Details

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.

Snapshots 1–3: the 1 ground state is nondegenerate and is annihilated by either operator or

Snapshots 4–6: the 3 state is transformed to 3 by

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.