Solve Hermite Equation & Find Its Use in Math Physics

[darkone1687]

I have a problem for a Math Physics course, that I was suppose to find the Hermite equation, find solutions to it, plug them into see if they work, and lastly find a use for the Hermite equation. I've done everything but I can't find a use for the equation I looked almost everywhere I would really applicate the help if someone could give me a use for the equation, it would be best if it was used in physics but if not it's alright. THANK YOU for the help!

 

[Mute]

Hermite Equation as in the differential equation producing the Hermite polynomials?

For the case of a particle in a harmonic potential (\frac{1}{2}m\omega^2x^2) in quantum mechanics, the Schrodinger Equation (once rescaled to be in terms of nondimensional variables) has solutions which are proportional to \exp[-x^2/2] times the Hermite polynomials.

http://en.wikipedia.org/wiki/Hermite_polynomials

 

[darkone1687]

I can't find the series solution to this equation can someone post it here please, also I was wondering how it fit in with shrodingers equation I'm suppose to somehow use Hermite to get to Schrodinger, or vice versa?

 

[HallsofIvy]

You say you've "done everything" but can't find the series solution?

I googled on "Hermite equation" and the very first website given,
http://www.sosmath.com/diffeq/series/series06/series06.html
gives the series solution!