1 Classical Mechanics 1
1.1 Newton's Laws,the Action,and the Hamiltonian 1
1.1.1 Newton's Law and Lagrange's Equations 1
1.1.2 Hamilton's Principle 2
1.1.3 Canonical Momenta and the Hamiltonian Formulation 4
1.2 Classical Space-Time Symmetries 6
1.2.1 The Space-Time Transformations 6
1.2.2 Translations 8
1.2.3 Rotations 9
1.2.4 Rotation Matrices 11
1.2.5 Symmetries and Conservation Laws 12
Problems 14
2 Fundamentals of Quantum Mechanics 19
2.1 The Superposition Principle 19
2.1.1 The Double-Slit Experiment 19
2.1.2 The Stern-Gerlach Experiment 22
2.2 The Mathematical Language of Quantum Mechanics 24
2.2.1 Vector Spaces 24
2.2.2 The Probability Interpretation 27
2.2.3 Linear Operators 27
2.2.4 Observables 30
2.2.5 Examples 31
2.3 Continuous Eigenvalues 35
2.3.1 The Dirac Delta Function 35
2.3.2 Continuous Observables 36
2.3.3 Fourier's Theorem and Representations of δ(x) 37
2.4 Canonical Commutators and the Schr？dinger Equation 38
2.4.1 The Correspondence Principle 38
2.4.2 The Canonical Commutation Relations 42
2.4.3 Planck's Constant 43
2.5 Quantum Dynamics 44
2.5.1 The Time-Translation Operator 44
2.5.2 The Heisenberg Picture 45
2.6 The Uncertainty Principle 47
2.7 Wave Functions 49
2.7.1 Wave Functions in Coordinate Space 49
2.7.2 Momentum and Translations 49
2.7.3 Schr？dinger's Wave Equation 52
2.7.4 Time-Dependent Free Particle Wave Functions 53
Problems 55
3 Stationary States 62
3.1 Elementary Examples 62
3.1.1 States with Definite Energy 62
3.1.2 A Two-State System 63
3.1.3 One-Dimensional Potential Problems 66
3.2 The Harmonic Oscillator 68
3.2.1 The Spectrum 69
3.2.2 Matrix Elements 71
3.2.3 The Ground-State Energy 72
3.2.4 Wave Functions 73
3.3 Spherically Symmetric Potentials and Angular Momentum 74
3.3.1 Spherical Symmetry 74
3.3.2 Orbital Angular Momentum as a Difierential Operator 75
3.3.3 The Angular Momentum Commutator Algebra 76
3.3.4 Classification of the States 80
3.4 Spherically Symmetric Potentials:Wave Functions 80
3.4.1 Spherical Coordinates and Spherical Harmonics 80
3.4.2 The Radial Wave Equation 82
3.5 Hydrogenlike Atoms 84
3.5.1 The Symmetries 84
3.5.2 The Energy Spectrum 86
3.5.3 The Radial Wave Functions 88
Problems 91
4 Symmetry Transformations on States 102
4.1 Introduction 102
4.1.1 Symmetries and Transformations 102
4.1.2 Groups of Transformations 103
4.1.3 Classical and Quantum Symmetries 105
4.2 The Rotation Group and Algebra 105
4.2.1 Representations of Groups 105
4.2.2 Representations of the Generators of Rotations 106
4.2.3 Generators in an Arbitrary Direction 107
4.2.4 Commutators of the Generators 108
4.2.5 Explicit Form of the Finite Dimensional Representations 111
4.2.6 Summary 112
4.3 Spin and Rotations in Quantum Mechanics 113
4.3.1 Rotations and Spinless Particles 113
4.3.2 Spin 114
4.3.3 The Spin-Zero Representation 116
4.3.4 The Spin-Half Representation 116
4.3.5 Euler Angles 117
4.3.6 The Spin-One Representation 119
4.3.7 Arbitrary j 119
4.4 Addition of Angular Momenta 120
