## The Courses

I took three courses on Quantum Mechanics at UofT, PHY356, PHY456, and PHY1520 (not counting the intro QM that was included in my undergrad engineering program), and have notes (of progressively higher quality) for each of those courses available below.

- Quantum Physics I (PHY356H1F), taught by Prof. Vatche Deyirmenjian, fall 2010. (263 pages)
- Quantum Physics II (PHY456H1F), taught 2011 by Prof. John E. Sipe, fall 2011. (281 pages). The notes for this class are available in a few forms:
- In paperback and hardcover (black and white) through kindle-direct-publishing (Amazon print on demand.)
- In color, for free, as a PDF.
- As an e-book on leanpub for $5+ (set your own price.) There is also a leanpub forum available for this book.

- Graduate Quantum Mechanics (PHY1520H) taught by Prof. Arun Paramekanti, fall 2015. (435 pages). The notes for this class are available in a few forms:
- In paperback or hardcover (black and white) through kindle-direct-publishing (Amazon print on demand.)
- As an e-book on leanpub for $5+ (set your own price.) There is also a leanpub forum available for this book.
- In color, for free, as a PDF.

The first two courses were taken as a non-degree student, and the last was taken with my UofT M.Eng program (a course based engineering master’s).

The grad quantum course was especially fun, and I enjoyed the chance to revisit the subject. This was the best round of my match against QM, and I came out much less bloody than the first two rounds.

Mathematica notebooks for each of these courses are also available:

I had a couple notebooks written in julia for phy1520 too (just plots I think), and they can be found on github.

## Contributing.

Should you wish to actively contribute typo fixes (or additions, editing, …) to any of these notes (phy356, phy456, phy1520), you can do so by contacting me, or by forking your own copy of the associated git repositories and building the book pdf from source, and submitting a subsequent merge request.

git clone [email protected]:peeterjoot/latex-notes-compilations.git peeterjoot cd peeterjoot # For phy356 and phy456 too, add: # figures/phy356-qmI figures/phy456-qmII phy356-qmI phy456-qmII for i in figures/phy1520-quantum phy1520-quantum julia mathematica latex ; do git submodule update --init $i (cd $i ; git checkout master) done export PATH=`pwd`/latex/bin:$PATH (cd phy356-qmI ; make) (cd phy456-qmII ; make) (cd phy1520-quantum ; make)

I reserve the right to impose dictatorial control over any editing and content decisions, and may not accept merge requests as-is, or at all. That said, I’ll probably not refuse reasonable suggestions or merge requests.

## Phy356 (QM I) Contents:

- Copyright
- Document Version
- Dedication
- Preface
- Contents
- List of Figures
- 1 Basic formalism
- 1.1 Dirac Adjoint notes
- 1.2 Lecture notes: Review
- 1.3 Problems
- 2 Commutator and time evolution
- 2.1 Rotations using matrix exponentials
- 2.2 On commutation of exponentials
- 2.3 Canonical Commutator
- 2.4 Generalized momentum commutator
- 2.5 Uncertainty principle
- 2.6 Size of a particle
- 2.7 Space displacement operator
- 2.8 Time evolution operator
- 2.9 Dispersion delta function representation
- 2.10 Unitary exponential sandwich
- 2.11 Lecture notes: Review
- 2.12 Problems
- 3 Dynamical equations
- 3.1 Lecture notes: Review
- 3.2 Problems
- 4 Free particles
- 4.1 Antisymmetric tensor summation identity
- 4.2 Question on raising and lowering arguments
- 4.3 Another question on raising and lowering arguments
- 4.4 Lecture notes: Review
- 4.5 Problems
- 5 Spin 1/2
- 5.1 Lecture notes: Review
- 5.2 Lecture: Orbital and Intrinsic Momentum
- 5.3 Problems
- 6 Gauge invariance, angular momentum and spin
- 6.1 Interaction with orbital angular momentum
- 7 Stern-Gerlach
- 7.1 Lecture: Stern Gerlach
- 7.2 Why do this (Dirac notation) math?
- 7.3 On section 5.11, the complete wavefunction
- 8 Lecture: Making Sense of Quantum Mechanics
- 8.0.1 Discussion
- 8.1 Projection operator
- 9 Bound state problems
- 9.1 Hydrogen like atom, and Laguerre polynomials
- 9.2 Examples
- 9.3 Lecture: Hydrogen atom
- 9.4 Problems
- 10 Harmonic oscillator
- 10.1 Setup
- 10.2 Relating states
- 10.3 Heisenberg picture
- 10.4 A couple comments on the Schr\366dinger picture
- 10.5 Back to the Heisenberg picture
- 10.6 Problems
- 11 Coherent states
- 11.1 interaction with a electric field
- 12 Rotations and angular momentum
- 12.1 Rotations (chapter 26)
- 12.2 Trig relations
- 12.3 Infinitesimal transformations
- 12.4 Verifying the commutator relations
- 12.5 General infinitesimal rotation
- 12.6 Position and angular momentum commutator
- 12.7 A note on the angular momentum operator exponential sandwiches
- 12.8 Trace relation to the determinant
- 12.9 Problems
- Bibliography

