Dual wavefunctions in two-dimensional quantum mechanics

TL;DR

This paper demonstrates that certain two-dimensional quantum systems have dual wavefunctions where amplitude and phase are interchangeable, revealing new symmetries and extending to optical analogues.

Contribution

It introduces a class of dual wavefunctions in 2D quantum mechanics with potential applications to optical systems, expanding understanding of wavefunction symmetries.

Findings

Dual solutions for harmonic oscillator and hydrogen atom systems.

Dual wavefunctions are solutions with vanishing Bohm potential.

Extension of dual wavefunctions to optical analogue systems.

Abstract

It is shown that the Schrodinger equation for a large family of pairs of two-dimensional quantum potentials possess wavefuctions for which the amplitude and the phase are interchangeable, producing two different solutions which are dual to each other. This is a property of solutions with vanishing Bohm potential. These solutions can be extended to three-dimensional systems. We explicitly calculate dual solutions for physical systems, such as the repulsive harmonic oscillator and the two-dimensional hydrogen atom. These dual wavefunctions are also solutions of an analogue optical system in the eikonal limit. In this case, the potential is related to the refractive index, allowing the study of this two-dimensional dual wavefunction solutions with an optical (analogue) system.