Form-preserving transformations of the Schrödinger equation

Abstract

Coordinate transformations of differential equations have long been studied in the context of mathematics and physics. They allow us to change a differential equation into one that is easier to solve. In nonrelativistic quantum mechanics, the time evolution of a wave function is determined by the time-dependent Schrodinger equation (TDSE). Since quantum mechanics must work in every nonrelativistic frame, there must be a TDSE for every set of coordinates one chooses to measure in. Coordinate transformations between two reference frames must then transform one Schrodinger equation into another. Called form-preserving transformations (FPTs), these transformations allow for many puzzling solutions to the TDSE and can be used for the efficient determination of symmetry groups.

In this work, we will determine the most general allowed FPT for the Schrodinger-Pauli equation of a spinless charged particle in N-dimensions. Furthermore, we show that the FPTs form a continuous Lie group, whose algebra is discussed in detail. Well-known symmetry groups such as the Galilean and Schrodinger groups are shown to be subgroups of the form-preserving group. We conclude with an analysis of FPTs in the phase-space formulation of quantum mechanics.