Laguerre's equation

The radial part of the hydrogen atom — how the electron's wavefunction depends on distance from the nucleus — reduces, after pulling out an exponential decay factor, to one named equation. Laguerre's equation is that equation: the equation whose polynomial solutions shape the radial structure of every atomic orbital.

In its basic form it is x y'' + (1 - x) y' + lambda y = 0 on the half-line x >= 0, with a regular singular point at the origin x = 0 (and an irregular point at infinity). Solving by series gives a one-step recurrence; demanding a solution that does not grow too fast at infinity forces lambda to be a non-negative integer n, and the series truncates into the Laguerre polynomial L_n(x). The hydrogen atom actually needs the associated Laguerre equation, x y'' + (alpha + 1 - x) y' + lambda y = 0, whose polynomial solutions L_n^{alpha}(x) carry an extra parameter alpha tied to the angular momentum.

Here the truncation condition is what quantizes the principal quantum number and hence the energy levels E_n proportional to -1/n^2 of hydrogen — the Rydberg formula falls out of requiring the radial series to terminate. The natural weight for Laguerre polynomials is e^{-x} on [0, infinity), which is exactly why they pair so well with the exponential tail of a bound electron. They also show up in the quantum oscillator in polar coordinates and in Gauss-Laguerre numerical integration of integrals weighted by e^{-x}.