## Phy456 (QM II) Contents:

- Copyright
- Document Version
- Dedication
- Preface
- Contents
- List of Figures
- 1 Approximate methods.
- 1.1 Approximate methods for finding energy eigenvalues and eigenkets.
- 1.2 Variational principle.
- 2 Perturbation methods.
- 2.1 States and wave functions.
- 2.2 Excited states.
- 2.3 Problems.
- 3 Time independent perturbation.
- 3.1 Time independent perturbation.
- 3.2 Issues concerning degeneracy.
- 3.3 Examples.
- 4 Time dependent perturbation.
- 4.1 Review of dynamics.
- 4.2 Interaction picture.
- 4.3 Justifying the Taylor expansion above (not class notes).
- 4.4 Recap: Interaction picture.
- 4.5 Time dependent perturbation theory.
- 4.6 Perturbation expansion.
- 4.7 Time dependent perturbation.
- 4.8 Sudden perturbations.
- 4.9 Adiabatic perturbations.
- 4.10 Adiabatic perturbation theory (cont.)
- 4.11 Examples.
- 5 Fermi’s golden rule.
- 5.1 Recap. Where we got to on Fermi’s golden rule.
- 5.2 Fermi’s Golden rule.
- 5.3 Problems.
- 6 WKB Method.
- 6.1 WKB (Wentzel-Kramers-Brillouin) Method.
- 6.2 Turning points..
- 6.3 Examples.
- 7 Composite systems.
- 7.1 Hilbert Spaces.
- 7.2 Operators.
- 7.3 Generalizations.
- 7.4 Recalling the Stern-Gerlach system from PHY354.
- 8 Spin and Spinors.
- 8.1 Generators.
- 8.2 Generalizations.
- 8.3 Multiple wavefunction spaces.
- 9 Two state kets and Pauli matrices.
- 9.1 Representation of kets.
- 9.2 Representation of two state kets.
- 9.3 Pauli spin matrices.
- 10 Rotation operator in spin space.
- 10.1 Formal Taylor series expansion.
- 10.2 Spin dynamics.
- 10.3 The hydrogen atom with spin.
- 11 Two spins, angular momentum, and Clebsch-Gordon.
- 11.1 Two spins.
- 11.2 More on two spin systems.
- 11.3 Recap: table of two spin angular momenta.
- 11.4 Tensor operators.
- 12 Rotations of operators and spherical tensors.
- 12.1 Setup.
- 12.2 Infinitesimal rotations.
- 12.3 A problem.
- 12.4 How do we extract these buried simplicities?
- 12.5 Motivating spherical tensors.
- 12.6 Spherical tensors (cont.)
- 13 Scattering theory.
- 13.1 Setup.
- 13.2 1D QM scattering. No potential wave packet time evolution.
- 13.3 A Gaussian wave packet.
- 13.4 With a potential.
- 13.5 Considering the time independent case temporarily.
- 13.6 Recap.
- 14 3D Scattering.
- 14.1 Setup.
- 14.2 Seeking a post scattering solution away from the potential.
- 14.3 The radial equation and its solution.
- 14.4 Limits of spherical Bessel and Neumann functions.
- 14.5 Back to our problem.
- 14.6 Scattering geometry and nomenclature.
- 14.7 Appendix.
- 14.8 Verifying the solution to the spherical Bessel equation.
- 14.9 Scattering cross sections.
- 15 Born approximation.
- A Harmonic oscillator Review.
- A.1 Problems.
- B Simple entanglement example.
- C Problem set 4, problem 2 notes.
- D Adiabatic perturbation revisited.
- E 2nd order adiabatically Hamiltonian.
- F Degeneracy and diagonalization.
- F.1 Motivation.
- F.2 A four state Hamiltonian.
- F.3 Generalizing slightly.
- G Review of approximation results.
- G.1 Motivation.
- G.2 Variational method.
- G.3 Time independent perturbation.
- G.4 Degeneracy.
- G.5 Interaction picture.
- G.6 Time dependent perturbation.
- G.7 Sudden perturbations.
- G.8 Adiabatic perturbations.
- G.9 WKB.
- H Clebsh-Gordan zero coefficients.
- H.1 Motivation.
- H.2 Recap on notation.
- H.3 The \(J_z\) action.
- I One more adiabatic perturbation derivation.
- I.1 Motivation.
- I.2 Build up.
- I.3 Adiabatic case.
- I.4 Summary.
- J Time dependent perturbation revisited.
- K Second form of adiabatic approximation.
- L Verifying the Helmholtz Green’s function.
- M Mathematica notebooks.
- Index
- Bibliography

### phy456 Changelog:

phy456.V0.1.11-3.pdf (Apr 22, 2023)

- URL updates.
- dmath purge.
- Figure updates (LaTeX captions replacing Mathematica captions, some hand drawn figures removed, PlotStyle -> Thick+Black, …)

phy456.V0.1.10-15 (May 13, 2020)

- First version posted to kindle-direct successfully.
- Switch to 6×9 layout.
- Add commas and periods to equations.
- Remove blank lines that cause additional undesired indenting (implied latex \par’s)
- Updated some of the figures.
- Fix missing figure captions.
- Some spell checking.

## Phy1520 (Grad Quantum) 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

### On the problem set solutions for Grad QM (phy1520)

These notes are *no longer redacted* and include whatever portions of the problem I completed, errors and all. In the event that any of the problem sets are recycled for future iterations of the course, students who are taking the course (all mature grad students pursuing science for the love of it, not for grades) are expected to act responsibly, and produce their own solutions, within the constraints provided by the professor.

### phy1520 Changelog:

phy1520.V0.1.9-3 (June 10, 2019)

- First version posted to kindle-direct successfully.
- Lots of 6×9 formatting fixes made.
- Add commas and periods to equations.
- Remove blank lines that cause additional undesired indenting (implied latex \par’s)

phy1520.V0.1.8-2 (May 5, 2019)

- Convert to 6×9 format for kdp upload.
- Fix missing dashes in lstlisting.
- Dedicate to my siblings.