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
    Preliminaries. A very brief introduction to classical and quantum mechanics.

  • Hilbert Space and Operators
    Definition of Hilbert Space. Examples. Dual Spaces. Dirac Notation and Continuum States. Linear Operators. Composite Systems and Tensor Products. Postulates of Quantum Mechanics. The Generalised Uncertainty Principle.

  • The Harmonic Oscillator
    Raising and Lowering Operators. Dynamics of Oscillators. Coherent States. A Brief Look at Anharmonic Oscillators. An Isotropic Oscillator in Three Dimensions.

  • Transformations and Symmetries
    Transformations of States and Operators. Spatial translations. Rotations. Translations Around a Circle. Spin. Time Translations and the Heisenberg Picture. Symmetries and Conservation Laws. Parity.

  • Angular Momentum
    Angular Momentum Eigenstates as Representations of SO(3). Rotation Spectrum of a Diatomic Molecule. Intrinsic Spin and Projective Representations. The Stern-Gerlach Experiment. Spin Matrices. Paramagnetic Resonance and MRI Scanners. Orbital Angular Momentum and Spherical Harmonics.

  • Addition of Angular Momentum
    Combining the Angular Momenta of Two States. Examples. Angular Momenta of Operators and the Wigner-Eckart Theorem. Dipole Moment Transitions. Dynamical Symmetries of the Isotropic Harmonic Oscillator and of the Hydrogen Atom. Angular Momentum in Hydrogen.

  • Identical Particles
    Bosons and Fermions. Pauli's Exclusion Principle. The Periodic Table. Exchange Transformations and Parity in the Center of Momentum Frame. Intrinsic Parity and Decay Processes.

  • Time Independent Perturbation Theory
    An Analytic Expansion. Fine Structure of Hydrogen. Hyperfine Structure and the 21cm Line. The Ground State of Helium. The Quadratic Stark Effect. Degenerate Perturbation Theory. The Linear Stark Effect. Convergence of Perturbation Theory.

  • Time Dependent Perturbation Theory
    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
    The Density Operator. Pure and Mixed States and the Bloch Sphere. Von Neumann Entropy. The Gibbs Distribution. Reduced Density Operators for Subsystems. Decoherence and Time Evolution of Reduced Density Operators. Decoherence and Measurement. The EPR Gedankenexperiment. Bell's Inequality and the CHSH Inequality.

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.