How to get position operator in momentum space?

[azoroth134]

Hi, I wish to get position operator in momentum space using Fourier transformation, if I simply start from here,

$$ <x>=\int_{-\infty}^{\infty} dx \Psi^* x \Psi $$

I could do the same with the momentum operator, because I had a derivative acting on |psi there, but in this case, How may I get the ih d/dp thing for the position operator, please give some hint

 

[strangerep]

Well, since you only wanted a hint...

Express $\Psi(x)$ in terms of its Fourier transform $\tilde\Psi(p)$ .

 

[azoroth134]

I already did that, I just don't know what to do after that, I don't have any derivative to perform on x or p

 

[blue_leaf77]

For a function $f(x)$ and its Fourier transform $F(k)$ (assuming it has one), we have the relation $f'(x) = FT[ikF(k)]$ and the inverse transform $ikF(k) = FT^{-1}[f'(x)]$. Using this how would you write $x\psi(x)$ in momentum space?

 

[strangerep]

I already did that [...]

Heh, then you should show your work. (Asking only vague questions makes it harder for others to help you constructively.)

 

[vanhees71]

...and do you have to do it with wave functions, or is Dirac notation allowed?