Foundations of Quantum Mechanics

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

1.4 Classical Mechanics

1.4.1 Definition of Newtonian Mechanics

1.4.2 Properties of Newtonian Mechanics

1.4.3 Hamiltonian Systems

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

1.6.2 Historical Overview

1.6.3 Equivariance

1.6.5 Delayed Choice Experiments

Summary

Exercises

References

2. Some Observables

2.1 Fourier Transform and Momentum

2.1.1 Fourier Transform

2.1.3 Momentum Operator

2.1.4 Tunnel Effect

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

2.3.1 Spinors and Pauli Matrices

2.3.2 The Pauli Equation

2.3.3 The Stern-Gerlach Experiment

2.3.4 Bohmian Mechanics with Spin

2.3.6 Is There an Actual Spin Vector?

2.3.7 Many-Particle Systems

2.3.9 Inverted Stern-Gerlach Magnet and Contextuality

Summary

Exercises

References

3.1 The Projection Postulate

3.1.1 Notation

3.1.2 The Projection Postulate

3.1.3 Projection and Eigenspace

3.2 The Measurement Problem

3.2.1 What the Problem Is

3.2.2 How Bohmian Mechanics Solves the Measurement P