

h = 6.625 × 10-27 erg·sec.

Additional insight into the significance of the quantum hypothesis is afforded by an alternative derivation of the radiation law, due to Einstein. Detailed reference to the electromagnetic radiation theory is avoided in this treatment, and the role played by statistical considerations is emphasized. Suppose that two states of energy E1 and E2 (E2 >E1) are available to an atom bathed in radiation of density uv. It is assumed that quantum jumps between these states can take place in three different ways: (1) spontaneous transitions, with the probability A12 that the atom jumps from state 2 to state 1, emitting a quantum, (2) absorption, with the probability B21uv that the atomic state changes from 1 to 2 with absorption of a quantum, and (3) induced emission, with probability B12uv, in which the state changes from 2 to 1 under the influence of the incident radiation. In thermal equilibrium, the probabilities that states 1 and 2 are occupied are proportional to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Therefore the equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1–14)

expresses the condition of equilibrium, namely,

(number of transitions from 2 to 1) = (number of transitions from 1 to 2)


(Continues...)

Excerpted from Quantum Mechanics by John L. Powell, Bernd Crasemann. Copyright © 1989 John L. Powell and Bernd Crasemann. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Chapter 1 Historical Origins of the Quantum Theory 1

1-1 Difficulties with classical models 1

1-2 Optical spectra 2

1-3 Blackbody radiation 3

1-4 The photoelectric effect 9

1-5 The Franck-Hertz experiment 11

1-6 The Rutherford atom 13

1-7 Stationary states of atoms 14

1-8 The correspondence principle 15

1-9 The Bohr atom 16

1-10 Spectroscopic series 17

1-11 Correction for finite mass of the nucleus 19

1-12 Quantization of the phase integral 21

1-13 Elliptic electron orbits 22

1-14 The particle in a box 26

1-15 The rigid rotator 27

1-16 The harmonic oscillator 29

1-17 Shortcomings of the old quantum theory 31

Chapter 2 Foundations of Wave Mechanics 35

2-1 Photons as particles: the Compton effect 35

2-2 Particle diffraction 38

2-3 Elements of Fourier analysis 41

2-4 Parseval's formula and the Fourier integral theorem 43

2-5 Fourier transforms; examples 47

2-6 Superposition of plane waves; time dependence 51

2-7 Wave packets and the Einstein-de Broglie relations 52

2-8 Wave functions for a free particle; the Schrödinger equation 56

2-9 Physical interpretation of the Schrödinger wave function 59

2-10 Expectation of a dynamical quantity 63

Chapter 3 Wave Packets and the Uncertainty Principle 69

3-1 Uncertainty of position and momentum 69

3-2 Exact statement and proof of the uncertainty principle for wave packets 71

3-3 An example of position-momentum uncertainty 73

3-4 Energy-time uncertainty 75

3-5 Monochromatic waves 75

3-6 The gaussian wave packet 77

3-7 Spread of the gaussian packet with time 79

3-8 General solution for time dependence of Ψ causality 82

Chapter 4 The Schrödinger Equation 86

4-1 Interaction among particles 86

4-2 Geometrical optics 89

4-3 Analogy between optics and mechanics 93

4-4 Principle of superposition of states 95

4-5 Probability current 96

4-6 Motion of wave packets 98

Chapter 5 Problems in One Dimension 102

5-1 Potential step 102

5-2 Potential barrier 107

5-3 Rectangular potential well 109

5-4 Degeneracy. Qualitative characteristics of the wave function 113

5-5 Theory of the Schrödinger equation. Linear independence 116

5-6 Properties of the zeros of Ψ Sturm's theorem 119

5-7 Bound states 122

5-8 Orthogonality 126

5-9 The linear harmonic oscillator 127

5-10 Hermite polynomials 132

5-11 Oscillator wave functions 135

5-12 Parity 139

5-13 The Wentzel-Kramers-Brillouin approximation 140

5-14 Penetration of a potential barrier by WKB approximation 147

Chapter 6 Operators and Eigenfunctions 158

6-1 Linear operators 158

6-2 Eigenfunctions and eigenvalues 161

