# Quantum Alchemy

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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".

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Contributed by: S. M. Blinder (August 2013)

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## Details

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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