quantum harmonic oscillator

Think of a mass on a spring, or a child on a swing: pull it from rest and a restoring force pulls it back, so it rocks back and forth at a steady rhythm. Now shrink that system down to the size of two atoms joined by a chemical bond, which acts like a spring between them. At that scale the back-and-forth can no longer take any energy it likes. The quantum harmonic oscillator is the model that describes such tiny springs with the rules of quantum mechanics.

More precisely, the quantum harmonic oscillator is a particle held by a restoring force that grows in proportion to how far it is pushed from its resting point — the same force law as an ideal spring. Solving the Schrödinger equation for it gives energy levels that are evenly spaced, like the rungs of a perfectly regular ladder, separated by a step set by Planck's constant and the spring's stiffness. The lowest rung sits above zero, giving the oscillator its zero-point energy.

The honest qualifier is that real molecular bonds are only approximately harmonic — stretch a bond far enough and it eventually snaps, which a true spring never does, so the higher levels actually crowd closer together rather than staying evenly spaced. Even so, the harmonic oscillator is the indispensable starting model for molecular vibrations, and it is the foundation for reading infrared and Raman spectra, where each absorbed frequency reports a vibrating bond.