After a consideration of basic quantum mechanics, this introduction aims at a side by side treatment of fundamental applications of the Schrödinger equation on the one hand and the applications of the path integral on the other. Different from traditional texts and using a systematic perturbation method, the solution of Schrödinger equations includes also those with anharmonic oscillator potentials, periodic potentials, screened Coulomb potentials and a typical singular potential, as well as the investigation of the large order behavior of the perturbation series. On the path integral side, after introduction of the basic ideas, the expansion around classical configurations in Euclidean time, such as instantons, is considered, and the method is applied in particular to anharmonic oscillator and periodic potentials. Numerous other aspects are treated on the way, thus providing the reader an instructive overview over diverse quantum mechanical phenomena, e.g. many! other potentials, Green’s functions, comparison with WKB, calculation of lifetimes and sojourn times, derivation of generating functions, the Coulomb problem in various coordinates, etc. All calculations are given in detail, so that the reader can follow every step.
Author(s): Harald J. W. Muller-Kirsten
Publisher: World Scientific Pub Co (
Year: 2003
Language: English
Pages: 826
Contents ......Page 6
Preface ......Page 16
1.1 Origin and Discovery of Quantum Mechanics ......Page 21
1.2 Contradicting Discretization: Uncertainties ......Page 27
1.3 Particle-Wave Dualism ......Page 32
1.4 Particle-Wave Dualism and Uncertainties ......Page 34
1.4.1 Further thought experiments ......Page 37
1.5 Bohr's Complementarity Principle ......Page 39
1.6 Further Examples ......Page 40
2.2 The Hamilton Formalism ......Page 43
2.3.1 Single particle consideration ......Page 49
2.3.2 Ensemble consideration ......Page 51
2.4 Expectation Values of Observables ......Page 54
2.5 Extension beyond Classical Mechanics ......Page 58
3.2 Hilbert Spaces ......Page 61
3.3 Operators in Hilbert Space ......Page 69
3.4 Linear Functionals and Distributions ......Page 73
3.4.1 Interpretation of distributions in physics ......Page 74
3.4.2 Properties of functionals and the delta distribution ......Page 75
4.1 Introductory Remarks ......Page 79
4.2 Ket and Bra States ......Page 80
4.3 Linear Operators Hermitian Operators ......Page 82
4.4 Observables ......Page 88
4.5 Representation Spaces and Basis Vectors ......Page 91
5.2 The Density Matrix ......Page 93
5.3 The Probability Density p(x t) ......Page 97
5.4 Schrodinger Equation and Liouville Equation ......Page 98
5.4.1 Evaluation of the density matrix ......Page 100
6.1 Introductory Remarks ......Page 103
6.2 The One-Dimensional Linear Oscillator ......Page 104
6.3 The Energy Representation of the Oscillator ......Page 110
6.4 The Configuration Space Representation ......Page 111
6.5.1 Derivation of the generating function ......Page 118
7.2 Time-dependent and Time-independent Cases ......Page 125
7.3 The Green's Function of a Free Particle ......Page 131
7.4 Green's Function of the Harmonic Oscillator ......Page 133
7.5.1 Wave packets ......Page 138
7.5.2 A particle's sojourn time T at the maximum ......Page 143
8.1 Introductory Remarks ......Page 149
8.2 Asymptotic Series versus Convergent Series ......Page 150
8.2.1 The error function and Stokes discontinuities ......Page 153
8.2.2 Stokes discontinuities of oscillator functions ......Page 159
8.3 Asymptotic Series from Differential Equations ......Page 163
8.4 Formal Definition of Asymptotic Expansions ......Page 166
8.5 Rayleigh-Schrodinger Perturbation Theory ......Page 167
8.6 Degenerate Perturbation Theory ......Page 172
8.7 Dingle-Miiller Perturbation Method ......Page 175
9.2 Reconsideration of Electrodynamics ......Page 181
9.3 Schrodinger and Heisenberg Pictures ......Page 186
9.4 The Liouville Equation ......Page 187
10.2 States and Observables ......Page 189
10.2.1 Uncertainty relation for observables A B ......Page 190
10.3 One-Dimensional Systems ......Page 193
10.3.1 The translation operator U(a) ......Page 196
10.4 Equations of Motion ......Page 198
10.5 States of Finite Lifetime ......Page 204
10.6 The Interaction Picture ......Page 205
10.7 Time-Dependent Perturbation Theory ......Page 209
10.8 Transitions into the Continuum ......Page 211
10.9 General Time-Dependent Method ......Page 215
