David Skinner: Principles of Quantum Mechanics

This is the site for the Part II course on Principles of Quantum Mechanics, offered to third year maths undergraduates at Cambridge. The notes here contain somewhat more than we'll have time to cover in the course.


Contents


  • Introduction:   PDF File
    Preliminaries. A very brief introduction to classical and quantum mechanics.
  • Hilbert Space:   PDF File
    Definition of Hilbert Space. Examples. Dual Spaces. Dirac notation and continuum states. Linear operators. Postulates of quantum mechanics. The generalised uncertainty principle.
  • The Harmonic Oscillator:   PDF File
    Raising and Lowering Operators. Dynamics of Oscillators. A brief look at anharmonic oscillators.
  • Transformations and Symmetries:   PDF File
    Transformations of states and operators. Spatial translations. Rotations. Translations around a circle. Spin. Time translations. The Heisenberg picture. Dynamics. Symmetries and conservation laws. Parity.
  • Angular Momentum:   PDF File
    Angular momentum eigenstates. Large rotations. Rotation spectra of diatomic molecules. Intrinsic Spin. The Stern-Gerlach Experiment. Spinors. Spin Matrices. Nuclear Magnetic Resonance. Orbital Angular Momentum. Wavefunctions for orbital angular momentum eigenstates.
  • Dynamical Symmetries:   PDF File
    The Isotropic Harmonic Oscillator in 3d. Enhanced Symmetry of the Isotropic Oscillator. Angular Momentum of the Harmonic Oscillator. Gross Structure of the Hydrogen Atom. The Quantum Runge-Lenz Vector. Angular Momentum in Hydrogen.
  • Composite Systems:   PDF File
    The Tensor Product of Hilbert spaces. Addition of Angular Momenta. Examples. Identical Particles. Pauli's Exclusion Principle. The Periodic Table. Intrinsic Parity and Decay Processes.
  • Time Independent Perturbation Theory:   PDF File
    An Analytic Expansion. Fine Structure of Hydgrogen. The Ground State of Helium. The Quadratic Stark Effect. Degenerate Perturbation Theory. The Linear Stark Effect. Validity of Perturbation Theory.
  • Time Dependent Perturbation Theory:   PDF File
    The Interaction Picture. Fermi's Golden Rule. Ionization by Monochromatic Light.Atomic Transitions and Their Selection Rules. Absorption, Stimulated Emission and Spontaneous Emission of Radiation. Einstein's Statistical Argument.
  • Interpretation of Quantum Mechanics:   PDF File
    Pure and Mixed States. The Density Operator. Reduced Density Operators for Subsystems. Von Neumann Entropy. Decoherence. The Copenhagen and Many Worlds Interpretations. Bell's Inequality. The EPR Gedankenexperiment.

Problem Sheets


Recommended Books

Any good quality Quantum Mechanics textbook will be appropriate for the course. I particularly recommend Weinberg Lectures on Quantum Mechanics, Sakurai Modern Quantum Mechanics, Dirac's classic Principles of Quantum Mechanics and you might also like to look at Binney & Skinner The Physics of Quantum Mechanics. An approach that takes a deeper look at the functional analysis aspects of Quantum Mechanics can be found in Hall Quantum Theory for Mathematicians.