Brian Blais’ Homemade Guide to the Harmonic Oscillator

Everyone who initially studies the harmonic oscillator is immediately barraged with examples of pendula and blocks with springs. If the only significance of the harmonic oscillator lay in these examples it would be pretty limiting. In physics, the harmonic oscillator is an extremely important problem, for a simple reason. Say, for example, you have a potential, V (x), for some system and you want to find the equations governing the motion of a particle under the influence of this potential. You might have a very difficult time solving this problem, especially if the potential is complicated. You then might think to look at the behavior of the particle around the minimum of this energy, which would be the behavior near equilibrium. One can do this simply by performing a Taylor expansion of the potential, V , about a minimum xo. For clarity we look only at one dimension.