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

and

, with

and

. The superpotential

is given by

. With

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*,

**58**(5), 1990 pp. 487–491.