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The one-dimensional harmonic oscillator

  1. Nov 5, 2004 #1
    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: Nov 5, 2004
  2. jcsd
  3. Nov 5, 2004 #2

    dextercioby

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    I can't give you any hint,just the result:
    http://functions.wolfram.com/PDF/HermiteH.pdf
     
  4. Nov 5, 2004 #3

    dextercioby

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    I think it's much more that in Abramowitz-Stegun.
     
  5. Nov 8, 2004 #4
    Hey
    Can you tell me that you mean with Aramowitz-Stegun?
     
  6. Nov 8, 2004 #5

    dextercioby

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    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...
     
  7. Nov 9, 2004 #6
    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...
     
    Last edited: Nov 9, 2004
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