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?
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