Hermite polynomial and transformation

by KFC
Tags: hermite, polynomial, transformation
 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 $$\mathcal{F} \left\{ \exp (-x^2/2) H_n(x) \right\} = (-i)^n \exp (-k^2/2) H_n(k)$$ Now I need to find the similar result in terms of momentum p, I know the relation between wavenumber and momentum is $$p = \hbar k$$ But I still cannot transform above result to that written in terms of p. Any clue?
 Sci Advisor P: 1,233 If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.
P: 369
 Quote by clem 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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