Quantum Mechanics THIRD EDITION

EUGEN MERZBACHER University of North Carolina at Chapel Hill

JOHN WILEY & SONS, INC. New York / Chichester / Weinheim

Brisbane / Singapore / Toronto

Contents

CHAPTER i Introduction to Quantum Mechanics 1

1. Quantum Theory and the Wave Nature of Matter 1

2. The Wave Function and its Meaning 4

Problems 10

CHAPTER 2 Wave Packets, Free Particle Motion, and the Wave Equation 12 1. The Principle of Superposition 12

2. Wave Packets and the Uncertainty Relations 14

3. Motion of a Wave Packet 18

4. The Uncertainty Relations and the Spreading of Wave Packets 20

5. The Wave Equation for Free Particle Motion 22

Problems 24

CHAPTER 3 The Schrödinger Equation, the Wave Function, and Operator Algebra 25 1. The Wave Equation and the Interpretation of tfi 25

2. Probabilities in Coordinate and Momentum Space 29

3. Operators and Expectation Values of Dynamical Variables 34

4. Commutators and Operator Algebra 38

5. Stationary States and General Solutions of the Wave Equation 41

6. The Virial Theorem 47

Problems 49

CHAPTER 4 The Principles of Wave Mechanics 51

1. Hermitian Operators, their Eigenfunctions and Eigenvalues 51

2. The Superposition and Completeness of Eigenstates 57

3. The Continuous Spectrum and Closure 60

4. A Familiär Example: The Momentum Eigenfunctions and the Free Particle 62

5. Unitary Operators. The Displacement Operator 68

6. The Charged Particle in an External Electromagnetic Field and Gauge Invariance 71

7. Galilean Transformation and Gauge Invariance 75

Problems 78

CHAPTER 5 The Linear Harmonie Oscillator 79

1. Preliminary Remarks 79

2. Eigenvalues and Eigenfunctions 81

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3. Study of the Eigenfunctions 84

4. The Motion of Wave Packets 89

Problems 90

CHAPTER 6 Sectionally Constant Potentials in One Dimension 92

1. The Potential Step 92

2. The Rectangular Potential Barrier 97

3. Symmetries and Invariance Properties 99

4. The Square Well 103

Problems 111

CHAPTER 7 The WKB Approximation 113

1. The Method 113

2. The Connection Formulas 116

3. Application to Bound States 121

4. Transmission Through a Barrier 125

5. Motion of a Wave Packet and Exponential Decay 131

Problems 134

CHAPTER 8 Variational Methods and Simple Perturbation Theory 135

1. The Calculus of Variations in Quantum Mechanics 135

2. The Rayleigh-Ritz Trial Function 139

3. Perturbation Theory of the Schrödinger Equation 142 4. The Rayleigh-Ritz Method with Nonorthogonal Basis Functions 146 5. The Double Oscillator 149 6. The Molecular Approximation 159

7. The Periodic Potential 165 Problems 176

CHAPTER 9 Vector Spaces in Quantum Mechanics 179

1. Probability Amplitudes and Their Composition 179

2. Vectors and Inner Products 186 3. Operators 188

4. The Vector Space of Quantum Mechanics and the Bra-Ket Notation 195 5. Change of Basis 199

6. Hubert Space and the Coordinate Representation 202

Problems 206

CHAPTER IO Eigenvalues and Eigenvectors of Operators, the Uncertainty Relations, and the Harmonie Oscillator 207 1. The Eigenvalue Problem for Normal Operators 207 2. The Calculation of Eigenvalues and the Construction of

Eigenvectors 209

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3. Variational Formulation of the Eigenvalue Problem for a Bounded Hermitian Operator 212

4. Commuting Observables and Simultaneous Measurements 214 5. The Heisenberg Uncertainty Relations 217

6. The Harmonie Oscillator 220 7. Coherent States and Squeezed States 225 Problems 231

CHAPTER 11 Angular Momentum in Quantum Mechanics 233

1. Orbital Angular Momentum 233

2. Algebraic Approach to the Angular Momentum Eigenvalue Problem 238 3. Eigenvalue Problem for Lz and L2. 242 4. Spherical Harmonics 248 5. Angular Momentum and Kinetic Energy 252 Problems 255

CHAPTER 12 Spherically Symmetrie Potentials 256

1. Reduction of the Central-Force Problem 256 2. The Free Particle as a Central-Force Problem 257 3. The Spherical Square Well Potential 262 4. The Radial Equation and the Boundary Conditions 263 5. The Coulomb Potential 265 6. The Bound-State Energy Eigenfunctions for the Coulomb Potential 270 Problems 275

CHAPTER 13 Scattering 278

1. The Cross Section 278 2. The Scattering of a Wave Packet 286 3. Green's Functions in Scattering Theory 290

4. The Born Approximation 295 5. Partial Waves and Phase Shifts 298 6. Determination of the Phase Shifts and Scattering Resonances 302 7. Phase Shifts and Green's Functions 308 8. Scattering in a Coulomb Field 310 Problems 314

