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. I will be lecturing this course for the first time in Michaelmas 2017, and the notes will appear here as they become available.


Contents


  • Introduction:   PDF File
    Preliminaries. A very brief introduction to classical and quantum mechanics.
  • Hilbert Space:   PDF File
    Definition of Hilbert Space. Dirac notation. Continuum states. Linear operators. Postulates of quantum mechanics. The generalised uncertainty principle.
  • Transformations and Symmetries:   PDF File
    Transformations of states and operators. Spatial translations. Rotations. Translations around a circle. Parity. Time translations. The Heisenberg picture. Symmetries and conservation laws. Dynamics.
  • Angular Momentum:   PDF File
    Angular momentum eigenstates. Rotation of diatomic molecules. Orbital angular momentum. Wavefunctions for orbital angular momentum eigenstates. Spin. SO(3), SU(2) and their representations. The Stern-Gerlach experiment. Precession in a magnetic field.
  • 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.
  • Dynamical Symmetries:   PDF File
    The Harmonic oscillator. Raising and lowering operators and SU(3) dynamical symmetry. Angular momentum of the harmonic oscillator. Dynamics of oscillators. Anharmonic oscillations. The Hydrogen atom. The Runge-Lenz operator. SU(2)×SU(2) dynamical symmetry of the gross structure of Hydrogen.
  • 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. Atomic transitions and their selection rules. Absorption, stimulated and spontaneous emission of photons.
  • Interpretation of Quantum Mechanics:
    The Copenhagen and Many Worlds interpretations. Entanglement. Pure and mixed states. The density matrix. The EPR gedankenexperiment. Bell inequalities and hidden variable theories.

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.