the quantum harmonic oscillator
The quantum harmonic oscillator is the single most useful exactly solvable model in all of physics. Take a particle in a parabolic bowl — a mass on an ideal spring — and let it be quantum. Because almost any smooth potential looks parabolic near its minimum, this one model approximates the low-energy behavior of an astonishing range of systems, from a vibrating molecule to a mode of the electromagnetic field.
Its Hamiltonian is H = p^2/(2m) + (1/2) m omega^2 x^2, and the time-independent Schrodinger equation for it has the remarkably clean spectrum E_n = (n + 1/2) hbar omega for n = 0, 1, 2, .... The levels are equally spaced by hbar omega, unlike the crowding levels of an atom, and the lowest level is not zero but E_0 = (1/2) hbar omega, the zero-point energy demanded by the uncertainty principle. The eigenfunctions are Hermite polynomials multiplied by a Gaussian envelope, psi_n(x) proportional to H_n(x) exp(-m omega x^2 / 2 hbar).
Why it is everywhere: near any potential minimum V(x) is approximately V(x_0) + (1/2) V''(x_0)(x - x_0)^2, so molecular vibrations, lattice vibrations (phonons), and small oscillations of any field reduce to harmonic oscillators. Quantizing the electromagnetic field mode by mode gives one oscillator per mode, and its quanta are photons — so the harmonic oscillator is quite literally the mathematical seed of quantum field theory. It is most elegantly solved not by wrestling with the differential equation but with ladder operators.
A diatomic molecule's stretching vibration behaves like a harmonic oscillator: it absorbs infrared light only at photon energies matching the level spacing hbar omega, and even in its ground state retains the zero-point energy (1/2) hbar omega.
Evenly spaced levels and an irreducible zero-point energy: the fingerprints of the quantum oscillator.
It is an idealization valid near a potential minimum, where the potential is well approximated as quadratic; real bonds are anharmonic, so at high excitation the equal spacing and exact Hermite solutions break down.