Solution for Simple Harmonic Oscillator (Xuen Zhen)

Claimed by Lim, Xuen Zhen (Spring 2022)

Introduction

One of the many situations which can be analyzed with the Schrodinger’s Equation is the one-dimensional simple harmonic oscillator. This situation draws an analogy between a quantum-mechanical particle’s oscillating movement and the movement of a common classical oscillator: an object attached to a spring. Like its classical spring counterpart described under Hooke's Law, a quantum harmonic oscillator has the force function [math]\displaystyle{ F = -k x }[/math] and the associated potential function [math]\displaystyle{ U = \frac{1}{2} k x^2 }[/math], with [math]\displaystyle{ k }[/math] being the force constant (spring constant in classical case). Despite the simplicity of a harmonic oscillator's arbitrary smooth potential, it acts as an important foundation to solving more complicated quantum systems due to it being one of the few quantum-mechanical systems with an exact, known analytical solution.

Mathematical Setup

We may use the time-independent Schrodinger's equation to represent the state of a quantum particle in the harmonic potential by substituting the potential [math]\displaystyle{ U }[/math] substituted with [math]\displaystyle{ \frac{1}{2} k x^2 }[/math].
[math]\displaystyle{ \frac{-\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{2} k x^2 \Psi = E \Psi }[/math]
The solution to this equation are the wave function [math]\displaystyle{ \Psi }[/math] and the energy function [math]\displaystyle{ E }[/math] that satisfies the above conditions.