DescriptionQuantum mechanically problems do not often have exact solutions, and it is often necessary to treat problems in `perturbation theory'; that is expanding our answer in some small parameter. One parameter which is always available in quantum theories is Planck's constant, which in everyday units is a ridiculously small number. The WKB or semi-classical approximation amounts to finding the leading order terms of a physical quantity, expressed as a power series in Planck's constant. The project will explore some of the fascinating mathematics associated with semi-classical approximation and its connexion to classical Hamiltonian mechanics. In the early formulation of quantum theory, Bohr-Sommerfeld quantisation involved assigning integer values to the `adiabatic invariants' of classical mechanics. The WKB approximation applied directly to Schrodinger's equation is intimately related to the Hamilton-Jacobi equation of classical mechanics, and had been discovered in the nineteenth century by Liouville and Green. In the path integral formulation of quantum mechanics the WKB approximation can be viewed as the method of steepest descent/stationary phase. The WKB approximation also has a number of important application in quantum field theory, from calculating classically forbidden tunneling effects to calculating scattering amplitudes and quantum corrections to the masses of soliton solutions. The project will explore a number of these applications in both quantum mechanics and quantum field theory.PrerequisitesQuantum Mechanics III (or Physics equivalent) is a prerequisite.ResourcesThe WKB approximation is covered in textbooks on Quantum Mechanics. There is a Wiki article if you want a quick look. A paper which goes in to considerably more depth is Berry and Mount. A good review of semi-classical methods in field theory is |
email: Peter Bowcock