Conformal Transformation of the Schr\"{o}dinger Equation for Central Potential Problems in Three-Dimensions

TL;DR

This paper demonstrates that a conformal transformation simplifies the Schr"{o}dinger equation for central potentials in three dimensions, removing potential fields and relating potential energy to ground state energy, applicable to hydrogen and harmonic oscillator.

Contribution

It extends a conformal transformation technique to the hydrogen atom, unifying it with the harmonic oscillator case and revealing a general potential-energy relationship.

Findings

Eliminates potential fields from the Schr"{o}dinger equation using conformal transformation.

Represents Coulomb and harmonic potentials as imaginary parts of complex time.

Derives a general relationship between potential energy and ground state energy.

Abstract

In a recent paper, it has been shown the Schr\"{o}dinger equation for the three-dimensional harmonic oscillator can be simplified through the use of an isometric conformal transformation. Here, it is demonstrated that the same transformation technique is also applicable to the Schr\"{o}dinger equation for the hydrogen atom. This approach has two interesting features. Firstly, it eliminates potential fields from the Schr\"{o}dinger equation. The Coulomb and harmonic binding terms are instead represented as imaginary parts of complex time. Secondly, the method leads to a general relationship between potential energy and ground state energy that encompasses both the hydrogen atom and the harmonic oscillator as special cases.