The one-dimensional harmonic oscillator

  • #1
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Main Question or Discussion Point

Dealing with the one-dimensional harmonic oscillator I'm trying to find a general formula for
[tex]
\int_{-\infty}^{\infty} \xi^p H_n(\xi) H_k(\xi) d\xi
[/tex]
there [tex]H_n(\xi) [/tex] and [tex]H_k(\xi)[/tex] are hermite polynomials and p is an integer ( [tex]p\geq 0[/tex]).
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:

Answers and Replies

  • #2
dextercioby
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eys_physics said:
Dealing with the one-dimensional harmonic oscillator I'm trying to find a general formula for
[tex]
\int_{-\infty}^{\infty} \xi^p H_n(\xi) H_k(\xi) d\xi
[/tex]
there [tex]H_n(\xi) [/tex] and [tex]H_k(\xi)[/tex] are hermite polynomials and p is an integer ( [tex]p\geq 0[/tex]).
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
 
  • #3
dextercioby
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I think it's much more that in Abramowitz-Stegun.
 
  • #4
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Hey
Can you tell me that you mean with Aramowitz-Stegun?
 
  • #5
dextercioby
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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...
 
  • #6
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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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