# The Hydrogen Atom in Parabolic Coordinates

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The Schrödinger equation for the hydrogen atom, (in atomic units ), can be separated and solved in parabolic coordinates as well as in the more conventional spherical polar coordinates . This is an indication of degeneracy in higher eigenstates and is connected to the existence of a "hidden symmetry", namely the Lie algebra associated with the Coulomb problem. Parabolic coordinates can be defined by , , with the same as in spherical coordinates. The wavefunction is separable in the form with . Here is a Whittaker function and , equal to the principal quantum number. Contour plots for the real part of the wavefunctions in the -plane are shown, including the values and . The nucleus is represented as a black dot. The corresponding energy eigenvalues are given by , independent of other quantum numbers (in the field-free nonrelativistic case).

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: the ground state

Snapshot 2: a - hybrid atomic orbital

Snapshot 3: state in an electric field

Reference: H. Bethe and E. R. Salpeter, *Quantum Mechanics of One- and Two-Electron Atoms*, New York: Academic Press, 1957 pp. 27–29 and 228–234.

## Permanent Citation