In this module we begin the task of marrying quantum mechanics with the other foundation stone of modern physics - special relativity. The tests of this theory have come from high energy particle scattering experiments but the implications are apparent even at low energies.
 
We first include magnetic fields into quantum systems via minimal coupling. We then develop an armoury of tools to obtain approximate solutions. We use semiclassical methods (WKB), the variational approach and finally perturbation theory. We apply perturbation theory to both time-dependent and time-independent problems. This naturally leads to ideas of scattering theory where we develop the Born approximation and then the partial wave analysis.
 
Finally we extend the Schrodinger equation to include special relativity via the Klein-Gordon and ultimately the Dirac equation. This leads to the notion of antiparticles and spin. The true understanding of these equations is only made apparent in quantum field theory which is a natural follow up for this module.

By the end of the module the student will be able to:

• Include magnetic fields into quantum systems;
• Develop and use the WKB method for obtaining approximate solutions to 1D quantum problems;
• Develop and use perturbation theory for time-independent, time-dependent and degenerate problems;
• Derive and use the Fermi golden rule;
• Understand basic ideas of scattering theory, differential cross-section and total cross-section;
• Develop and use the Born approximation and partial wave analysis;
• Derive the Klein-Gordon and Dirac equations from first principles;
• Derive the probability current and density from the Klein-Gordon equation and discuss the perceived difficulties;
• Solve these equations and physically understand the solutions including the origin of spin and antiparticles.