Graduate Quantum Mechanics
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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.

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About the Author

Peeter Joot
Peeter Joot

Peeter Joot is a math and physics enthusiast with a love of geometric algebra.

Peeter's education includes a B.A.Sc from UofT (1997 Engineering Science - computer engineering), 2019 UofT M.Eng (ECE electromagnetics), a lot of self-study, and non-degree study of most of the interesting 4th year UofT undergrad physics courses.

Peeter's day job is software development. He has over 20 years of experience with low level systems programing, operating system and hardware abstraction and exploitation, concurrency, and large scale refactoring. Peeter works for LzLabs and now has fun with PL/I, COBOL, JCL, VSAM, CICS and other legacy subsystems sure to scare away most developers of his generation.

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
  • Bibliography

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