The Hydrogen Atom in Parabolic Coordinates
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).
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
https://demonstrations.wolfram.com/TheHydrogenAtomInParabolicCoordinates/
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