Introduction
This course is given to all second year physicists
and is examined on paper A3 at the end of the second year.
For the academic year 2015/2016, the lecture course began in Michaelmas Term 2015 (12 lectures) with the remainder of the lectures in Hilary Term 2016 (15 lectures). [I gave this course from 2013 until 2018.]
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.
