Author Cook, David B
Title Probability And Schrodinger's Mechanics
Imprint Singapore : World Scientific Publishing Company, 2002
©2002
book jacket
Descript 1 online resource (343 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Note Intro -- Contents -- Preface -- Organisation -- Part 1 Preliminaries -- Chapter 1 Orientation and Outlook -- 1.1. General Orientation -- 1.2. Materialism -- 1.3. Materialism and Realism -- 1.4. Logic -- 1.5. Mathematics -- 1.6. Reversing Abstraction -- 1.7. Definitions, Laws of Nature and Causality -- 1.8. Foundations -- 1.9. Axioms -- 1.10. An Interpreted Theory -- Part 2 Probabilities -- Chapter 2 Simple Probabilities -- 2.1. Colloquial and Mathematical Terminology -- 2.2. Probabilities for Finite Systems -- 2.2.1. An Example: The Faces of a Cube -- 2.2.2. Dice: Statistical Methods of Measure -- 2.2.3. Loaded Dice: Statistical Methods of Measure -- 2.2.4. Standard Dice and Conservation Laws -- 2.3. Probability and Statistics -- 2.3.1. An Extreme Example -- 2.4. Probabilities in Deterministic Systems -- 2.5. The Referent of Probabilities and Measurement -- 2.5.1. Single System or Ensemble? -- 2.5.2. The Collapse of the Distribution -- 2.5.3. Hidden Variables -- 2.6. Preliminary Summary -- Chapter 3 A More Careful Look at Probabilities -- 3.1. Abstract Objects -- 3.2. States and Probability Distributions -- 3.2.1. The Propensity Interpretation -- 3.3. The Formal Definition of Probability -- 3.3.1. A Premonition -- 3.4. Time-Dependent Probabilities -- 3.5. Random Tests -- 3.6. Particle-Distribution Probabilities -- Part 3 Classical Mechanics -- Chapter 4 The Hamilton-Jacobi Equation -- 4.1. Historical Connections -- 4.2. The H-J Equation -- 4.3. Solutions of the H-J Equation -- 4.3.1. Cartesian Coordinates -- 4.3.2. Spherical Polar Coordinates -- 4.3.3. Comparisons -- 4.3.4. Cylindrical Coordinates -- 4.4. Distribution of Trajectories -- 4.5. Summary -- Appendix 4.A Transformation Theory -- Chapter 5 Angular Momentum -- 5.1. Coordinates and Momenta -- 5.2. The Angular Momentum "Vector" -- 5.3. The Poisson Brackets and Angular Momentum
5.4. Components of the Angular Momentum "Vector" -- 5.5. Conclusions for Angular Momentum -- Part 4 Schrödinger's Mechanics -- Chapter 6 Prelude: Particle Diffraction -- 6.1. History -- 6.1.1. The Experiment -- 6.1.2. The Explanations -- 6.2. The Wave Theory -- 6.3. The Particle Theory -- 6.4. A Simple Case -- 6.5. Experimental Verification -- 6.6. The Answer to a Rhetorical Question -- 6.7. Conclusion -- Chapter 7 The Genesis of Schrödinger's Mechanics -- 7.1. Lagrangians, Hamiltonians, Variation Principles -- 7.1.1. Equations and Identities -- 7.2. Replacing the Hamilton-Jacobi Equation -- 7.3. Generalising the Action S -- 7.3.1. Changing the Notation for Action -- 7.3.2. Interpreting the Change -- 7.4. Schrödinger's Dynamical Law -- 7.4.1. Position Probability and Energy Distributions -- 7.4.2. The Schrodinger Condition -- 7.5. Probability Distributions? -- 7.6. Summary of Basic Principles -- Chapter 8 The Schrödinger Equation -- 8.1. The Variational Derivation -- 8.2. Some Interpretation -- 8.3. The Boundary Conditions -- 8.4. The Time-Independent Schrödinger Equation -- Appendix 8.A Schrödinger's First Paper of 1926 -- Chapter 9 Identities: Momenta and Dynamical Variables -- 9.1. Momentum Definitions and Distributions -- 9.2. Abstract Particles of Constant Momentum -- 9.3. Action and Momenta in Schrödinger's Mechanics -- 9.4. Momenta and Kinetic Energy -- 9.5. Boundary Conditions -- 9.5.1. Constant Momenta and Kinetic Energy -- 9.5.2. Solution of the Schrodinger Equation -- 9.6. The "Particle in a Box" and Cyclic Boundary Conditions -- Chapter 10 Abstracting the Structure -- 10.1. The Idea of Mathematical Structure -- 10.1.1. A Pitfall of Abstraction: The Momentum Operator -- 10.2. States and Hilbert Space -- 10.3. The Real Use of Abstract Structures -- Part 5 Interpretation from Applications -- Chapter 11 The Quantum Kepler Problem
