Solution for Simple Harmonic Oscillator

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 smooth parabolic 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] 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.

Deriving the Solution

For quantum harmonic oscillator, there are no boundaries between different regions, so one condition for the wave function is it must approach 0 as [math]\displaystyle{ }[/math] and [math]\displaystyle{ }[/math]. A simple general wave function that satisfies this requirement is [math]\displaystyle{ }[/math]. We begin the derivation with finding the first and second order differential of the general wave equation.
[math]\displaystyle{ }[/math]
[math]\displaystyle{ }[/math]
Substituting the differential equations into the time-independent Schrodinger equation produces
[math]\displaystyle{ }[/math]
One common point of confusion to be aware of is that this is not an equation to be solved for [math]\displaystyle{ x }[/math]. [math]\displaystyle{ x }[/math] is posing as a free variable here, and the solution is one that makes the Schrodinger equation true for any value of x.

Applications

While a one-dimensional quantum harmonic oscillator with smooth potential is virtually inexistant in the nature, there exists some systems that behave akin to one, such as a vibrating diatomic molecule. A complex potential with arbitrary smooth curve can also usually be approximated as a harmonic potential near its stable equilibrium point, also known as the minimum.