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Chapter 1: Introduction and Prerequisites Next

\frontmatter

Advanced Quantum Mechanics II
PHYS 40202
version 0.99

Niels Walet
Spring/Summer 2012

Last changed on 5/2/2012

Table of Contents

Chapter 1: Introduction and Prerequisites

Section 1.1: Prerequisite courses

Chapter 2: Summary of Prerequisites

Section 2.1: Basic QM

Section 2.2: Linear Algebra

Section 2.3: Other Physics

Chapter 3: Symmetries

Section 3.1: Why symmetries?

Subsection 3.1.1: Linearity

Section 3.2: Examples of Symmetries

Subsection 3.2.1: Translational invariance

Subsubsection 3.2.1.1: Time translations

Subsection 3.2.2: Rotational invariance

Subsection 3.2.3: Galilean invariance

Subsection 3.2.4: Discrete symmetries

Subsubsection 3.2.4.1: Space inversion/parity

Subsubsection 3.2.4.2: Time reversal

Subsubsection 3.2.4.3: Space inversion without momentum inversion

Subsection 3.2.5: Lorentz and Poincaré invariance

Subsubsection 3.2.5.1: Boosts

Subsubsection 3.2.5.2: Full Lorentz symmetry

Subsubsection 3.2.5.3: Poincaré invariance

Subsection 3.2.6: Active and passive view of transformations

Section 3.3: Consequences for Quantum Mechanics

Subsection 3.3.1: Role of symmetry

Subsection 3.3.2: Active and passive view

Subsection 3.3.3: Symmetry generators

Subsubsection 3.3.3.1: Translations: momentum

Subsection 3.3.4: Discrete symmetries

Subsubsection 3.3.4.1: Space inversion

Subsubsection 3.3.4.2: Time reversal

Subsection 3.3.5: Simultaneous diagonalisation of operators

Subsection 3.3.6: Lorentz Boosts

Section 3.4: Unitary operators

Subsection 3.4.1: Exponent of generators

Subsection 3.4.2: Unitarity

Subsubsection 3.4.2.1: Unitarity of the transformation operators

Subsection 3.4.3: Translation in space

Subsection 3.4.4: Translation in time

Subsection 3.4.5: Rotations

Subsubsection 3.4.5.1: Euler angles

Subsubsection 3.4.5.2: Representation matrices

Subsubsection 3.4.5.3: Angular momentum states and ladder operators

Subsubsection 3.4.5.4: L=1

Subsubsection 3.4.5.5: Spin

Chapter 4: Charged Particles and Electromagnetic Fields

Section 4.1: Introduction

Section 4.2: Hamiltonian

Subsection 4.2.1: Lagrangian approach

Subsection 4.2.2: Canonical momentum

Subsection 4.2.3: Quantum Hamiltonian

Subsection 4.2.4: Conservation of momentum

Section 4.3: Gauge transformations and Gauge invariance

Subsection 4.3.1: Gauge transformation

Subsection 4.3.2: Unitary transformations and gauge changes

Subsection 4.3.3: Phase is unobservable

Section 4.4: Landau Levels

Subsection 4.4.1: (Integer) Quantum Hall Effect

Section 4.5: Atoms in laser fields: dipole coupling

Section 4.6: Internal degrees of freedom: spin

Section 4.7: Aharonov-Bohm effect 1: Bound states

Chapter 5: Path Integrals

Section 5.1: Sum of paths

Subsection 5.1.1: Free particle

Subsection 5.1.2: General approach

Section 5.2: Interpretation

Section 5.3: Free particle revisited

Section 5.4: General Gaussian integrals

Section 5.5: Harmonic oscillator

Subsection 5.5.1: The determinant

Subsection 5.5.2: The classical action

Subsection 5.5.3: The complete continuum limit

Section 5.6: Time-dependent oscillator

Section 5.7: Semiclassical limit

Subsection 5.7.1: Classical limit

Subsection 5.7.2: Asymptotic analysis of integrals

Subsection 5.7.3: Gaussian expansion of path integrals

Section 5.8: Slits

Subsection 5.8.1: Single slit

Subsection 5.8.2: Aharonov-Bohm II

Section 5.9: Brownian Motion

Section 5.10: Time-ordered Perturbation Theory

Section 5.11: Partition sum

Section 5.12: Perturbation theory for the ground state energy

Section 5.13: Potential scattering

Section 5.14: Numerical calculations

Chapter 6: Relativistic wave equations

Section 6.1: Probability currents

Section 6.2: Klein-Gordon equation

Subsection 6.2.1: relativistic energy momentum relation

Subsection 6.2.2: Klein-Gordon wave equation

Subsection 6.2.3: Some simple solutions

Subsection 6.2.4: Lorentz invariance and external potentials

Subsection 6.2.5: Probability and currents

Subsection 6.2.6: Issues

Section 6.3: Dirac equation

Subsection 6.3.1: Using minimal coupling

Subsubsection 6.3.1.1: The Pauli-Schrödinger equation

Subsubsection 6.3.1.2: Relativistic form

Subsection 6.3.2: The Classical solution: Linearisation of the energy momentum relationship

Subsection 6.3.3: Plane wave solutions

Subsubsection 6.3.3.1: Positive energy solutions

Subsubsection 6.3.3.2: Negative energy solutions

Subsubsection 6.3.3.3: Spin?

Subsubsection 6.3.3.4: Zero rest mass: helicity basis

Subsubsection 6.3.3.5: Charge conjugation

Subsubsection 6.3.3.6: Completeness

Subsection 6.3.4: Probability

Subsection 6.3.5: Klein Paradox

Subsection 6.3.6: Lorentz transformations and external field

Subsubsection 6.3.6.1: Coupling to external fields

Subsubsection 6.3.6.2: Lorentz covariance

Subsubsection 6.3.6.3: Finite transformations

Subsection 6.3.7: Non-relativistic limit

Subsection 6.3.8: Zitterbewegung

Section 6.4: Graphene

Subsection 6.4.1: Tight binding model

Subsection 6.4.2: Dirac points--Dirac equation

Subsection 6.4.3: Klein paradox and graphene

Subsubsection 6.4.3.1: Simple potential barrier

Subsubsection 6.4.3.2: Picture of general result

Subsubsection 6.4.3.3: Experimental data

Section 6.5: Spherical symmetry

Subsection 6.5.1: Role of vector spherical harmonics

Subsection 6.5.2: Radial equations

Subsection 6.5.3: Hydrogen-like atom

Subsubsection 6.5.3.1: Relativistic approach

Appendix A: Useful Mathematics

Section A.1: Vector algebra

Section A.2: Coordinate systems

Subsection A.2.1: Cartesian coordinates

Subsection A.2.2: Spherical coordinates

Subsection A.2.3: Cylindrical coordinates

Subsection A.2.4: Guide to other choices

Section A.3: Gaussian Integrals

Section A.4: Vector calculus

Section A.5: Matrix algebra

Section A.6: Relativity-tensor notation

Section A.7: Commutators

Section A.8: Anticommutators

Section A.9: Standard Matrices

\mainmatter

Chapter 1: Introduction and Prerequisites Next