The Schrödinger equation for a nonrelativistic hydrogenic atom has the form , with use of atomic units and infinite nuclear mass (, ). In our specific examples we take for the hydrogen atom itself. The solution is separable in spherical polar coordinates: . The are spherical harmonics; their transformation properties are well documented and we need not consider them further. We use the value in our illustrations of atomic orbitals. For bound states with , the normalized radial function can be expressed , where is an associated Laguerre polynomial and .

It is simpler to work with the reduced radial function , which obeys the pseudo one-dimensional differential equation . We consider the two "alchemical" transformations and , which have the following actions: and . The first operator, for example, turns a -orbital into a -orbital, while the second turns it into a -orbital.

This Demonstration shows a sequence of steps that can convert the ground state orbital into a chosen orbital with user specified values of and , up to and . Plots of the radial functions and are also shown.