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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 are 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. The real part of the wavefunctions is plotted in the plane including the values and . The wavefunction is positive in the blue regions, negative in the white regions. The nucleus is shown as a black dot. The corresponding energy eigenvalues are given by , independent of other quantum numbers (in the field-free nonrelativistic case).
The hydrogen atom in a constant electric field ℰ along the direction is also separable in parabolic coordinates and can thus be used to treat the Stark effect. The functions and are more complicated but can be obtained by perturbation expansions. To first order, the Stark effect energies are given by . One atomic unit of electric field ℰ is equivalent to V/m. The presence of an electric field is shown by a red arrow.

SNAPSHOTS

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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.
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