Commutator Relations: [x,p]=ih, Proof of p=-iħ∂/∂x+f(x)

  • #1
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given that [x,p]=ih, show that if x=x, p has the representation p=-iħ∂/∂x+f(x) where f(x) is an arbitrary function of x
 
  • #2
alisa, you're supposed to show an attempted solution at the problem. That goes for your other threads as well.

given that [x,p]=ih, show that if x=x,

What do you mean " if x=x". x=x by definition.
 
  • #3
What do you mean " if x=x". x=x by definition.

It's the coordinate representation in which the Hilbert space is [itex] L^{2}(\mathbb{R},dx) [/itex]. The "x" operator is realized by a multiplication by "x". She's asked to prove that the most general representation of the momentum operator in this Hilbert space is the one written there.
 
  • #4
OK, so then it should read something like "If [itex]\hat{x}|\psi>=x|\psi>[/itex]...", right?
 
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  • #5
Exactly. That's the spectral equation, but nonetheless, yes.
 

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