Hermite polynomial and transformation

[KFC]

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?

 

[Meir Achuz]

If you FT a product of two functions of x, you should get a convolution in k space, not a simple product.

 

[KFC]

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