Renormalization in non-relativistic quantum mechanics

A problem of non-relativistic quantum mechanics solved using regularization and renormalization techniques is presented in this thesis. After a general introduction of these techniques, they are applied to a problem in classical electromagnetism and to the bound state of a single quantum particle subjected to a two-dimensional delta-function potential, that is divergent if computed naively solving directly the Schroedinger equation or using the theory of propagators. The regularization techniques used are the cutoff regularization and the dimensional one and they both leads to the same outcome. An effective field theory approach, in which the potential is regularized through the real space scheme, is also presented. After regularization has been performed, the potential is renormalized re-defining the coupling constant. The running of the renormalized coupling constant is also found, i.e. the renormalization group equation.