Special Functions in Physics

Differential Equations · Axiom Academy

One equation, three geometries — the quantum harmonic oscillator, the hydrogen atom, and a particle boxed in a circle each hand you a different named special function. Every bound quantum system solves the same equation, — but separating it by the system's symmetry always leaves behind an ordinary differential equation whose solution is a special function . Drive all three below. The oscillator well — Hermite polynomials A mass on a quantum spring can only sit at specific energies. Drag the level n and watch the real wavefunction ride up the parabolic well — its shape is always a Gaussian times a Hermite polynomial. The hydrogen atom — Legendre & Laguerre polynomials Switch to spherical symmetry and the SAME separation trick splits the equation into three pieces: the angular part solves to associated Legendre functions, the radial part to associated Laguerre polynomials. Drag the quantum numbers and watch the orbital reshape. A particle boxed in a circle — Bessel functions Swap the boundary from a sphere to a circular wall — a particle in a circular well, or a drumhead pinned at the rim — and separation now leaves behind Bessel's equation. The wall forces the wavefunction to zero, and Bessel functions only hit zero at specific points — that's what quantizes the modes. Special functions aren't arbitrary — they're what falls out of separation of variables once you match the equation to the problem's symmetry. Same recipe, different geometry, different name.

This is the written version of the interactive lesson above. See the full Differential Equations course.