Advances in Quantum Mechanics

Cover......Page 1
Section 1 The Classical-Quantum Correspondence......Page 2
©......Page 3
Contents......Page 6
Preface......Page 10
1.1. Conjugate variables......Page 14
1.2. Conjugate coordinate systems......Page 15
1.3. The interplay between conjugate variables......Page 16
1.4. Conjugate motions......Page 17
1.5. Poisson brackets and commutators......Page 19
1.6. The commutator as a derivation and its consequences......Page 21
2. Quantum systems......Page 23
3. Time evolution using energy and time eigenstates......Page 27
References......Page 31
Chapter 2 Classical and Quantum Correspondence in Anisotropic Kepler Problem......Page 34
Chapter 3 Charathéodory’s “Royal Road” to the Calculus of Variations: A Possible Bridge Between Classical and Quantum Physics......Page 52
Chapter 4 The Improvement of the Heisenberg Uncertainty Principle......Page 78
Section 2 The Schrödinger Equation......Page 90
Chapter 5 Schrödinger Equation as a Hamiltonian System, Essential Nonlinearity, Dynamical Scalar Product and some Ideas of Decoherence......Page 92
Chapter 6 Schrödinger Equation and (Future) Quantum Physics......Page 116
Chapter 7 Quantum Damped Harmonic Oscillator......Page 144
Section 3 Path Integrals......Page 168
Chapter 8 The Schwinger Action Principle and Its Applications to Quantum Mechanics......Page 170
1. Introduction......Page 194
2. Generalized path integral......Page 197
2.1. Expansion of the path integral......Page 200
2.1.2. Temperature is not zero......Page 202
2.2. Paths through N-slit grating......Page 204
2.2.2. Definition of new working parameters......Page 206
3. Wave function behind the grating......Page 207
4. Bohmian trajectories and variance of momenta and positions along paths......Page 212
4.1. Dispersion of trajectories and the uncertainty principle......Page 215
5. Concluding remarks......Page 218
Acknowledgement......Page 219
References......Page 220
Chapter 10 Quantum Intentionality and Determination of Realities in the Space-Time Through Path Integrals and Their Integral Transforms......Page 224
Section 4 Perturbation Theory......Page 256
Chapter 11 Convergence of the Neumann Series for the Schrödinger Equation and General Volterra Equations in Banach Spaces......Page 258
1. Introduction......Page 280
2. Quantum correction term......Page 281
2.1. Wigner-Kirkwood expansion......Page 282
2.2. Free energy......Page 286
3. Framework......Page 287
3.1. Potentials......Page 292
4. Results......Page 294
5. Conclusion......Page 298
6. Applications......Page 299
References......Page 300
1. Introduction......Page 304
2.1. Heisenberg’ equation of motion and Ehrenfest theorem......Page 305
2.2. Quantized Hamilton dynamics and quantal cumulant dynamics......Page 306
2.3. Energy conservation law and least uncertainty state......Page 309
2.4. Distribution function and joint distribution......Page 311
2.5. Extension to multi-dimensional systems......Page 313
3.1. Application to molecular vibration......Page 315
3.2. Proton transfer reaction in guanine-cytosine base pair......Page 318
3.3. Quantal structural transition of finite clusters......Page 320
4. Summary......Page 323
Author details......Page 324
References......Page 325
Chapter 14 Unruh Radiation via WKB Method......Page 328
Section 5 Foundations of Quantum Mechanics......Page 344
Chapter 15 A Basis for Statistical Theory and Quantum Theory......Page 346
Chapter 16 Relational Quantum Mechanics......Page 372
1. Introduction......Page 382
2. More about photons......Page 385
3. On the arrow of time......Page 387
4. More about the hydrogen atom......Page 389
5. Quantum chemistry......Page 394
6. Conclusion......Page 403
References......Page 404
1. Introduction......Page 406
2. The universal computational principle's D2 A–causal computation......Page 408
3. The exhaustiveness of the universal consciousness principle for all natural phenomena......Page 410
4. The universal computational principle's paradigmatic shift: Transcending cartesian dualism......Page 420
5. The CUFT's sixth postulate: 'Ontological relativism'......Page 424
6. The seventh postulate: “Universal Consciousness Spectrum (UCS)”......Page 426
7. Critical Predictions of the 'Universal Consciousness Spectrum'......Page 427
