  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.