11.1. Two Interacting Particles -- 11.2. Quantum Kepler Problem in a Plane -- 11.3. Abstract and Concrete Hydrogen Atoms -- 11.4. The Kepler Problem in Three Dimensions -- 11.5. The Separation of the Schrödinger Equation -- 11.6. Commuting Operators and Conservation -- 11.7. The Less Familiar Separations -- 11.7.1. The Everyday Solutions -- 11.8. Conservation in Concrete and Abstract Systems -- 11.9. Conclusions from the Kepler Problem -- 11.9.1. Concrete Objects and Symmetries -- Appendix 11.A Hamiltonians by Substitution? -- Chapter 12 The Harmonic Oscillator and Fields -- 12.1. The Schrödinger Equation for SHM -- 12.2. SHM Details -- 12.3. Factorisation Method -- 12.4. Interpreting the SHM Solutions -- 12.5. Vibrations of Fields and "Particles" -- 12.5.1. Phonons and Photons -- 12.6. Second Quantisation -- Chapter 13 Perturbation Theory and Epicycles -- 13.1. Perturbation Theories in General -- 13.2. Perturbed Schrödinger Equations -- 13.3. Polarisation of Electron Distribution -- 13.4. Interpretation of Perturbation Theory -- 13.5. Quantum Theory and Epicycles -- 13.6. Approximations to Non-existent Functions -- 13.7. Summary for Perturbation Theory -- Chapter 14 Formalisms and "Hidden" Variables -- 14.1. The Semi-empirical Method -- 14.2. The Chemical Bond -- 14.3. Dirac's Spin "Hamiltonian" -- 14.4. Interpretation of the Spin Hamiltonian -- Part 6 Disputes and Paradoxes -- Chapter 15 Measurement at the Microscopic Level -- 15.1. Recollection: Concrete and Abstract Objects -- 15.2. Statistical Estimates of Probabilities -- 15.2.1. von Neumann's Theory of Measurement -- 15.3. Measurement as "State Preparation" -- 15.4. Heisenberg's Uncertainty Principle -- 15.4.1. Measurement and Decoherence -- 15.5. Measurement Generalities -- Appendix 15.A Standard Deviations of Conjugate Variables -- Chapter 16 Paradoxes -- 16.1. The Classical Limit
16.1.1. The Ehrenfest Relations -- 16.2. The Einstein-Podolsky-Rosen (EPR) Paradox -- 16.2.1. The EPR Original -- 16.2.2. Bohm's Modification -- 16.2.3. Bell's Inequality and Theorem -- 16.3. Bell's Assumptions -- 16.3.1. Lessons from EPR -- 16.3.2. Density of Spin and EPR -- 16.4. Zero-Point Energy -- Chapter 17 Beyond Schrödinger's Mechanics? -- 17.1. An Interregnum? -- 17.2. The Avant-Garde -- 17.3. The Break with the Past -- 17.4. Classical and Quantum Mechanics -- Index
This book addresses some of the problems of interpreting Schrödinger'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 Schrödinger'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 Schrödinger's mechanics on God, his alleged mind or, indeed, minds at all. The author has taken the naïve 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: Preliminaries: Orientation and Outlook; Probabilities: Simple Probabilities; A More Careful Look at Probabilities; Classical Mechanics: The Hamilton-Jacobi Equation; Angular Momentum; Schrödinger's Mechanics: Prelude: Particle Diffraction; The Genesis of Schrödinger's Mechanics; The Schrödinger Equation; Identities: Momenta and Dynamical Variables; Abstracting the Structure; Interpretation from Applications: The Quantum Kepler Problem; The Harmonic Oscillator and Fields; Perturbation Theory and Epicycles; Formalisms and "Hidden" Variables; Disputes and Paradoxes: Measurement at the Microscopic Level; Paradoxes; Beyond Schrödinger's Mechanics?. Readership: Physical scientists interested in quantum theory, philosophers of science, and students of scientific philosophy
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Cook, David B Probability And Schrodinger's Mechanics Singapore : World Scientific Publishing Company,c2002 9789812381910
Subject Quantum theory.;Probabilities
Electronic books
Alt Author Leong, H.T
Doyle, Anthony