Overview
The Harmonic Oscillator is a class problem in Physics. It represents the movement of a weight on a spring. The equation governing its dynamics in classical physics is
{% \frac{d ^2y(x)}{dx^2} + kx^2 = 0 %}
It represents a Kinetic Energy term {% \frac{d ^2y(x)}{dx^2} %} and a potential term {% kx^2 %}.
Translated into the framework of the Schrodinger Equation, it becomes
{% (- \frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + \frac{1}{2}k x^2) \psi = E \psi %}
(here we have )