Supersymmetric quantum mechanics can be applied to the solution of the hydrogenic radial equation, treated as a pseudo-one-dimensional problem in the variable
with effective Hamiltonians denoted
. There exist two partner Hamiltonians for each value of
, which can be written
. The superpotential
is given by
as defined above,
. The lowest-energy eigenstate of
has no partner eigenstate, but all higher-energy eigenstates have degenerate supersymmetric partners. These can be labeled by increasing values of the principal quantum number
, beginning with
. The composite pattern for all
-values leads to the characteristic degeneracies for
in a pure Coulomb field, associated with a higher symmetry than would be implied by spherical invariance alone.
Reference: A. Valance, T. J. Morgan, and H. Bergeron, "Eigensolution of the Coulomb Hamiltonian via Supersymmetry," American Journal of Physics
(5), 1990 pp. 487–491.