The Lie group reduction of two-dimensional Schrodinger equation

The Schrodinger equation describes how the quantum state of some physical systems change with respect to time. In this thesis, we consider a two-dimensional Schrodinger equation and we obtain the most general form of Lie-reduced equation that is solvable. By applying a general Lie group method for a class of infinitesimal transformations, we obtain the reduced system of equations and the group generators under which the original equation remains invariant. Furthermore, the related characteristic equations are constructed and the set of constraints that allows those generators to be valid is given. We have shown that the two-dimensional Schrodinger equation can be converted to a steady state Fokker-Planck equation by using a class of transformations. We have concluded that the solutions of such an equation vanish on the boundary of any infinite domain.