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Advanced Quantum Mechanics 2
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Chapter 1: Introduction and Prerequisites
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\frontmatter
Advanced Quantum Mechanics II
PHYS 40202
version 0.99
Niels Walet
Spring/Summer 2012
Last changed on 7/3/2012
Table of Contents
Chapter 1: Introduction and Prerequisites
Section 1.1: Prerequisite courses
Bibliography
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: Coupling of light to atoms and nuclei: multipole couplings
Section 4.6: Internal degrees of freedom: spin
Section 4.7: The Pauli-Schrödinger equation
Section 4.8: 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: Semi-classical 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: 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
Index
\mainmatter
Chapter 1: Introduction and Prerequisites
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