Hermite's equation
/ Hermite: air-MEET /
The quantum harmonic oscillator — a particle in a parabolic potential, the model for a vibrating molecular bond and for every field mode in quantum field theory — has a Schrodinger equation that, after stripping off a Gaussian factor, reduces to one clean equation. Hermite's equation is that equation: the polynomial heart of the oscillator's wavefunctions.
It is y'' - 2x y' + 2 lambda y = 0, with no finite singular points (every point on the real line is ordinary), so you solve it by an ordinary power series about x = 0. The recurrence a_{k+2} = 2(k - lambda)/((k+2)(k+1)) a_k shows the structure plainly: for a general lambda the series runs forever and grows like e^{x^2}, which when multiplied back by the Gaussian wavefunction factor still blows up — physically forbidden. The only escape is for the series to terminate, which happens exactly when lambda is a non-negative integer n; then a_{n+2} = 0 and the solution becomes the Hermite polynomial H_n(x) of degree n.
That termination condition is the cleanest example in all of physics of how a boundary condition quantizes a continuous parameter: requiring a normalizable (finite-energy) wavefunction forces n to be an integer, and the oscillator's energy levels E = (n + 1/2) times h-bar omega fall straight out. H_0 = 1, H_1 = 2x, H_2 = 4x^2 - 2, and so on are exactly the polynomial parts of the oscillator's ground, first-excited, and higher states. Hermite functions also appear in probability (Gaussian moments, the Edgeworth expansion) and in signal processing.
For lambda = 2 the equation y'' - 2x y' + 4 y = 0 truncates to give H_2(x) = 4x^2 - 2. The full oscillator wavefunction is this polynomial times the Gaussian e^{-x^2/2}; the polynomial supplies the two nodes (sign changes) of the second-excited state.
The polynomial degree n equals the number of nodes — directly the quantum number.
Two normalisation conventions for H_n coexist (the 'physicists' Hermite polynomials with weight e^{-x^2} and the 'probabilists' He_n with weight e^{-x^2/2}). They differ by a scaling of x and a constant factor — always check which one a formula assumes.