8. The scientific implications of the CUFT's universal consciousness principle......Page 428
8.1. A singular 'A–causal' universal consciousness principle computation of all inductive and deductive 'X–Y' relationships......Page 429
8.2. The "universal consciousness principle's computational program"......Page 431
9. Theoretical ramifications of the universal consciousness principle......Page 435
10. The CUFT's eighth postulate: The 'universal consciousness reality'......Page 438
References......Page 446
Chapter 19 Emergent un-Quantum Mechanics......Page 448
Chapter 20 The Wigner-Heisenberg Algebra in Quantum Mechanics......Page 488
Chapter 21 New System-Specific Coherent States by Supersymmetric Quantum Mechanics for Bound State Calculations......Page 510
Section 6 Quantization and Entanglement......Page 530
Chapter 22 Quantum Dating Market......Page 532
1. Introduction......Page 554
2.1. Euler’s axioms......Page 555
2.2. Eulerian principles of change of state for single bodies......Page 556
2.3. Eulerian principles of change of state for Hamiltonian systems......Page 557
3. Quantization as selection problem — I. Derivation of the stationary Schrödinger equation......Page 558
3.1.2. Einsteinian selection problem between CM and non-CM......Page 559
3.2. Non-classical representation of the potential and kinetic energies......Page 560
3.3. The stationary Schrödinger equation......Page 561
4.1. The linear oscillator......Page 563
4.2. The mathematically distinguished solutions......Page 564
4.3. The physically distinguished solutions......Page 565
4.4. The non-classical potential energy and the tunnel effect......Page 566
5.1. The time dependence of the stationary states......Page 567
5.2. The equation-of-state-change......Page 568
6. Summary and conclusions......Page 569
Author details......Page 571
References......Page 572
Chapter 24 Entanglement, Nonlocality, Superluminal Signaling and Cloning......Page 576
Chapter 25 The Husimi Distribution: Development and Applications......Page 606
Section 7 Quantum Information and Related Topics......Page 632
2. Quasiclosed ensembles......Page 634
3. Adiabatic approximation......Page 636
4. Estimation of the number G(0) of different quasiclosed ensembles......Page 638
5. Model spectrum for the description of configurational excitations......Page 641
6. On the temperature ranges of melting and softening......Page 644
7. Admissible states......Page 645
8. Conclusions......Page 647
References......Page 649
1. Introduction......Page 652
2. Information-theoretical measures and complexities......Page 656
3. Aminoacids......Page 659
3.1. Physical and information-theoretical properties......Page 661
3.2. Homochirality......Page 664
4.1. Codons......Page 666
4.2. Physical and information-theoretical properties......Page 668
5. Concluding remarks......Page 675
References......Page 676
1. Introduction......Page 682
2. A singular 'a-causal' universal consciousness principle computation of all inductive and deductive 'x-y' relationships......Page 683
3. The "universal consciousness principle's computational program"......Page 685
4. Theoretical ramifications of the universal consciousness principle......Page 689
References......Page 692
Chapter 29 Shannon Informational Entropies and Chemical Reactivity......Page 694
Chapter 30 A Novel Isospectral Deformation Chain in Supersymmetric Quantum Mechanics......Page 718
1. Introduction......Page 734
2. Covariant total derivative......Page 739
3. Fractal space-time and the motion equation of free particles in the dissipative approximation......Page 742
3.1. Solving the Schrödinger type equation by means of the WKBJ approximation method......Page 745
3.2. Velocity potential γ (x) and the bound states......Page 747
3.3. Velocity potential γ (x) and the quantum barrier......Page 760
4. Casimir type effect in scale relativity theory......Page 763
5.1. The dissipative approximation......Page 776
5.1.2. Anomalous “diffusion” type equation. Weibull relation......Page 777
5.1.3. The correspondence between theoretical model and experimental results......Page 779
5.2. The dispersive approximation......Page 780
5.2.1. The correspondence between theoretical model and experimental results......Page 783
6. Conclusions......Page 785
Author details......Page 786
References......Page 787