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Operators, switching basis.

  1. Nov 6, 2011 #1
    1. The problem statement, all variables and given/known data

    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: Nov 6, 2011
  2. jcsd
  3. Nov 6, 2011 #2


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    <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.
  4. Nov 6, 2011 #3
    Great. Thanks so much.
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