The Hydrogen Orbitals
A nice and simple quantum mechanical screensaver

A hydrogen atom is a relatively simple system, having a single proton and a single electron bound to the proton. Even in quantum mechanics, this two-body problem has an analytic solution without requiring the construction of atomic orbital basis sets. Indeed, the hydrogen atomic orbitals may be used as a basis, though it is not often done in practice for computational reasons.

The wavefunction of the bound electron in a hydrogen atom may be obtained by separation of variables in a spherical polar coordinate system, like so:

More specifically, for the energy and angular momentum quantum numbers $n$, $l$ and $m$, the wavefunction has the form:

where the normalization factor equals:

The next term equals the generalized Laguerre polynomial:

And the final term equals the spherical harmonic equation:

Finally, the term in the spherical harmonic equation represents the associated Legendre polynomial:

While these equations seem messy, they are directly computable, given a few simple data structures for handling polynomial expressions.

The screensaver randomly generates a set of valid quantum numbers, and then
calculates slices of the wavefunction probability density and phase. I also
actually implemented an octree-based representation of the wavefunction
that could be used in a marching cubes algorithm for displaying contours,
but the slices were so pretty, I didnâ€™t finish it. **:)**

Of course, this page would be pretty useless without source code and final product, right?