# Hydrogenic Radial Functions via Supersymmetry

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An application of supersymmetric quantum mechanics enables all the bound-state radial functions for the hydrogen atom to be evaluated using first-order differential operators, without any explicit reference to Laguerre polynomials.

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Contributed by: S. M. Blinder (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## 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.

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