Table of Contents

A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states.

So far, we have concentrated on problems that were analytically solvable, such as the simple harmonic oscillator, the hydrogen atom, and square well type potentials. In fact, we shall soon be confronted with situations where an exact analytic solution is unknown: more general potentials, or atoms with more than one electron. To make progress in these cases, we need approximation methods.

The Schrödinger equation for realistic systems quickly becomes unwieldy, and analytical solutions are only available for very simple systems - the ones we have described as fundamental systems in this module. Numerical approaches can cope with more complex problems, but are still (and will remain for a good while) limited by the available computer power. Perturbation theory is one such approximation that is best used for small changes to a known system, whereby the Hamiltonian is modified.

Almost everything we know about nuclei and elementary particles has been discovered in scattering experiments, from Rutherford’s surprise at finding that atoms have their mass and positive charge concentrated in almost point-like nuclei, to the more recent discoveries, on a far smaller length scale, that protons and neutrons are themselves made up of apparently point-like quarks.