Operators, switching basis.

  • #1

Homework Statement

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

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

It's easy to prove that [tex]<x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'}[/tex]
(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?
Last edited:
  • #2
<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.
  • #3
Great. Thanks so much.

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