Symmetries of Schroedinger equation with scalar and vector potentials

TL;DR

This paper classifies Lie symmetries of time-dependent Schrödinger equations with scalar and vector potentials, revealing new symmetry structures and mappings, including a connection between free particles and oscillators.

Contribution

It provides a comprehensive classification of symmetries for Schrödinger equations with scalar and vector potentials, including new mappings and equivalence relations.

Findings

Complete classification of symmetries for these equations.

Identification of a mapping between free Schrödinger equation and repulsive oscillator.

Specification of symmetry and equivalence relations.

Abstract

Using the algebraic approach Lie symmetries of time dependent Schroedinger equations for charged particles interacting with superpositions of scalar and vector potentials are classified. Namely, all the inequivalent equations admitting symmetry transformations with respect to continuous groups of transformations are presented. This classification is completed and includes the specification of symmetries and admissible equivalence relations for such equations. In particular, a simple mapping between the free Schroedinger equation and the repulsive oscillator is found which has a clear group-theoretical sense.