# 13.5: The Simple Harmonic Oscillator

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One important potential energy function is the Simple Harmonic Oscillator, or SHO. This is the potential energy of a spring (so long as you don’t stretch of squish the spring too much). It also turns out to be a decent approximation, at least for lower energy levels, for a number of quantum systems. One such system is the vibrational energy states of a Hydrogen molecule $$H_{2}$$. The form of this potential, in one dimension, is:

$V(x)=\frac{1}{2} m \omega^{2} x^{2}\tag{13.9}$

Here, $$m$$ is the mass of the particle moving in the potential. $$\omega$$ is the “natural frequency of oscillation” for the potential; for a classical spring, it would correspond to $$2 \pi / T$$, where $$T$$ is the period of oscillations. (Of course, for a classical spring, the system could also have any energy!)

The solution to the one dimensional Schrödinger equation for this potential gives the following energies for the energy eigenstates:

$E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega\tag{13.10}$

where $$n$$ is an integer 0, 1, 2, . . .. As written, this potential is an infinitely high potential $$(V(x)$$ just keeps going up as $$x$$ gets farther and farther from 0.) As such, there are an infinite number of allowed energy levels. Of course, as an approximation to a real physical system, usually the approximation will get worse and worse as $$x$$ gets farther and farther from 0, which means that the solutions less and less of a good approximation to the real energy system for higher and higher energy levels.

This page titled 13.5: The Simple Harmonic Oscillator is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pieter Kok via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.