# The one-dimensional harmonic oscillator

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.

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## Answers and Replies

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

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

dextercioby
Science Advisor
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:
"I.S.Gradshteyn/I.M.Ryzhik:<<Table of Integrals,Series and Products>>,Corrected and Enlarged Edition,Academic Press Inc.,1980".Also famous.

But it's much easier with the "functions.wolfram.com" website.
I think it's free...

Hey, I doubt that is the integral you wish to calculate for in dealing with the oscilator in QM you always have a gaussian in there as the weighing function. Anyway, try integrating by parts...

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