QHO

Quantum Physics

This is the potential of a quantum harmonic oscillator, and it applies to all the equations below

When the potential is that of a harmonic oscillator, the wave eigenfunctions of energy $\displaystyle E_{n}$ takes the above form
You may also substitute $\displaystyle k=m\omega ^{2}$ to get $\displaystyle U(x)=\frac{1}{2}m\omega ^{2}x^{2}$
$\displaystyle \xi\equiv \sqrt{ \frac{m\omega}{\hbar} }x$ for substitution
$\displaystyle \left( \frac{m\omega}{\pi \hbar} \right)^{\frac{1}{4}}\frac{1}{\sqrt{ 2^{n}n! }}$ is the normalization factor
$\displaystyle H_{n}(\xi)$ is a Hermite polynomial
Desmos Demo of Different Standing Waves

$\displaystyle \hat{a}_{+}$ is the raising operator
$\displaystyle {\psi}_{0}(x)$ is the ground state and is equal to $\displaystyle\left( \frac{m\omega}{\pi \hbar} \right)^{1/4}e^{-\frac{m\omega}{2\hbar}x^{2}}$

General wave function of a harmonic oscillator
$\displaystyle h(\xi)=a_{0}+a_{1}\xi+a_{2}\xi ^{2}+\ldots=\sum_{j = 0}^{\infty}a_{n}\xi^{n},a_{n+2}=\frac{2n-k+1}{(n+1)(n+2)}a_{n}$

Shows how energy is quantized in a harmonic oscillator
$\displaystyle E_{n}$ is the energy of $\displaystyle n$th energy state
$\displaystyle \hbar$ is the reduced planck constant
$\displaystyle \omega$ is the angular frequency of oscillation
$\displaystyle {\left\langle{K}\right\rangle}$ is the average kinetic energy of the particle
$\displaystyle {\left\langle{U}\right\rangle}$ is the average potential energy of the particle

$\displaystyle \frac{ \mathrm{d}^2 \psi}{ \mathrm{d} x^{2}}=\frac{2m}{\hbar}[V(x)-E]\psi$