4.4.1 Spin and Orbital Angular Momentum 120
4.4.2 Two Simple Examples 121
4.5 Clebsch-Gordan Coefficients 122
4.5.1 Definition of the Coefficients 122
4.5.2 Spin Half+Spin Half 123
4.5.3 Spin Half+Angular Momentum One 125
4.5.4 Spin Half+Angular Momentum l 126
4.5.5 The General Rule 127
4.5.6 Recursion Relation for the Coefficients 129
4.5.7 The Clebsch-Gordan Series 129
Problems 130
5 Symmetry Transformations on Operators 138
5.1 Vector Observables 138
5.1.1 Symmetries,Lifetimes,and Selection Rules 138
5.1.2 Vector Operators Under Rotations 140
5.1.3 Spherical Components of Vector Observables 141
5.1.4 Selection Rules for Matrix Elements ofVectors 142
5.2 Tensor Observables 144
5.2.1 Cartesian Tensor Operators 144
5.2.2 Spherical Tensor Components 146
5.2.3 Higher Rank Spherical Tensors 147
5.2.4 Selection Rules and the Wigner-Eckart Theorem 148
5.3 Discrete Symmetries 151
5.3.1 Reflections and Parity 151
5.3.2 Reversal of the Direction of Motion 153
5.3.3 Identical Particles 157
5.4 Internal Symmetries:Isospin 159
Problems 164
6 Interlude 171
6.1 External Magnetic Fields 171
6.1.1 Natural Units 171
6.1.2 Gauge Invariance 172
6.1.3 Constant Magnetic Fields and Landau Levels 173
6.1.4 Magnetic Moment 176
6.1.5 The Hydrogen Atom in a Magnetic Field 177
6.2 The Density Matrix 178
6.2.1 Definition 178
6.2.2 Example:Thermodynamic Equilibrium 180
6.2.3 Example:Spin-Half Systems 181
6.2.4 Spin Magnetic Resonance 182
6.3 Neutrino Interference 185
6.3.1 Neutrinos 185
6.3.2 Neutrino Mixing 186
6.3.3 Neutrino Oscillations and the Mass Splitting 187
6.3.4 Solar Neutrinos 189
6.3.5 Neutrino Oscillations in Matter 189
6.4 Measurements in Quantum Mechanics 191
6.4.1 Wave-Function Collapse 191
6.4.2 The EPR Paradox 191
6.4.3 Bell's Inequality 193
Problems 195
7 Approximation Methods for Bound States 202
7.1 Bound-State Perturbation Theory 202
7.1.1 The Perturbation Expansion 202
7.1.2 Example:Harmonic Oscillator 205
7.2 Static External Electric Fields 206
7.2.1 Perturbation of the First Excited Level 207
7.2.2 Polarizabilitv of the Ground State 208
7.3 Fine Structure of the Hydrogen Atom 212
7.3.1 The Spin-Orbit Coupling 212
7.3.2 Correction to Energy Levels 214
7.3.3 The Relativistic Kinetic Energy Correction 215
7.3.4 The Fine Structure of the Hydrogen Atom 216
7.3.5 External Magnetic Field Again 217
7.3.6 The Hyperfine Structure of the Hydrogen Atom 218
7.4 Other Atoms 220
7.4.1 The Ground State of Helium 220
7.4.2 The Perturbation Method for the Helium Atom 222
7.5 The Variational Method 223
7.5.1 The General Method 223
7.5.2 The Helium Atom 225
7.5.3 The Eigenvalue-Variational Scheme 228
7.6 Molecules 229
7.6.1 The Born-Oppenheimer Approximation 229
7.6.2 The Hydrogen Molecular Ion 233
7.7 The WKB Method 237
7.7.1 Turning Points and Connection Formulas 238
7.7.2 The Linear Approximation 240
7.7.3 Bound States 243
7.7.4 Tunneling through a Barrier 247
Problems 248
8 Potential Scattering 264
8.1 Introduction 264
8.1.1 Kinematics of Scattering 264