CHAPTER U The Principles of Quantum Dynamics 315

1. The Evolution of Probability Amplitudes and the Time Development Operator 315

2. The Pictures of Quantum Dynamics 319

3. The Quantization Postulates for a Particle 323 4. Canonical Quantization and Constants of the Motion 326

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5. Canonical Quantization in the Heisenberg Picture 331

6. The Forced Harmonie Oscillator 335

Problems 342

CHAPTER 15 The Quantum Dynamics of a Particle 344

1. The Coordinate and Momentum Representations 344

2. The Propagator in the Coordinate Representation 348

3. Feynman's Path Integral Formulation of Quantum Dynamics 355

4. Quantum Dynamics in Direct Product Spaces and Multiparticle Systems 358

5. The Density Operator, the Density Matrix, Measurement, and Information 363

Problems 370

CHAPTER 16 The Spin 372

1. Intrinsic Angular Momentum and the Polarization of ip waves 372

2. The Quantum Mechanical Description of the Spin 377

3. Spin and Rotations 381

4. The Spin Operators, Pauli Matrices, and Spin Angular Momentum 385

5. Quantum Dynamics of a Spin System 390

6. Density Matrix and Spin Polarization 392

7. Polarization and Scattering 399

8. Measurements, Probabilities, and Information 403

Problems 408

CHAPTER n Rotations and Other Symmetry Operations 410

1. The Euclidean Principle of Relativity and State Vector Transformations 410

2. The Rotation Operator, Angular Momentum, and Conservation Laws 413

3. Symmetry Groups and Group Representations 416

4. The Representations of the Rotation Group 421

5. The Addition of Angular Momenta 426

6. The Clebsch-Gordan Series 431

7. Tensor Operators and the Wigner-Eckart Theorem 432

8. Applications of the Wigner-Eckart Theorem 437

9. Refiection Symmetry, Parity, and Time Reversal 439

10. Local Gauge Symmetry 444

Problems 448

CHAPTER 18 Bound-State Perturbation Theory 451

1. The Perturbation Method 451

2. Inhomogeneous Linear Equations 453

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3. Solution of the Perturbation Equations 455 4. Electrostatic Polarization and the Dipole Moment 459 5. Degenerate Perturbation Theory 463 6. Applications to Atoms 467 7. The Variational Method and Perturbation Theory 473 8. The Helium Atom 476 Problems 480

CHAPTER 19 Time-Dependent Perturbation Theory 482

1. The Equation of Motion in the Interaction Picture 482

2. The Perturbation Method 485 3. Coulomb Excitation and Sum Rules 487 4. The Atom in a Radiation Field 491 5. The Absorption Cross Section 495

6. The Photoelectric Effect 501 7. The Golden Rule for Constant Transition Rates 503 8. Exponential Decay and Zeno's Paradox 510 Problems 515

CHAPTER 20 The Formal Theory of Scattering 517

1. The Equations of Motion, the Transition Matrix, the S Matrix, and the Cross Section 517

2. The Integral Equations of Scattering Theory 521

3. Properties of the Scattering States 525 4. Properties of the Scattering Matrix 527 5. Rotational Invariance, Time Reversal Symmetry, and the S Matrix 530 6. The Optical Theorem 532 Problems 533

CHAPTER 21 Identical Particles 535

1. The Indistinguishability of and the State Vector Space for Identical Particles 535

2. Creation and Annihilation Operators 538 3. The Algebra of Creation and Annihilation Operators 540 4. Dynamical Variables 544

5. The Continuous One-Particle Spectrum and Quantum Field Operators 546

6. Quantum Dynamics of Identical Particles 549 Problems 552

CHAPTER 22 Applications to Many-Body Systems 555

1. Angular Momentum of a System of Identical Particles 555 2. Angular Momentum and Spin One-Half Boson Operators 556

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3. First-Order Perturbation Theory in Many-Body Systems 558

4. The Hartree-Fock Method 560

5. Quantum Statistics and Thermodynamics 564

Problems 567

CHAPTER 23 Photons and the Electromagnetic Field 569

1. Fundamental Notions 569

2. Energy, Momentum, and Angular Momentum of the Radiation Field 573

3. Interaction with Charged Particles 576

4. Elements of Quantum Optics 580

5. Coherence, Interference, and Statistical Properties of the Field 583

Problems 591

CHAPTER 24 Relativistic Electron Theory 592

1. The Electron-Positron Field 592

2. The Dirac Equation 596

3. Relativistic Invariance 600

4. Solutions of the Free Field Dirac Equation 606

5. Charge Conjugation, Time Reversal, and the PCT Theorem 608

6. The One-Particle Approximation 613

7. Dirac Theory in the Heisenberg picture 617

8. Dirac Theory in the Schrödinger Picture and the Nonrelativistic Limit 621

9. Central Forces and the Hydrogen Atom 623

Problems 629

APPENDIX 630

1. Fourier Analysis and Delta Functions 630

2. Review of Probability Concepts 634

3. Curvilinear Coordinates 638

4. Units and Physical Constants 640

REFERENCES 642

INDEX 647