Wolfram Demonstrations Project

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

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.

Contributed by: S. M. Blinder

Automatic Animation

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

Snapshots 4–6: the 3

state is transformed to 3

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.

Supersymmetry for the Square-Well Potential (Wolfram Demonstrations Project)

"Hydrogenic Radial Functions via Supersymmetry" from the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/HydrogenicRadialFunctionsViaSupersymmetry/

Contributed by: S. M. Blinder