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{d^2 \Psi}{d 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{ x → +∞ }[/math] and [math]\displaystyle{ x → -∞ }[/math]. A simple general wave function that satisfies this requirement is [math]\displaystyle{ \Psi (x) = A e^{-ax^2} }[/math]. We begin the derivation with finding the second order differential of the general wave equation.

[math]\displaystyle{ \frac{d \Psi}{d x} = -2 a x (A e^{-ax^2}) = -2 a x \Psi }[/math]

[math]\displaystyle{ \frac{d^2 \Psi}{d x^2} = -2 a (A e^{-ax^2})-2 a x (-2 a x)(A e^{-ax^2}) = (-2 a+4a^2x^2)A e^{-ax^2} = (-2 a+4a^2x^2)\Psi }[/math]

Substituting the differential equation into the time-independent Schrodinger equation produces

[math]\displaystyle{ \frac{-\hbar^2}{2m} (-2 a+4a^2x^2)\Psi + \frac{1}{2} k x^2 \Psi = E \Psi }[/math]
[math]\displaystyle{ \frac{\hbar^2 a}{m}\Psi+\frac{2 a^2 \hbar^2 x^2}{m}\Psi + \frac{1}{2} k x^2 \Psi = E \Psi }[/math]
[math]\displaystyle{ \frac{\hbar^2 a}{m}+\frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2 = E }[/math]

One common misconception 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 [math]\displaystyle{ x }[/math]. To allow this equation to be consistent for any [math]\displaystyle{ x }[/math], the coefficients to [math]\displaystyle{ x^2 }[/math] must cancel out, leaving the remaining constants to be equal to each other.

[math]\displaystyle{ \frac{\hbar^2 a}{m}+ }[/math]

[math]\displaystyle{ \frac{2 a^2 \hbar^2}{m}x^2 + \frac{1}{2} k x^2 }[/math]

[math]\displaystyle{ = E }[/math]

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.