View source for Solution for Simple Harmonic Oscillator (Xuen Zhen)

<b>Claimed by Lim, Xuen Zhen (Spring 2022)</b>
===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> F = -k x </math> and the associated potential function <math> U = \frac{1}{2} k x^2 </math>, with <math> 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> U </math> substituted with <math> \frac{1}{2} k x^2 </math>.
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<math>\frac{-\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + \frac{1}{2} k x^2 \Psi = E \Psi</math>
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The solution to this equation are the wave function <math> \Psi </math> and the energy function <math> E </math> that satisfies the above conditions.

===Deriving the Solution===

===Applications===