Foundations of Quantum Mechanics

Preface

1. Waves and Particles

1.1 Overview

1.2 The Schrodinger Equation

1.3 Unitary Operators in Hilbert Space

1.3.1 Existence and Uniqueness of Solutions of the Schrodinger Equation

1.3.2 The Time Evolution Operators

1.3.3 Unitary Matrices and Rotations

1.3.4 Inner Product

1.3.5 Abstract Hilbert Space

1.4 Classical Mechanics

1.4.1 Definition of Newtonian Mechanics

1.4.2 Properties of Newtonian Mechanics

1.4.3 Hamiltonian Systems

1.5 The Double Slit Experiment

1.5.1 Classical Predictions for Particles and Waves

1.5.2 Actual Outcome of the Experiment

1.5.3 Feynman's Discussion

1.6 Bohmian Mechanics

1.6.1 Definition of Bohmian Mechanics

1.6.2 Historical Overview

1.6.3 Equivariance

1.6.4 The Double Slit Experiment in Bohmian Mechanics

1.6.5 Delayed Choice Experiments

Summary

Exercises

References

2. Some Observables

2.1 Fourier Transform and Momentum

2.1.1 Fourier Transform

2.1.2 Momentum

2.1.3 Momentum Operator

2.1.4 Tunnel Effect

2.2 Operators and Observables

2.2.1 Heisenberg's Uncertainty Relation

2.2.2 Self-Adjoint Operators

2.2.3 The Spectral Theorem

2.2.4 Conservation Laws in Quantum Mechanics

2.3 Spin

2.3.1 Spinors and Pauli Matrices

2.3.2 The Pauli Equation

2.3.3 The Stern-Gerlach Experiment

2.3.4 Bohmian Mechanics with Spin

2.3.5 Is an Electron a Spinning Ball?

2.3.6 Is There an Actual Spin Vector?

2.3.7 Many-Particle Systems

2.3.8 Representations of SO(3)

2.3.9 Inverted Stern-Gerlach Magnet and Contextuality

Summary

Exercises

References

3. Collapse and Measurement

3.1 The Projection Postulate

3.1.1 Notation

3.1.2 The Projection Postulate

3.1.3 Projection and Eigenspace

3.1.4 Remarks

3.2 The Measurement Problem

3.2.1 What the Problem Is

3.2.2 How Bohmian Mechanics Solves the Measurement P