Probability and Schrödinger's mechanics / David B. Cook.

Format:

Book

Author/Creator:

Cook, David B.

Language:

English

Subjects (All):

Quantum theory.

Probabilities.

Physical Description:

xviii, 323 pages : illustrations ; 24 cm

Place of Publication:

River Edge, N.J. : World Scientific, [2002]

Summary:

This book addresses some of the problems of interpreting Schrodinger's mechanics -- the most complete and explicit theory falling under the umbrella of "quantum theory." The outlook is materialist ("realist") and stresses the development of Schrodinger's mechanics from classical theories and its close connections with (particularly) the Hamilton-Jacobi theory. Emphasis is placed on the concepts and use of the modern objective measure-theoretic) probability theory. The work is free from any mention of the bearing of Schrodinger's mechanics on God, his alleged mind or, indeed, minds at all. The author has taken the naive view that this mechanics is about the structure and dynamics of atomic and sub-atomic systems since he has been unable to trace any references to minds, consciousness or measurements in the foundations of the theory.

Contents:

1.3. Materialism and Realism 8

1.4. Logic 9

1.5. Mathematics 12

1.6. Reversing Abstraction 13

1.7. Definitions, Laws of Nature and Causality 15

1.8. Foundations 18

1.9. Axioms 20

1.10. An Interpreted Theory 21

Part 2 Probabilities 23

2.2. Probabilities for Finite Systems 27

2.2.1. An Example: The Faces of a Cube 29

2.2.2. Dice: Statistical Methods of Measure 31

2.2.3. Loaded Dice: Statistical Methods of Measure 34

2.2.4. Standard Dice and Conservation Laws 35

2.3. Probability and Statistics 39

2.3.1. An Extreme Example 40

2.4. Probabilities in Deterministic Systems 41

2.5. The Referent of Probabilities and Measurement 45

2.5.1. Single System or Ensemble? 47

2.5.2. The Collapse of the Distribution 49

2.5.3. Hidden Variables 50

Chapter 3 A More Careful Look at Probabilities 53

3.1. Abstract Objects 53

3.2. States and Probability Distributions 55

3.2.1. The Propensity Interpretation 56

3.3. The Formal Definition of Probability 58

3.3.1. A Premonition 62

3.4. Time-Dependent Probabilities 63

3.5. Random Tests 66

3.6. Particle-Distribution Probabilities 67

Part 3 Classical Mechanics 69

Chapter 4 The Hamilton-Jacobi Equation 71

4.1. Historical Connections 71

4.2. The H-J Equation 73

4.3. Solutions of the H-J Equation 76

4.3.1. Cartesian Coordinates 78

4.3.2. Spherical Polar Coordinates 79

4.3.3. Comparisons 81

4.3.4. Cylindrical Coordinates 83

4.4. Distribution of Trajectories 84

Appendix 4.A Transformation Theory 89

Chapter 5 Angular Momentum 99

5.1. Coordinates and Momenta 99

5.2. The Angular Momentum "Vector" 101

5.3. The Poisson Brackets and Angular Momentum 106

5.4. Components of the Angular Momentum "Vector" 107

5.5. Conclusions for Angular Momentum 109

Part 4 Schrodinger's Mechanics 111

Chapter 6 Prelude: Particle Diffraction 113

6.1. History 113

6.1.1. The Experiment 114

6.1.2. The Explanations 114

6.2. The Wave Theory 115

6.3. The Particle Theory 116

6.5. Experimental Verification 120

6.6. The Answer to a Rhetorical Question 120

Chapter 7 The Genesis of Schrodinger's Mechanics 123

7.1. Lagrangians, Hamiltonians, Variation Principles 123

7.1.1. Equations and Identities 125

7.2. Replacing the Hamilton-Jacobi Equation 126

7.3. Generalising the Action S 128

7.3.1. Changing the Notation for Action 129

7.3.2. Interpreting the Change 131

7.4. Schrodinger's Dynamical Law 134

7.4.1. Position Probability and Energy Distributions 135

