Graduate Quantum Mechanics
Graduate Quantum Mechanics
Notes and problems from 2015 UofT PHY1520H
About the Book
This book contains notes for the Winter 2015 session of the University of Toronto Graduate Quantum Mechanics course (PHY1520H), taught by Prof. Arun Paramekanti.
Table of Contents
- 1 Fundamental concepts
- 1.1 Classical mechanics
- 1.2 Quantum mechanics
- 1.3 Transformation from a position to momentum basis
- 1.4 Matrix interpretation
- 1.5 Time evolution
- 1.6 Review: Basic concepts
- 1.7 Average of an observable
- 1.8 Left observables
- 1.9 Pure states vs. mixed states
- 1.10 Entropy when density operator has zero eigenvalues
- 1.11 Problems
- 2 Quantum Dynamics
- 2.1 Classical Harmonic Oscillator
- 2.2 Quantum Harmonic Oscillator
- 2.3 Coherent states
- 2.4 Coherent state time evolution
- 2.5 Expectation with respect to coherent states
- 2.6 Coherent state uncertainty
- 2.7 Quantum Field theory
- 2.8 Charged particle in a magnetic field
- 2.9 Gauge invariance
- 2.10 Diagonalizating the Quantum Harmonic Oscillator
- 2.11 Constant magnetic solenoid field
- 2.12 Lagrangian for magnetic portion of Lorentz force
- 2.13 Problems
- 3 Dirac equation in 1D
- 3.1 Construction of the Dirac equation
- 3.2 Plane wave solution
- 3.3 Dirac sea and pair creation
- 3.4 Zitterbewegung
- 3.5 Probability and current density
- 3.6 Potential step
- 3.7 Dirac scattering off a potential step
- 3.8 Problems
- 4 Symmetries in quantum mechanics
- 4.1 Symmetry in classical mechanics
- 4.2 Symmetry in quantum mechanics
- 4.3 Translations
- 4.4 Rotations
- 4.5 Time-reversal
- 4.6 Problems
- 5 Theory of angular momentum
- 5.1 Angular momentum
- 5.2 Schwinger's Harmonic oscillator representation of angular momentum operators.
- 5.3 Representations
- 5.4 Spherical harmonics
- 5.5 Addition of angular momentum
- 5.6 Addition of angular momenta (cont.)
- 5.7 Clebsch-Gordan
- 5.8 Problems
- 6 Approximation methods
- 6.1 Approximation methods
- 6.2 Variational methods
- 6.3 Variational method
- 6.4 Perturbation theory (outline)
- 6.5 Simplest perturbation example.
- 6.6 General non-degenerate perturbation
- 6.7 Stark effect
- 6.8 van der Walls potential
- 6.9 Problems
- A Useful formulas and review
- B Odds and ends
- B.1 Schwartz inequality in bra-ket notation
- B.2 An observation about the geometry of Pauli x,y matrices
- B.3 Operator matrix element
- B.4 Generalized Gaussian integrals
- B.5 A curious proof of the Baker-Campbell-Hausdorff formula
- B.6 Position operator in momentum space representation
- B.7 Expansion of the squared angular momentum operator
- C Julia notebooks
- D Mathematica notebooks
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