Principles Of Quantum Mechanics

Cover......Page 1
Title......Page 3
ISBN 0-306-44790-8......Page 4
Dedications......Page 5
Preface to the First Edition......Page 9
Prelude......Page 13
Contents......Page 15
1.1. Linear Vector Spaces: Basics......Page 19
1.2. Inner Product Spaces......Page 25
1.3. Dual Spaces and the Dirac Notation......Page 29
1.3.1. Expansion of Vectors in an Orthonormal Basis......Page 30
1.3.2. Adjoint Operation......Page 31
Gram-Schmidt Theorem......Page 32
Schwarz and Triangle Inequalities......Page 34
1.4. Subspaces......Page 35
1.5. Linear Operators......Page 36
1.6. Matrix Elements of Linear Operators......Page 38
The Adjoint of an Operator......Page 43
Hennitian, Anti-Hermitian, and Unitary Operators......Page 45
1.7. Active and Passive Transformations......Page 47
1.8. The Eigenvalue Problem......Page 48
The Characteristic Equation and the Solution to the Eigenvalue Problem......Page 50
Degeneracy......Page 56
Diagonalization of Hermitian Matrices......Page 58
Simultaneous Diagonalization of Two Hermitian Operators......Page 61
The Propagator......Page 69
The Normal Modes......Page 70
1.9. Functions of Operators and Related Concepts......Page 72
Derivatives of Operators with Respect to Parameters......Page 73
1.10. Generalization to Infinite Dimensions......Page 75
Operators in Infinite Dimensions......Page 81
2.1. The Principle of Least Action and Lagrangian Mechanics......Page 93
2.2. The Electromagnetic Lagrangianz......Page 101
2.3. The Two-Body Problem......Page 103
2.5. The Hamiltonian Formalism......Page 104
2.6. The Electromagnetic Force in the Hamiltonian Scheme......Page 108
2.7. Cyclic Coordinates, Poisson Brackets, and Canonical Transformations......Page 109
Canonical Transformations......Page 110
Active Transformations......Page 115
2.8. Symmetries and Their Consequences......Page 116
A Useful Relation Between S and E......Page 122
3.1. Particles and Waves in Classical Physics......Page 125
3.2. An Experiment with Waves and Particles (Classical)......Page 126
3.3. The Double-Slit Experiment with Light......Page 128
3.5. Conclusions......Page 130
4.1. The Postulatest......Page 133
4.2. Discussion of Postulates 1-111......Page 134
Collapse of the State Vector......Page 140
How to Test Quantum Theory......Page 143
Expectation Value......Page 145
The Uncertainty......Page 146
Compatible and Incompatible Variables......Page 147
The Density Matrix-a Digression3......Page 151
Generalization to More Degrees of Freedom......Page 159
Setting Up the Schrodinger Equation......Page 161
General Approach to the Solution......Page 163
Choosing a Basis for Solving Schrodinger\'s Equation......Page 167
5.1. The Free Particle......Page 169
Time Evolution of the Gaussian Packet......Page 172
Some General Features of Energy Eigenfunctions......Page 174
5.2. The Particle in a Box......Page 175
5.3. The Continuity Equation for Probability......Page 182
5.4. The Single-Step Potential: A Problem in Scatteringt......Page 185
5.5. The Double-Slit Experiment......Page 193
5.6. Some Theorems......Page 194
06.\rThe Classical Limit......Page 197
7.1. Why Study the Harmonic Oscillator?......Page 203
7.2. Review of the Classical Oscillator......Page 206
7.3. Quantization of the Oscillator (Coordinate Basis)......Page 207
7.4. The Oscillator in the Energy Basis......Page 220
7.5. Passage from the Energy Basis to the X Basis......Page 234
8.1. The Path Integral Recipe......Page 241
8.2. Analysis of the Recipe......Page 242
8.3. An Approximation to U(t) for a Free Particle......Page 243
8.4. Path Integral Evaluation of the Free-Particle Propagator......Page 244
8.5. Equivalence to the Schrodinger Equation......Page 247
8.6. Potentials of the Form V = a + bx + cx2 + dx + exx8......Page 249
9.2. Derivation of the Uncertainty Relations......Page 255
9.3. The Minimum Uncertainty Packet......Page 257
9.4. Applications of the Uncertainty Principle......Page 259
9.5. The Energy-Time Uncertainty Relation......Page 263
The Two-Particle Hilbert Space......Page 265
VIB2 AS a Direct Product Space......Page 266
The Direct Product Revisited......Page 271
Evolution of the Two-Particle State Vector......Page 272
N Particles in One Dimension......Page 276
10.2. More Particles in More Dimensions......Page 277
The Classical Case......Page 278
Two-Particle Systems-Symmetric and Antisymmetric States......Page 279
Bosons and Fermions......Page 281
Bosonic and Fermionic Hilbert Spaces......Page 283
Determination of Particle Statistics......Page 287
Systems of N Identical Particles......Page 289
When Can We Ignore Symmetrization and Antisymmetrization?......Page 291
11.2. Translational Invariance in Quantum Theory......Page 297
Translation in Terms of Passive Transformations......Page 302
A Digression on the Analogy with Classical Mechanicst......Page 304
Finite Translations......Page 307
A Digression on Finite Canonical and Unitary Transformations$......Page 308