7.4.2. The Schrodinger Condition 136

7.5. Probability Distributions? 139

Chapter 8 The Schrodinger Equation 147

8.1. The Variational Derivation 147

8.3. The Boundary Conditions 156

8.4. The Time-Independent Schrodinger Equation 158

Appendix 8.A Schrodinger's First Paper of 1926 161

Chapter 9 Identities: Momenta and Dynamical Variables 179

9.1. Momentum Definitions and Distributions 179

9.2. Abstract Particles of Constant Momentum 180

9.3. Action and Momenta in Schrodinger's Mechanics 182

9.4. Momenta and Kinetic Energy 186

9.5. Boundary Conditions 189

9.5.1. Constant Momenta and Kinetic Energy 190

9.5.2. Solution of the Schrodinger Equation 191

9.6. The "Particle in a Box" and Cyclic Boundary Conditions 192

Chapter 10 Abstracting the Structure 195

10.1. The Idea of Mathematical Structure 195

10.1.1. A Pitfall of Abstraction: The Momentum Operator 198

10.2. States and Hilbert Space 201

10.3. The Real Use of Abstract Structures 204

Part 5 Interpretation from Applications 207

Chapter 11 The Quantum Kepler Problem 209

11.1. Two Interacting Particles 210

11.2. Quantum Kepler Problem in a Plane 211

11.3. Abstract and Concrete Hydrogen Atoms 212

11.4. The Kepler Problem in Three Dimensions 214

11.5. The Separation of the Schrodinger Equation 216

11.6. Commuting Operators and Conservation 218

11.7. The Less Familiar Separations 221

11.7.1. The Everyday Solutions 223

11.8. Conservation in Concrete and Abstract Systems 223

11.9. Conclusions from the Kepler Problem 227

11.9.1. Concrete Objects and Symmetries 231

Appendix 11.A Hamiltonians by Substitution? 233

Chapter 12 The Harmonic Oscillator and Fields 237

12.1. The Schrodinger Equation for SHM 237

12.2. SHM Details 239

12.3. Factorisation Method 241

12.4. Interpreting the SHM Solutions 242

12.5 Vibrations of Fields and "Particles" 244

12.5.1 Phonons and Photons 248

12.6. Second Quantisation 249

Chapter 13 Perturbation Theory and Epicycles 251

13.1. Perturbation Theories in General 251

13.2. Perturbed Schrodinger Equations 252

13.3. Polarisation of Electron Distribution 255

13.4. Interpretation of Perturbation Theory 256

13.5. Quantum Theory and Epicycles 258

13.6. Approximations to Non-existent Functions 259

13.7. Summary for Perturbation Theory 261

Chapter 14 Formalisms and "Hidden" Variables 263

14.1. The Semi-empirical Method 263

14.2. The Chemical Bond 264

14.3. Dirac's Spin "Hamiltonian" 267

14.4. Interpretation of the Spin Hamiltonian 268

Part 6 Disputes and Paradoxes 271

Chapter 15 Measurement at the Microscopic Level 273

15.1. Recollection: Concrete and Abstract Objects 273

15.2. Statistical Estimates of Probabilities 275

15.2.1. von Neumann's Theory of Measurement 278

15.3. Measurement as "State Preparation" 281

15.4. Heisenberg's Uncertainty Principle 284

15.4.1. Measurement and Decoherence 286

15.5. Measurement Generalities 287

Appendix 15.A Standard Deviations of Conjugate Variables 289

Chapter 16 Paradoxes 291

16.1. The Classical Limit 291

16.1.1. The Ehrenfest Relations 293

16.2. The Einstein-Podolsky-Rosen (EPR) Paradox 294

16.2.1. The EPR Original 295

16.2.2. Bohm's Modification 297

16.2.3. Bell's Inequality and Theorem 298

16.3. Bell's Assumptions 300

16.3.1. Lessons from EPR 303

16.3.2. Density of Spin and EPR 304

16.4. Zero-Point Energy 307

Chapter 17 Beyond Schrodinger's Mechanics? 311

17.1. An Interregnum? 311

17.2. The Avant-Garde 313

17.3. The Break with the Past 314

17.4. Classical and Quantum Mechanics 315.

Notes:

Includes bibliographical references and index.

ISBN:

9812381910

OCLC:

50948838