QUANTUM MECHANICS II
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QUANTUM MECHANICS II

Notes and problems from UofT PHY456H1F 2012

About the Book

These are my personal lecture notes for the Fall 2011, University of Toronto Quantum mechanics II course (PHY456H1F), taught by Prof. John E Sipe.

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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

  • 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

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