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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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
[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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+