\(\text{Solve for}\,\psi\\ \frac{1}{2}(-\frac{\hbar^2}{m}\Delta+-\frac{1}{2}k\hat{x}^2)\psi=E\psi\)

I believe this is the Shrodinger equation in operator form for a quantum harmonic oscillator.

https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

thanks :)