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Hermite polynomial and transformation

  1. Feb 15, 2009 #1


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    In the chapter of quantum harmonic oscillator, we use the Hermite polynomial a lot. And the fourier transformation of Hermite polynomial (in wavenumber space) gives

    [tex]\mathcal{F} \left\{ \exp (-x^2/2) H_n(x) \right\} = (-i)^n \exp (-k^2/2) H_n(k)[/tex]

    Now I need to find the similar result in terms of momentum p, I know the relation between wavenumber and momentum is

    [tex]p = \hbar k[/tex]

    But I still cannot transform above result to that written in terms of p. Any clue?
  2. jcsd
  3. Feb 16, 2009 #2


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    If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.
  4. Feb 16, 2009 #3


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    But I am talking about a special case: Hermite polynomial, so in this case, the above equation is correct. They are just simple product in k space
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