Advanced Quantum Mechanics

Judith McGovern

September 26, 2012

This is the web page for Advanced Quantum Mechanics (PHYS30201) for the session 2012/13.

The notes are not intended to be fully self-contained, but summarise lecture material and give pointers to textbooks.

NEW: The full pdf file of the notes is here.

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This course and other resources.
Operator methods in Quantum Mechanics
 1.1 Postulates of Quantum Mechanics
 1.2 Position and Momentum Representations
 1.3 The Stern-Gerlach experiment
 1.4 Ehrenfest’s Theorem and the Classical Limit
Approximate methods I: variational method and WKB
 2.1 Variational methods: ground state
 2.2 Variational methods: excited states
 2.3 Variational methods: the helium atom
 2.4 WKB approximation
  2.4.1 WKB approximation for bound states
  2.4.2 WKB approximation for tunnelling
Approximate methods II: Time-independent perturbation theory
 3.1 Formalism
  3.1.1 Simple examples of perturbation theory
 3.2 Example of degenerate perturbation theory
 3.3 The fine structure of hydrogen
 3.4 The Zeeman effect: hydrogen in an external magnetic field
 3.5 The Stark effect: hydrogen in an external electric field
Approximate methods III: Time-dependent perturbation theory
 4.1 Formalism
  4.1.1 Perturbation which is switched on slowly
  4.1.2 Sudden perturbation
 4.2 Oscillatory perturbation and Fermi’s golden rule
 4.3 Emission and absorption of radiation
  4.3.1 Einstein’s A and B coefficients
 4.4 Radiative decay of 2p state of hydrogen
 4.5 Finite width of excited state
 4.6 Selection rules
 4.7 Heisenberg vs Schrödinger pictures
Approximate methods IV: Scattering theory
 5.1 Preliminaries
 5.2 The Born approximation
 5.3 Phase Shifts
  5.3.1 Hard sphere scattering
  5.3.2 Scattering from a finite square barrier or well
Quantum Measurement
 6.1 The Einstein-Poldosky-Rosen “paradox” and Bell’s inequalities
Mathematical background and revision
 A.1 Vector Spaces
  A.1.1 Direct Products
 A.2 Angular Momentum
 A.3 Hydrogen wave functions
 A.4 Harmonic oscillators, creation and annihilation operators
 A.5 The Helium atom
 A.6 Spherical Bessel functions
 A.7 Airy functions
 A.8 Properties of δ-functions
 A.9 Gaussian integrals
 A.10 Density of states, periodic boundary conditions and black-body radiation
 A.11 Checking units and scaling
 A.12 Units in EM
Problem Sheets