Hydrogen atom as a four-dimensional oscillator

A coordinate transformation which exhibits the rotational invariance of the hydrogen atom in four-dimensional Hilbert space is introduced. The coordinates are shown to be directly related to the spherical polar and parabolic coordinates in position space. With the use of the transformation, the Schrödinger equation for the hydrogen atom left-multiplied by $4r$ is transformed into one for a four-dimensional harmonic oscillator. Solutions are obtained and related to the hydrogenic wave functions. Group-theoretical implications of the transformation and its application to the hydrogen Stark problem are briefly discussed.