Regularisation, Renormalisation and Anomalies in Quantum Mechanics
This bachelor thesis studies three different singular potentials: the one dimensional δ-potential and its derivative, δ , as well as the two dimensional δ (2) -potential. The latter two potentials require regularisation and renormalisation and δ (2) demonstrates anomalous symmetry breaking: the breaking of a symmetry (in this case scale invariance) through regularisation. The δ-potential in one dimension does not require regularisation but it is done anyway, to show that it produces the same results. Bound states and scattering coefficients are calculated with boundary conditions and Fourier transforms and it is regularised using a finite well. The δ needs both regularisation and renormalisation. The potential is regularised by a finite well and barrier (called the ’threshold potential’), by two approaching δ-functions and by using Fourier transforms. The potential strength is renormalised in both cases to keep the energy levels, which are physical observables, finite and independent of any parameters connected to the regularisation. The two dimesional δ (2) -function’s bound states are regularised with Fourier transforms, while both bound states and scattering coefficients are regularised by a finite well.
LU TP 15-20