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Hydrogenic Radial Functions via Supersymmetry

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.
The nonrelativistic hydrogen-like system with atomic number and assumed infinite nuclear mass satisfies the Schrödinger equation , in atomic units . Separation of variables in spherical polar coordinates gives . Defining the reduced radial function , the radial equation can be expressed as . For the case (the 1, 2, 3, 4, … states) the radial function has the nodeless form .
As shown in the Details section, operators and are defined with the effect of lowering or raising the quantum number by 1, respectively, when applied to the radial function , namely, and . The constants are most easily determined after the fact by the normalization conditions .
In this Demonstration, you can plot any radial function with to and show the result of applying (red curve) or (blue curve). The plots pertain to the case . You can also choose to view the results as formulas (with variable ) or on an energy-level diagram.

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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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