# Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

## Main Question or Discussion Point

How are Hermite Polynomials related to the solutions to the Schrodinger equation for a harmonic oscillator? Are they the solutions themselves, or are the solutions to the equation the product of a Hermite polynomial and an exponential function?

Thanks!!!

Related Quantum Physics News on Phys.org
Wavefunction of the n-th excited state of the harmonic oscillator is equal to the n-th Hermite polynomial times ${\large e^{-\frac{x^{2}}{4l}}}$ where $l = \sqrt{\frac{\hbar}{2m\omega}}$

jtbell
Mentor
The Hermite polynomials (or any polynomials for that matter) cannot be solutions of the SE over all x because they are not normalizable over all x.

Being a solution to the Schrodinger equation and being normalizable are two different things.