Solutions to the Harmonic Oscillator Equation and Hermite Polynomials

[*FaerieLight*]

How are Hermite Polynomials related to the solutions to the Schrödinger 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!

 

[saim_]

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]

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.

 

[matonski]

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