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

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