Implications of Translational Invariances......Page 310
11.3. Time Translational Invariance......Page 312
11.4. Parity Invariance......Page 315
11.5. Time-Reversal Symmetry......Page 319
12.1. Translations in Two Dimensions......Page 323
12.2. Rotations in Two Dimensions......Page 324
Explicit Construction of U[R]......Page 325
12.3. The Eigenvalue Problem of L,......Page 331
Solutions to Rotationally Invariant Problems......Page 333
12.4. Angular Momentum in Three Dimensions......Page 336
12.5. The Eigenvalue Problem of L* and L,......Page 339
Finite Rotations$......Page 347
Angular Momentum Eigenfunctions in the Coordinate Basis......Page 351
12.6. Solution of Rotationally Invariant Problems......Page 357
General Properties of UEl......Page 360
The Free Particle in Spherical Coordinates$......Page 364
Connection with the Solution in Cartesian Coordinates......Page 367
The Isotropic Oscillator......Page 369
13.1. The Eigenvalue Problem......Page 371
The Energy Levels......Page 372
The Wave Functions......Page 374
13.2. The Degeneracy of the Hydrogen Spectrum......Page 377
Numerical Estimates......Page 379
Comparison with Experiment......Page 384
13.4. Multielectron Atoms and the Periodic Table......Page 387
14.2. What is the Nature of Spin?......Page 391
14.3. Kinematics of Spin......Page 392
Explicit Forms of Rotation Operators......Page 401
14.4. Spin Dynamics......Page 403
Orbital Magnetic Moment in Quantum Theory......Page 405
Spin Magnetic Moment......Page 407
Paramagnetic Resonance......Page 410
Negative Absolute Temperature (Optional Digression)......Page 412
14.5. Return of Orbital Degrees of Freedom......Page 415
The Stern-Gerlach (SG) Experiment......Page 417
15.1. A Simple Example......Page 421
15.2. The General Problem......Page 426
Clebsch-Gordan (CG) Coefficients......Page 430
Addition of L and S......Page 432
The Modified Spectroscopic Notation......Page 433
15.3. Irreducible Tensor Operators......Page 434
Tensor Operators......Page 435
15.4. Explanation of Some \"Accidental\" Degeneracies......Page 439
Hydrogen......Page 440
The Oscillator......Page 441
The Free-Particle Solutions......Page 444
16.1. The Variational Method......Page 447
16.2. The Wentzel-Kramers-Brillouin Method......Page 453
Connection with the Path Integral Formalism......Page 456
Tunneling Amplitudes......Page 462
Bound States......Page 463
17.1. The Formali......Page 469
17.2. Some Examples......Page 472
Selection Rules......Page 476
17.3. Degenerate Perturbation Theory......Page 482
Fine Structure......Page 484
18.1. The Problem......Page 491
18.2. First-Order Perturbation Theory......Page 492
The Sudden Perturbation......Page 495
The Adiabatic Perturbation......Page 496
The Periodic Perturbation......Page 500
18.3. Higher Orders in Perturbation Theory$......Page 502
The Interaction Picture......Page 503
The Heisenberg Picture......Page 508
Classical Electrodynamics......Page 510
The Potentials in Quantum Theory......Page 514
Photoelectric Effect in Hydrogen......Page 517
Field Quantization$......Page 524
Spontaneous Decay......Page 535
19.1. Introduction......Page 541
19.2. Recapitulation of One-Dimensional Scattering and Overview......Page 542
19.3. The Born Approximation (Time-Dependent Description)......Page 547
19.4. Born Again (The Time-Independent Description)......Page 552
Validity of the Born Approximation......Page 561
19.5. The Partial Wave Expansion......Page 563
A Model Calculation of 6,: The Hard Sphere......Page 567
Resonances......Page 568
19.6. Two-Particle Scattering......Page 573
Passage to the Lab Frame......Page 577
Scattering of Identical Particles......Page 578
20.1. The Free-Particle Dirac Equation......Page 581
20.2. Electromagnetic Interaction of the Dirac Particle......Page 584
The Electron Spin and Magnetic Moment......Page 585
Hydrogen Fine Structure......Page 587
20.3. More on Relativistic Quantum Mechanics......Page 592
21.Path Integrals: Part II\r......Page 599
21.1. Derivation of the Path Integral......Page 600
The Landau Levels......Page 605
The Berry Phase......Page 610
Coherent State Path Integral......Page 625
21.2. Imaginary Time Formalism......Page 631
Path Integral for the Imaginary Time Propagator......Page 632
Tunneling by Path Integrals: Well, well!......Page 634
Spontaneous Symmetry Breaking......Page 638
Imaginary Time Path Integrals and Quantum Statistical Mechanics......Page 642
Relation to Classical Statistical Mechanics......Page 644
Spin Coherent States and Path Integral......Page 654
Fermion Oscillator and Coherent States......Page 658
The Fermionic Path Integral......Page 664
21.4. Summary......Page 670
Bibliography......Page 671
A.1. Matrix Inversion......Page 673
A.2. Gaussian Integrals......Page 677
A.3. Complex Numbers......Page 678
Answers to Selected Exercises......Page 683
Mnemonics for Hydrogen......Page 687
Index......Page 689