8.1.2 Scattering and Wave Functions 265
8.2 The Scattering Amplitude 269
8.2.1 Equation for the Scattering Amplitude 269
8.2.2 The Born Series 270
8.2.3 Spherically Symmetric Potentials 270
8.2.4 The Optical Theorem 272
8.2.5 The Refractive Index 273
8.3 Partial Waves 275
8.3.1 Expansion of a Plane Wave in a Legendre Series 276
8.3.2 Partial Wave Expansion of？(r) 278
8.3.3 Calculation of the Phase Shift 280
8.4 The Radial Wave Function 281
8.4.1 The Integral Equation 281
8.4.2 Partial Wave Green's Functions 281
8.4.3 Scattering by an Impenetrable Sphere 283
Problems 284
9 Transitions 288
9.1 Transitions in an External Field 288
9.1.1 Time-Dependent Perturbations 288
9.1.2 The Semiclassical Method 288
9.2 The Transition Matrix 291
9.2.1 The Transition Matrix 292
9.2.2 The Lippmann-Schwinger Equation 295
9.2.3 Relation to the Scattering Amplitude 297
9.3 Scattering and Cross Sections 298
9.3.1 The Scattering Matrix 298
9.3.2 The Transition Probability 299
9.3.3 Cross Sections 300
9.3.4 Scattering of Electrons by Atoms 302
9.3.5 Scattering with Recoil 303
9.3.6 Identical Particle Scattering 305
9.4 Decays of Excited States 307
9.4.1 Lowest-Order Transition Rates 308
9.4.2 Time Dependence of the Initial State 310
9.4.3 Distribution of the Final States 314
Problems 316
10 Further Topics in Quantum Dynamics 324
10.1 Path Integration 324
10.1.1 The Propagator as an Integral over Paths 324
10.1.2 The Free Particle Propagator 327
10.1.3 The Harmonic Oscillator 328
10.1.4 The Euclidean Formalism 331
10.1.5 The Ground-State Energy 332
10.2 Path Integration:Some Applications 334
10.2.1 The Born Series 334
10.2.2 External Fields and Gauge Invariance 337
10.2.3 The Aharonov-Bohm Effect 338
10.3 Berry's Phase 341
10.3.1 Origin of the Phase 341
10.3.2 Example:Electron in a Precessing Magnetic Field 343
10.3.3 The General Formula 344
10.3.4 Two States near a Degeneracy 347
10.3.5 Fast and Slow Coordinates 348
10.3.6 The Aharonov-Bohm Effect Again 351
Problems 352
11 The Quantized Electromagnetic Field 356
11.1 The Classical Electromagnetic Field Hamiltonian 356
11.1.1 Maxwell's Equations and the Transverse Gauge Condition 356
11.1.2 The Independent Modes 358
11.1.3 The Classical Hamiltonian 360
11.1.4 The Canonical Coordinates 361
11.2 The Quantized Radiation Field 362
11.2.1 The Heisenberg Picture 362
11.2.2 Canonical Quantization 362
11.2.3 Photons 364
11.3 Properties of the Quantum Electromagnetic Field 365
11.3.1 The Momentum of the Field 365
11.3.2 The Angular Momentum of the Field 366
11.3.3 The Photon Spin 367
11.4 Electromagnetic Decays of Excited States 369
11.4.1 The Unperturbed Hamiltonian 369
11.4.2 The Vector Potential Interaction 369
11.4.3 The Spin Interaction 370
11.4.4 The Rate for Photon Emission 370
11.4.5 Multipole Matrix Elements 372
11.5 Examples 373
11.5.1 Decay of the 2P State of Atomic Hydrogen 373
11.5.2 Hyperfine Emission 374
11.6 Absorption and Stimulated Emission of Radiation 378
11.6.1 Periodic Boundary Conditions 378
