The Harmonic Oscillator

Abstract

The harmonic oscillator is one of the most important systems of physics. It occurs almost everywhere where vibration is found—from the ideal pendulum to quantum field theory. Among other things, the reason is that the parabolic oscillator potential is a good approximation of a general potential V(x), if we consider small oscillations around a stable equilibrium position $x_{0}$. Thus, in this case we can approximate V(x) by the first terms of the Taylor series:

$$ V\left( x\right) =V\left( x_{0}\right) +\left( x-x_{0}\right) V^{\prime }\left( x_{0}\right) + \frac{1}{2}\left( x-x_{0}\right) ^{2}V^{\prime \prime }\left( x_{0}\right) +\cdots $$