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Quantum Alchemy

Schrödinger [1, 2] made use of a factorization method on the hydrogen atom radial equation to show that all solutions can be generated starting with the ground state. Such procedures are now usually categorized as supersymmetric quantum mechanics. In an earlier publication [3], we dubbed this modern form of alchemy: "quantum alchemy".
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.

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The supersymmetric operator is given by . For example, (so that ). To apply the ladder operator for principal quantum numbers, we must first express the radial function in the form , where Then the quantum number is increased by 1 in the operation , where the square brackets represent the operator . For example, operating on a -orbital, for which , , , and , we find . The last expression reduces to , after setting
References
[1] E. Schrödinger, "A Method of Determining Quantum-Mechanical Eigenvalues and Eigenfunctions," Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 46, 1940 pp. 9–16.
[2] E. Schrödinger, "Further Studies on Solving Eigenvalue Problems by Factorization," Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, 46, 1941 pp. 183–206.
[3] S. M. Blinder, "Quantum Alchemy: Transmutation of Atomic Orbitals," Journal of Chemical Education, 78(3), 2001 pp. 391–394.
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