11.6.2 Absorption 379
11.6.3 Stimulated Emission 381
11.6.4 The Blackbody Formula 382
11.7 Scattering of Photons by Atoms 383
11.7.1 The Photoelectric Effect 383
11.7.2 Elastic Scattering of Photons 385
11.7.3 Scattering by a Free Electron 391
11.8 The Casimir Effect 394
11.8.1 The Ground-State Energy of the Electromagnetic Field 394
11.8.2 The Casimir Force with an Elementary Cutoff 397
11.8.3 The General Calculation 400
Problems 402
12 Relativistic Wave Equations 407
12.1 Lorentz Transformations 407
12.1.1 Four-Vectors and Tensors 407
12.1.2 Lorentz Transformations 408
12.1.3 Spin 411
12.2 Vector and Scalar Fields 413
12.2.1 The Electromagnetic Field 413
12.2.2 The Klein-Gordon Equation 414
12.3 Relativistic Spin-Half Equations 417
12.3.1 Two Component Spin-Half Equations 417
12.3.2 The Dirac Equation 420
12.3.3 Free Particle Solutions 421
12.3.4 Probability Current and Hole Theory 423
12.4 Dirac Electron in an Electromagnetic Field 423
12.4.1 Second-Order Form of the Dirac Equation 423
12.4.2 The Gyromagnetic Ratio 424
12.4.3 The Nonrelativistic Limit and the Fine Structure 425
12.5 The Dirac Hydrogen Atom 428
12.5.1 Second-Order Equation 428
12.5.2 Spherically Symmetric Potentials 431
12.5.3 Radial Equations 432
12.5.4 The Hydrogen Atom 433
Problems 435
13 Identical Particles 438
13.1 Nonrelativistic Identical-Particle Systems 438
13.1.1 Creation and Annihilation Operators for Bosons 438
13.1.2 Creation and Annihilation Operators for Fermions 443
13.2 Elementary Applications 444
13.2.1 Ideal Gas Distributions 444
13.2.2 Ideal Electron Gas 446
13.2.3 Collapsed Stars 448
13.3 Relativistic Spinless Particles 453
13.3.1 The Neutral Scalar Field 453
13.3.2 The Classical Theory 453
13.3.3 The Quantum Theory 455
13.3.4 Charged Particles 456
13.4 The Quantized Dirac Field 458
13.4.1 The Dirac Action 458
13.4.2 The Plane Wave Expansion 459
13.5 Interacting Relativistic Fields 462
13.5.1 Normal Ordering 462
13.5.2 Example:The φ4 interaction 463
Problems 465
APPENDICES 470
A Mathematical Tools 470
A.1 Miscellaneous Tools 470
A.1.1 The Dirac Delta Function 470
A.1.2 The Levi-Civita Symbol 472
A.1.3 Some Integrals 472
A.1.4 The Trapezoidal Approximation Series 474
A.2 Special Functions 476
A.2.1 Gamma Function 476
A.2.2 Legendre Polynomials 479
A.2.3 Solutions to the Free Radial Equation 484
A.2.4 Hermite Polynomials 487
A.2.5 Bessel Functions 489
A.3 Orthogonal Curvilinear Coordinates 491
A.3.1 Vector Calculus in Orthogonal Curvilinear Coordinates 491
A.3.2 Hydrogen Atom in Parabolic Coordinates 495
A.3.3 Elliptic Coordinates 498
B Rotation Matrices 500
B.1 Rotation Matrices—Ⅰ 500
B.1.1 Rotation Matrices and Spherical Harmonics 500
B.1.2 The Explicit Form of the Rotation Matrices 502
B.1.3 The Projection Theorem 507
B.2 Rotation Matrices—Ⅱ 508
B.2.1 Averages over Products of Rotation Matrices 508
B.2.2 The Wigner-Eckart Theorem Again 510
C SU(3) 512
C.1 The Group and Algebra 512
C.2 Some Representations 513
D References 516
Index 519