11.2 Separation of Variables Angular Momentum ......Page 219
11.2.1 Separation of variables ......Page 225
11.3 Representation of Rotation Group ......Page 226
11.4 Angular Momentum:Angular Representation ......Page 230
11.5 Radial Equation for Hydrogen-like Atoms ......Page 233
11.6.1 The eigenvalues ......Page 235
11.6.2 Laguerre polynomials: Various definitions in use! ......Page 239
11.6.3 The eigenfunctions ......Page 243
11.6.4 Hydrogen-like atoms in parabolic coordinates ......Page 247
11.7 Continuous Spectrum of Coulomb Potential ......Page 254
11.7.1 The Rutherford formula ......Page 257
11.8 Scattering of a Wave Packet ......Page 259
11.9 Scattering Phase and Partial Waves ......Page 263
12.1 Introductory Remarks ......Page 269
12.2 Continuity Equation and Conditions ......Page 270
12.3 The Short-Range Delta Potential ......Page 271
12.4 Scattering from a Potential Well ......Page 274
12.5 Degenerate Potentials and Tunneling ......Page 279
13.2 The Freely Falling Particle: Quantization ......Page 285
13.2.1 Superposition of de Broglie waves ......Page 286
13.2.2 Probability distribution at large times ......Page 290
13.3 Stationary States ......Page 292
13.4 The Saddle Point or Stationary Phase Method ......Page 296
14.1 Introductory Remarks ......Page 301
14.2 Classical Limit and Hydrodynamics Analogy ......Page 302
14.3.1 The approximate WKB solutions ......Page 306
14.3.2 Turning points and matching of WKB solutions ......Page 310
14.3.3 Linear approximation and matching ......Page 313
14.4 Bohr-Sommerfeld-Wilson Quantization ......Page 317
14.5 Further Examples ......Page 321
15.1 Introductory Remarks ......Page 327
15.2 The Power Potential ......Page 328
15.3 The Three-Dimensional Wave Function ......Page 335
16.1 Introductory Remarks ......Page 339
16.2 Regge Trajectories ......Page 342
16.3 The S-Matrix ......Page 348
16.4 The Energy Expansion ......Page 349
16.5 The Sommerfeld-Watson Transform ......Page 350
16.6 Concluding Remarks ......Page 356
17.1 Introductory Remarks ......Page 359
17.2.1 The Floquet exponent ......Page 361
17.2.2 Four types of periodic solutions ......Page 370
17.3.1 Preliminary remarks ......Page 373
17.3.2 The solutions ......Page 374
17.3.3 The eigenvalues ......Page 381
17.3.4 The level splitting ......Page 383
17.4.1 Introduction ......Page 391
17.4.2 Solutions and eigenvalues ......Page 393
17.4.3 The level splitting ......Page 395
17.4.4 Reduction to Mathieu functions ......Page 397
17.5 Concluding Remarks ......Page 398
18.1 Introductory Remarks ......Page 399
18.2.1 Defining the problem ......Page 402
18.2.2 Three pairs of solutions ......Page 404
18.2.3 Matching of solutions ......Page 411
18.2.4 Boundary conditions at the origin ......Page 413
18.2.5 Boundary conditions at infinity ......Page 416
18.2.6 The complex eigenvalues ......Page 422
18.3.1 Defining the problem ......Page 425
18.3.2 Three pairs of solutions ......Page 427
18.3.3 Matching of solutions ......Page 432
18.3.4 Boundary conditions at the minima ......Page 434
18.3.5 Boundary conditions at the origin ......Page 437
18.3.6 Eigenvalues and level splitting ......Page 444
18.3.7 General Remarks ......Page 447
19.1 Introductory Remarks ......Page 455
19.2.1 Preliminary considerations ......Page 456
19.2.2 Small h2 solutions in terms of Bessel functions ......Page 458
19.2.3 Small h2 solutions in terms of hyperbolic functions ......Page 461
19.2.4 Notation and properties of solutions ......Page 462
19.2.5 Derivation of the S-matrix ......Page 466
19.2.6 Evaluation of the S-matrix ......Page 475
19.2.7 Calculation of the absorptivity ......Page 478
19.3.1 Preliminary remarks ......Page 480
19.3.2 The Floquet exponent for large h2 ......Page 481
19.3.3 Construction of large-h2 solutions ......Page 484
19.3.4 The connection formulas ......Page 486
19.3.5 Derivation of the S-matrix ......Page 488
19.4 Concluding Remarks ......Page 490
20.1 Introductory Remarks ......Page 491
20.2 Cosine Potential: Large Order Behaviour ......Page 496
20.3.1 The decaying ground state ......Page 499
20.3.2 Decaying excited states ......Page 506
20.3.3 Relating the level splitting to imaginary E ......Page 513
20.3.4 Recalculation of large order behaviour ......Page 514
20.4 Cosine Potential: A Different Calculation ......Page 515
20.5.1 The inverted double well ......Page 520
20.5.2 The double well ......Page 521
