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Hermite polynomial and transformation

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KFC
#1
Feb15-09, 10:23 PM
P: 369
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?
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clem
#2
Feb16-09, 08:17 AM
Sci Advisor
P: 1,269
If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.
KFC
#3
Feb16-09, 08:30 AM
P: 369
Quote Quote by clem View Post
If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.
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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