Quantum Mechanics

Preface v 

Chapter 1 Foundations 1 
1.1 Faraday, Thomson, and Electrons 2 
1.2 Spectra, Radiation, and Planck 4 
1.3 The Rutherford-Bohr Atom 14 
1.4 de Broglie Matter-Waves 23 
Summary 28 
Problems 29 

Chapter 2 Schrodinger's Equation 35 
2.1 The Classical Wave Equation 36 
2.2 The Time-Independent Schrodinger Equation 41 
2.3 The Time-Dependent Schrodinger Equation 45 
2.4 Interpretation of 1/1: Probabilities and Boundary Conditions 49 
Summary 56 
Problems 56 

Chapter 3 Solutions of Schrodinger's Equation in One Dimension 59 

Part I: Potential Wells 60 
3.1 Concept of a Potential Well 60 
3.2 The Infinite Potential Well 62 
3.3 The Finite Potential Well 70 
3.4 Finite Rectangular Well-Even Solutions 79 
3.5 Number of Bound States in a Finite Potential Well 81 
3.6 Sketching Wavefunctions 85 
Part II: Potential Barriers and Scattering 88
3.7 Potential Barriers 88 
3.8 Penetration of Arbitrarily Shaped Barriers 94 
3.9 Alpha-Decay as a Barrier Penetration Effect 97 
3.10 Scattering by One-Dimensional Potential Wells 103 
Summary 106 
Problems 108 

Chapter 4 Operators, Expectation Values, and the Uncertainty Principle 113 
4.1 Properties of Operators 114 
4.2 Expectation Values 118 
4.3 The Uncertainty Principle 126 
4.4 Commutators and Uncertainty Relations 131 
4.5 Ehrenfest's Theorem 134 
4.6 The Orthogonality Theorem 137 
4.7 The Superposition Theorem 139 
4.8 Constructing a Time-Dependent Wave Packet 141 
4.9 The Virial Theorem 146 
Summary 153 
Problems 154 

Chapter 5 The Harmonic Oscillator 159 
5.1 A Lesson in Dimensional Analysis 160 
5.2 The Asymptotic Solution 164 
5.3 The Series Solution 165 
5.4 Hermite Polynomials and Harmonic Oscillator Wavefunctions 172 
5.5 Comparing the Classical and Quantum Simple Harmonic Oscillators 176 
5.6 Raising and Lowering Operators (Optional) 179 
Summary 186 
Problems 187 

Chapter 6 Schrodinger's Equation in Three Dimensions and an Introduction to the Quantum Theory of Angular Momentum 191 
6.1 Separation of Variables: Cartesian Coordinates 192 
6.2 Spherical Coordinates 201 
6.3 Angular Momentum Operators 203 
6.4 Separation of Variables in Spherical Coordinates: Central Potentials 209 
6.5 Angular Wavefunctions and Spherical Harmonics 211 
6.5.1 Solution of the $\Phi$ Equation 212 
6.5.2 Solution of the e Equation 215 
6.5.3 Spherical Harmonics 220 
Summary 225 
Problems 227 

Chapter 7 Central Potentials 231 
7.1 Introduction 231 
7.2 The Infinite Spherical Well 234 
7.3 The Finite Spherical Well 236 
7.4 The Coulomb Potential 240 
7.5 Hydrogen Atom Probability Distributions 253 
7.5.1 The (1, 0, 0) State of Hydrogen 254 
7.5.2 The (2, 0, 0) and Other States of Hydrogen 257 
7.6 The Effective Potential 264 
7.7 Some Philosophical Remarks 266 
Summary 267 
Problems 268 

Chapter 8 Further Developments with Angular Momentum and Multiparticle Systems 275 
8.1 Angular Momentum Raising and Lowering Operators 275 
Section 8.1 Problems 282 
8.2 The Stern-Gerlach Experiment: Evidence for Quantized 
Angular Momentum and Electron Spin 282 
Section 8.2 Problems 287 
8.3 Diatomic Molecules and Angular Momentum 287 
Section 8.3 Problems 291 
8.4 Identical Particles, Indistinguishability, and the Pauli Exclusion Principle 291 
Section 8.4 Problems 297 

Chapter 9 Approximation Methods 299 
9.1 The WKB Method 299 
9.2 The Superposition Theorem Revisited 305 
9.3 Perturbation Theory 309 
9.4 The Variational Method 327 
9.5 Improving the Variational Method (Optional) 336 
Summary 340 
Problems 343 

Chapter 10 Numerical Solution of Schrodinger's Equation 351 
10.1 Atomic Units 352 
10.2 A Straightforward Numerical Integration Method 353 
Problems 365 

Chapter 11 A Sampling of Results from Time-Dependent Quantum Mechanics: Transition Rates and Probabilities 369 
11.1 Transition Frequencies 369 
11.2 Transition Rules 373 
11.3 The Sudden Approximation 374 
Summary 376 
Problems 376 

Appendix A Miscellaneous Derivations 379 
A.l Heisenberg's Uncertainty Principle 379 
A.2 Explicit Series Form for Associated Legendre Functions 384 
A.3 Proof that $Y_{\ell,-m}=(-1)^m Y_{\ell,m}*$ 386 
A.4 Radial Nodes in Hydrogen Wavefunctions 391 

Appendix B Answers to Selected Problems 393 

Appendix C Integrals and Trigonometric Identities 401 

Appendix D Physical Constants 405 

Notes and References 407 

Index 411