20.6 General Remarks ......Page 522
21.1 Introductory Remarks ......Page 523
21.2 Path Integrals and Green's Functions ......Page 524
21.3.1 Configuration space representation ......Page 530
21.3.2 Momentum space represenation ......Page 533
21.4 Including V in First Order Perturbation ......Page 534
21.5 Rederivation of the Rutherford Formula ......Page 538
21.6 Path Integrals in Dirac's Notation ......Page 544
21.7 Canonical Quantization from Path Integrals ......Page 553
22.1 Introductory Remarks ......Page 557
22.2 The Constant Classical Field ......Page 559
22.3 Soliton Theories in One Spatial Dimension ......Page 564
22.4 Stability of Classical Configurations ......Page 569
22.5 Bogomol'nyi Equations and Bounds ......Page 574
22.6 The Small Fluctuation Equation ......Page 577
22.7 Existence of Finite-Energy Solutions ......Page 584
22.8 Ginzburg-Landau Vortices ......Page 590
22.9 Introduction to Homotopy Classes ......Page 594
22.10 The Fundamental Homotopy Group ......Page 599
23.2 Instantons and Anti-Instantons ......Page 603
23.3 The Level Difference ......Page 612
23.4.1 The fluctuation equation ......Page 616
23.4.2 Evaluation of the functional integral ......Page 623
23.4.3 The Faddeev-Popov constraint insertion ......Page 629
23.4.4 The single instanton contribution ......Page 633
23.4.5 Instanton-anti-instanton contributions ......Page 634
23.5 Concluding Remarks ......Page 638
24.1 Introductory Remarks ......Page 639
24.2 The Bounce in a Simple Example ......Page 645
24.3.1 The bounce solution ......Page 651
24.3.2 The single bounce contribution ......Page 655
24.3.3 Evaluation of the single bounce kernel ......Page 657
24.3.4 Sum over an infinite number of bounces ......Page 661
24.4 Inverted Double Well: Constant Solutions ......Page 664
24.5 The Cubic Potential and its Complex Energy ......Page 665
25.1 Introductory Remarks ......Page 669
25.2.1 Periodic configurations ......Page 670
25.2.2 The fluctuation equation ......Page 679
25.2.3 The limit of infinite period ......Page 683
25.3.1 Periodic configurations ......Page 684
25.3.2 The fluctuation equation ......Page 687
25.3.3 The limit of infinite period ......Page 689
25.4.1 Periodic configurations ......Page 690
25.4.2 The fluctuation equation ......Page 691
25.5 Conclusions ......Page 693
26.1 Introductory Remarks ......Page 695
26.2.1 Periodic configurations and the double well ......Page 696
26.2.2 Transition amplitude and Feynman kernel ......Page 698
26.2.3 Fluctuations about the periodic instanton ......Page 699
26.2.4 The single periodic instanton contribution ......Page 704
26.2.5 Sum over instanton-anti-instanton pairs ......Page 708
26.3.1 Periodic configurations and the cosine potential ......Page 710
26.3.2 Transition amplitude and Feynman kernel ......Page 713
26.3.3 The fluctuation equation and its eigenmodes ......Page 714
26.3.4 The single periodic instanton contribution ......Page 716
26.3.5 Sum over instanton-anti-instanton pairs ......Page 720
26.4.1 Periodic configurations and the inverted double well ......Page 722
26.4.2 Transition amplitude and Feynman kernel ......Page 725
26.4.3 The fluctuation equation and its eigenmodes ......Page 726
26.4.4 The single periodic bounce contribution ......Page 728
26.4.5 Summing over the infinite number of bounces ......Page 730
26.5 Concluding Remarks ......Page 734
27.1 Introductory Remarks ......Page 735
27.2 Constraints: How they arise ......Page 737
27.2.1 Singular Lagrangians ......Page 740
27.3 The Hamiltonian of Singular Systems ......Page 743
27.4 Persistence of Constraints in Course of Time ......Page 746
27.5 Constraints as Generators of a Gauge Group ......Page 747
27.6 Gauge Fixing and Dirac Quantization ......Page 754
27.7 The Formalism of Dirac Quantization ......Page 756
27.8 Dirac Quantization of Free Electrodynamics ......Page 760
27.9.1 The method of Faddeev and Jackiw ......Page 765
28.1 Introductory Remarks ......Page 773
28.2 Relating Period to Temperature ......Page 775
28.3 Crossover in Previous Cases ......Page 776
28.3.1 The double well and phase transitions ......Page 777
28.3.2 The cosine potential and phase transitions ......Page 779
28.4 Crossover in a Simple Spin Model ......Page 780
28.5 Concluding Remarks ......Page 791
29 Summarizing Remarks ......Page 793
Appendix A Properties of Jacobian Elliptic Functions ......Page 795
Bibliography ......Page 799
Index ......Page 817
