Operators, switching basis.

[novop]

Homework Statement

I need to prove that, $$<p'|\hat{x}p> = i\hbar\frac{d}{dp'}\delta{p-p'}$$

i.e. find the position operator in the momentum basis p for p'...

It's easy to prove that $$<x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'}$$
(position operator in position basis for x')
since I can use the fact that the operator x is hermitian. But what about for the first problem? Any hints?

 

[dextercioby]

<p'|X|p> = int_x dx <p'|X|x><x|p> = int_x dx x <p'|x> <x|p> = int_x dx x (1/√2πhbar) e^-ixp/hbar <x|p> =-hbar/i int_x dx ∂/∂p' <p'|x> <x|p> = ihbar ∂/∂p' ... = what you need.

 

[novop]

Great. Thanks so much.