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

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.