# The one-dimensional harmonic oscillator

## Main Question or Discussion Point

Dealing with the one-dimensional harmonic oscillator I'm trying to find a general formula for
$$\int_{-\infty}^{\infty} \xi^p H_n(\xi) H_k(\xi) d\xi$$
there $$H_n(\xi)$$ and $$H_k(\xi)$$ are hermite polynomials and p is an integer ( $$p\geq 0$$).
I can found the answer for p=0 and p=1 but I can't find the formula for a general p so I need some hint how to do it.

Last edited:

Related Quantum Physics News on Phys.org
dextercioby
Homework Helper
eys_physics said:
Dealing with the one-dimensional harmonic oscillator I'm trying to find a general formula for
$$\int_{-\infty}^{\infty} \xi^p H_n(\xi) H_k(\xi) d\xi$$
there $$H_n(\xi)$$ and $$H_k(\xi)$$ are hermite polynomials and p is an integer ( $$p\geq 0$$).
I can found the answer for p=0 and p=1 but I can't find the formula for a general p so I need some hint how to do it.

I can't give you any hint,just the result:
http://functions.wolfram.com/PDF/HermiteH.pdf

dextercioby
Homework Helper
I think it's much more that in Abramowitz-Stegun.

Hey
Can you tell me that you mean with Aramowitz-Stegun?

dextercioby
Homework Helper
eys_physics said:
Hey
Can you tell me that you mean with Aramowitz-Stegun?
It's "Milton Abramowitz and Irene A.Segun:<<Handbook of Mathematical Functions>>,Dover Publications Inc.,NewYork".Any edition.Famous book among physicists.
A better book for the integrals part is obviously: