LaTeX Demo: 1D Harmonic Oscillator

The time-independent Schrödinger equation: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2\,\psi \,=\, E\,\psi. $$ Energies: $E_n = \hbar\omega\left(n+\tfrac12\right)$.

Normalized ground state: $$ \psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}}. $$