Wave Functions for Quantum Harmonic Oscillators

Submicroscopic harmonic oscillators are popular quantum physics problems because harmonic oscillators are relatively simple systems — the force that keeps a particle bound here is proportional to the distance that the particle is from the equilibrium point.

Here’s the harmonic oscillator at its simplest:

where

And here’s the next-higher state:

In general, you can use the following equation for the wave functions, where Hn is a hermite polynomial:

where