Gauge Variant Symmetries for the Schr\"odinger Equation

TL;DR

This paper explores gauge variant symmetries in the Schr"odinger equation, demonstrating that different gauge choices lead to solutions differing only by a phase, and analyzes the symmetry structure of the harmonic oscillator.

Contribution

It introduces a framework for understanding gauge variant symmetries in the Schr"odinger equation and constructs solutions based on these symmetries, highlighting their phase differences.

Findings

All Schr"odinger equations for the harmonic oscillator have the same Lie symmetries.

Solutions differ only by a gauge-dependent phase.

The gauge choice influences the phase but not the symmetry structure.

Abstract

The last multiplier of Jacobi provides a route for the determination of families of Lagrangians for a given system. We show that the members of a family are equivalent in that they differ by a total time derivative. We derive the Schr\"odinger equation for a one-degree-of-freedom system with a constant multiplier. In the sequel we consider the particular example of the simple harmonic oscillator. In the case of the general equation for the simple harmonic oscillator which contains an arbitrary function we show that all Schr\"odinger equations possess the same number of Lie point symmetries with the same algebra. From the symmetries we construct the solutions of the Schr\"odinger equation and find that they differ only by a phase determined by the gauge.