The Quantum Cookbook: Mathematical Recipes for the Foundations for Quantum Mechanics
The Quantum Cookbook: Mathematical Recipes for the Foundations for Quantum Mechanics
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Abstract
The Quantum Cookbook provides a unique bridge between popular exposition and formal textbook presentation, based on derivations of the iconic equations of quantum mechanics. This is a book for curious readers with some background in physics and sufficient mathematical capability. It aims not to teach readers how to do quantum mechanics but rather helps them to understand how to think about quantum mechanics. Each derivation is presented as a ‘recipe’ with listed ingredients, including standard results from the mathematician’s toolkit, set out in a series of easy-to-follow steps. The recipes have been written sympathetically, for readers who—like the author—will often struggle to follow the logic of a derivation which misses out steps that are ‘obvious’, or which use techniques that readers are assumed to know. The simple truth is that quantum mechanics did not suddenly materialize overnight in the minds of its creators, fully formed, complete with all its axioms and principles. It was instead tortured from much more familiar classical physical descriptions, over a period of decades, as physicists struggled to interpret a series of ever more baffling experimental results. Only later was a much higher level of abstraction introduced into quantum mechanics, in an attempt to establish a secure mathematical foundation that would eradicate all the confusing classical misconceptions inherited from its birth and early childhood. Whilst there are some obvious exceptions, for the most part these derivations are triumphs of physical intuition over mathematical rigour and consistency.
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Front Matter
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Prologue
What’s Wrong with This Picture? The Description of Nature at the End of the Nineteenth Century
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1
Planck’s Derivation of E=hν: The Quantization of Energy
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2
Einstein’s Derivation of E=mc2: The Equivalence of Mass and Energy
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3
Bohr’s Derivation of the Rydberg Formula: Quantum Numbers and Quantum Jumps
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4
De Broglie’s Derivation of λ = h/p: Wave–Particle Duality
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5
Schrödinger’s Derivation of the Wave Equation: Quantization as an Eigenvalue Problem
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6
Born’s Interpretation of the Wavefunction: Quantum Probability
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7
Heisenberg, Bohr, Robertson, and the Uncertainty Principle: The Interpretation of Quantum Uncertainty
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8
Heisenberg’s Derivation of the Pauli Exclusion Principle: The Stability of Matter and the Periodic Table
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9
Dirac’s Derivation of the Relativistic Wave Equation: Electron Spin and Antimatter
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10
Dirac, Von Neumann, and the Derivation of the Quantum Formalism: State Vectors in Hilbert Space
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11
Von Neumann and the Problem of Quantum Measurement: The ‘Collapse of the Wavefunction’
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12
Einstein, Bohm, Bell, and the Derivation of Bell’s Inequality: Entanglement and Quantum Non-locality
- Epilogue A Game of Theories: The Quantum Representation of Reality
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End Matter
- Appendix 1 Cavity Modes
- Appendix 2 Lorentz Transformation for Energy and Linear Momentum
- Appendix 3 Energy Levels of the Hydrogen Atom
- Appendix 4 Orthogonality of the Hydrogen Atom Wavefunctions
- Appendix 5 The Integral Cauchy-Schwarz Inequality
- Appendix 6 Orbital Angular Momentum in Quantum Mechanics
- Appendix 7 A Very Brief Introduction to Matrices
- Appendix 8 A Simple Local Hidden Variables Theory
- Index
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