Quantum mechanics

374 Pages·2012·3.037 MB·English

by  Hecht, Eugene, Peleg, Yoav, Pnini, Reuven, Zaarur, Elyahu

Quantum
Mechanics
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Quantum
Mechanics
Second Edition
Yoav Peleg, Ph.D.
Reuven Pnini, Ph.D.
Elyahu Zaarur, M.Sc.
Eugene Hecht, Ph.D.
Schaum’s Outline Series
New York Chicago San Francisco Lisbon London
Madrid Mexico City Milan New Delhi San Juan
Singapore Sydney Toronto
YOAV PELEG, Ph.D., received his doctorate in physics from the Technion Institute of Technology in Haifa, Israel. He has published to date
dozens of articles, mostly in the area of general relativity and quantum cosmology. Currently he is working as a researcher with Motorola
Israel.
REUVEN PNINI, Ph.D., received his doctorate in physics from the Technion Institute of Technology in Haifa, Israel. He has published to
date several articles, mostly in the area of condensed matter physics. He is currently the Chief Scientifi c Editor of Rakefet Publishing Ltd.
ELYAHU ZAARUR, M.Sc., received his master of science in physics from the Technion Institute of Technology in Haifa, Israel. He has
published more than a dozen books on physics. He is currently the Managing Director of Rakefet Publishing Ltd.
EUGENE HECHT, Ph.D., is a full-time member of the Physics Department of Adelphi University in New York. He has authored nine books
including Optics, 4th edition, published by Addison Wesley, which has been the leading text in the fi eld, worldwide, for over three decades.
Dr. Hecht has also written Schaum’s Outline of Optics and Schaum’s Outline of College Physics.
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Preface
The main purpose of this second edition of Quantum Mechanics is to make an already fine book more usable
for the student reader. Accordingly, a great deal of effort has been given to simplifying and standardizing the
notation. For example, a number of modern QM textbooks now distinguish operators from other quantities
by placing a cap (^) over the corresponding symbol for the operator. This simple emendation can nonethe-
less be very helpful to the student and that practice has been adopted throughout this edition. Similarly
I have avoided using the same symbol to represent different quantities, inasmuch as this can be unduly
confusing. Wherever necessary, discussions have been extended and the prose has been clarified. The all-
but-unavoidable typographical and other minor first-edition errors have been corrected. Additionally, all of
the art has been redrawn to improve visual readability, content, clarity, and accuracy. A substantial number
of new introductory-level solved problems have been added to ensure that the student can gain a good grasp
of the basics before approaching a more challenging range of questions. Indeed, it is my intention to add
more such problems in future editions.
If you have any comments or suggestions, or favorite problems you’d like to share, send them along to
Prof. E. Hecht, Physics Department, Adelphi University, Garden City, NY 11530 or if you prefer e-mail,
[email protected].
EUGENE HECHT
Freeport, NY
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Contents
CHAPTER 1 Introduction 1
1.1 The Particle Nature of Electromagnetic Radiation 1.2 Quantum Particles
1.3 Wave Packets and the Uncertainty Relation
CHAPTER 2 Mathematical Background 13
2.1 The Complex Field C 2.2 Vector Spaces over C 2.3 Linear Operators
and Matrices 2.4 Eigenvectors and Eigenvalues 2.5 Fourier Series and the
Fourier Transform 2.6 The Dirac Delta Function
CHAPTER 3 The Schrödinger Equation and Its Applications 25
3.1 Wavefunctions of a Single Particle 3.2 The Schrödinger Equation
3.3 Particle in a Time-Independent Potential 3.4 Scalar Product of Wave-
functions: Operators 3.5 Probability Density and Probability Current
CHAPTER 4 The Foundations of Quantum Mechanics 61
4.1 Introduction 4.2 Postulates in Quantum Mechanics 4.3 Mean
Value and Root-Mean-Square Deviation 4.4 Commuting Observables
4.5 Function of an Operator 4.6 Hermitian Conjugation 4.7 Discrete and
Continuous State Spaces 4.8 Representations 4.9 The Time Evolution
4.10 Uncertainty Relations 4.11 The Schrödinger and Heisenberg Pictures
CHAPTER 5 Harmonic Oscillator 98
5.1 Introduction 5.2 The Hermite Polynomials 5.3 Two- and Three-
Dimensional Harmonic Oscillators 5.4 Operator Methods for a Harmonic
Oscillator
CHAPTER 6 Angular Momentum 117
6.1 Introduction 6.2 Commutation Relations 6.3 Lowering and Raising
Operators 6.4 Algebra of Angular Momentum 6.5 Differential Represen-
tations 6.6 Matrix Representation of an Angular Momentum 6.7 Spherical
Symmetry Potentials 6.8 Angular Momentum and Rotations
CHAPTER 7 Spin 145
7.1 Definitions 7.2 Spin 1/2 7.3 Pauli Matrices 7.4 Lowering and
Raising Operators 7.5 Rotations in the Spin Space 7.6 Interaction with
a Magnetic Field
vii
viii Contents
CHAPTER 8 Hydrogen-like Atoms 164
8.1 A Particle in a Central Potential 8.2 Two Interacting Particles 8.3 The
Hydrogen Atom 8.4 Energy Levels of the Hydrogen Atom 8.5 Mean
Value Expressions 8.6 Hydrogen-like Atoms
CHAPTER 9 Particle Motion in an Electromagnetic Field 179
9.1 The Electromagnetic Field and Its Associated Potentials 9.2 The Hamil-
tonian of a Particle in the Electromagnetic Field 9.3 Probability Density and
Probability Current 9.4 The Magnetic Moment 9.5 Units
CHAPTER 10 Solution Methods in Quantum Mechanics—Part A 204
10.1 Time-Independent Perturbation Theory 10.2 Perturbation of a Non-
degenerate Level 10.3 Perturbation of a Degenerate State 10.4 Time-
Dependent Perturbation Theory
CHAPTER 11 Solution Methods in Quantum Mechanics—Part B 232
11.1 The Variational Method 11.2 Semiclassical Approximation (The WKB
Approximation)
CHAPTER 12 Numerical Methods in Quantum Mechanics 249
12.1 Numerical Quadrature 12.2 Roots 12.3 Integration of Ordinary
Differential Equations
CHAPTER 13 Identical Particles 264
13.1 Introduction 13.2 Permutations and Symmetries of Wavefunctions
13.3 Bosons and Fermions
CHAPTER 14 Addition of Angular Momenta 273
14.1 Introduction 14.2 {Jˆ2,Jˆ2,Jˆ2,Jˆ } Basis 14.3 Clebsch–Gordan
1 2 z
Coefficients
CHAPTER 15 Scattering Theory 296
15.1 Cross Section 15.2 Stationary Scattering States 15.3 Born
Approximation 15.4 Partial Wave Expansions 15.5 Scattering of Identical
Particles
CHAPTER 16 Semiclassical Treatment of Radiation 330
16.1 The Interaction of Radiation with Atomic Systems 16.2 Time-
Dependent Perturbation Theory 16.3 Transition Rate 16.4 Multipole
Transitions 16.5 Spontaneous Emission
Contents ix
APPENDIX Mathematical Appendix 347
A.1 Fourier Series and Fourier Transform A.2 The Dirac d-Function
A.3 Hermite Polynomials A.4 Legendre Polynomials A.5 Associated
Legendre Functions A.6 Spherical Harmonics A.7 Associated Laguerre
Polynomials A.8 Spherical Bessel Functions
Index 355