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Quantum Mechanics

Introduction

This course is given to all second year physicists and is examined on paper A3 at the end of the second year. The lecture course begins in Michaelmas Term 2013 (12 lectures) with the remainder of the lectures in Hilary Term 2014 (15 lectures).

In this course we introduce the subject of quantum mechanics.

Synopsis

Probabilities and probability amplitudes. Interference, state vectors and the bra-ket notation, wavefunctions. Hermitian operators and physical observables, eigenvalues and expectation values. The effect of measurement on a state; collapse of the wave function. Successive measurements and the uncertainty relations. The relation between simultaneous observables, commutators and complete sets of states.

The time-dependent Schrödinger equation. Energy eigenstates and the time-independent Schroedinger equation. The time evolution of a system not in an energy eigenstate. Wave packets in position and momentum space.

Probability current density.

Wave function of a free particle and its relation to de Broglie's hypothesis and Planck's relation. Particle in one-dimensional square-well potentials of finite and infinite depth. Scattering off, and tunnelling through, a one-dimensional square potential barrier. Circumstances in which a change in potential can be idealised as steep; [Non examinable: Use of the WKB approximation.]

The simple harmonic oscillator in one dimension by operator methods. Derivation of energy eigenvalues and eigenfunctions and explicit forms of the eigenfunctions for n=0,1 states.

Amplitudes and wave functions for a system of two particles. Simple examples of entanglement.

Commutation rules for angular momentum operators including raising and lowering operators, their eigenvalues (general derivation of the eigenvalues of L2 and L not required), and explicit form of the spherical harmonics for l=0,1 states. Rotational spectra of simple diatomic molecules.

Representation of spin-1/2 operators by Pauli matrices. The magnetic moment of the electron and precession in a homogeneous magnetic field. The Stern-Gerlach experiment. The combination of two spin-1/2 states into S=0,1; [non-examinable: Derivation of states of well-defined total angular momentum using raising and lowering operators]. Rules for combining angular momenta in general (derivation not required). [Non-examinable: Spectroscopic notation.]

Hamiltonian for the gross structure of the hydrogen atom. Centre of mass motion and reduced particle. Separation of the kinetic-energy operator into radial and angular parts. Derivation of the allowed energies; principal and orbital angular-momentum quantum numbers; degeneracy of energy levels. Functional forms and physical interpretation of the wavefunctions for n<3.

Course-related links
Textbooks
  • Recommended textbooks are listed in the first course handout.
Quantum-related links
  • to come
(Updated: October 2013)
Email: s dot blundell at physics.ox.ac.uk