6-3 The operator formalism in quantum mechanics 161

6-4 The operator (h/i)(d/dx) 163

6-5 Orthogonal systems 164

6-6 Expansion in eigenfunctions 166

6-7 Hermitian operators 169

6-8 Simultaneous eigenfunctions; commutators 171

6-9 The parity operator 175

6-10 The fundamental commutation rule 178

6-11 Equations of motion 181

6-12 Commutation rules and the uncertainty principle 182

6-13 Remark on the correspondence principle 184

Chapter 7 Spherically Symmetric Systems 188

7-1 The Schrödinger equation for spherically symmetric potentials 188

7-2 Spherical harmonics 190

7-3 Degeneracy; angular momentum 200

7-4 The three-dimensional harmonic oscillator 205

7-5 Laguerre polynomials 211

7-6 Many-particle systems 216

7-7 The hydrogen atom 220

Chapter 8 Theory of Scattering 237

8-1 General remarks 237

8-2 Wave functions for a free particle in spherical polar coordinates 238

8-3 Expansion of a plane wave in spherical harmonics 245

8-4 Scattering by a short-range central field 247

8-5 Method of partial waves 250

8-6 Dependence of δ1 on l and E 254

8-7 Low-energy scattering 257

8-8 Scattering by an attractive square potential well 261

8-9 Theoretical connection between scattering and bound states; S-matrix 264

8-10 General formulation of scattering theory 266

8-11 Green's function 268

8-12 Integral equation for the wave function 270

8-13 Connection with partial-wave analysis 271

8-14 The Born approximation 272

8-15 Atomic scattering of electrons 274

Chapter 9 Matrix Mechanics 282

9-1 Introduction 282

9-2 Linear vector spaces 283

9-3 Orthonormai systems 286

9-4 Linear transformations 287

9-5 Matrices 290

9-6 Change of basis 296

9-7 Hermitian operators; diagonalization 299

9-8 Degenerate matrices 308

9-9 Infinite-dimensional spaces 310

9-10 Hilbert space 311

9-11 Abbreviated notation for matrix elements 315

9-12 Involutions and projection operators 316

9-13 Restatement of quantum-mechanical assumptions 319

9-14 The one-dimensional harmonic oscillator in matrix mechanics 321

Chapter 10 Angular Momentum and Spin 333

10-1 Rotations in three-dimensional space; angular momentum 333

10-2 Rotations of the coordinate frame 337

10-3 Diagonalization of J2 and Jz 340

10-4 Explicit forms of the angular momentum matrices 344

10-5 Effect of a magnetic field 345

10-6 The classical Zeeman effect 351

10-7 Quantum theory of the normal Zeeman effect 353

10-8 Electron spin 355

10-9 Electronic states in a central field 360

10-10 Addition of angular momenta 361

10-11 The p-states of an electron 364

10-12 Spin states for two particles of spin one-half 365

10-13 Other operators. Selection rules 368

Chapter 11 Perturbation Theory 381

11-1 Introduction 381

11-2 Perturbation of nondegenerate stationary states 381

11-3 Example: enharmonic oscillator 387

11-4 Perturbation of degenerate stationary states 390

11-5 Atomic Zeeman levels; Russell-Saunders-coupling 394

11-6 The variational method 396

11-7 Time-dependent perturbations; transition probability 401

11-8 Constant perturbation 404

11-9 Transitions to the continuum 406

11-10 The Born approximation 408

11-11 Perturbation, harmonic in time. Radiative transitions 414

11-12 Atomic radiation 420

11-13 Zeeman effect-polarization 425

11-14 Forbidden transitions. Higher multipoles 436

Chapter 12 Identical Particles 446

12-1 Principle of indistinguishability of identical particles 446

12-2 Statistics of identical particles 452

12-3 The helium atom 454

12-4 The Pauli exclusion principle 459

12-5 Scattering of identical particles 462

Appendix 471

A-1 Complex integration and the theory of residues 471

A-2 Parseval's formula 476

A-3 Schwarz's inequality 477

A-4 Functional transformations; Dirac delta function 478

A-5 The eikonal equation in geometrical optics 482

A-6 Derivation of Eq. (4-11) from Maxwell's equations 484

A-7 Exponential form for unitary matrices and derivation of Eq. (11-21) 484

A-8 Physical constants and conversion factors 486

Index 487

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