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

[alisa]

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

 

[quantumdude]

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.

 

[dextercioby]

What do you mean " if x=x". x=x by definition.

It's the coordinate representation in which the Hilbert space is L^{2}(\mathbb{R},dx). 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.

 

[quantumdude]

OK, so then it should read something like "If \hat{x}|\psi>=x|\psi>...", right?

 

[dextercioby]

Exactly. That's the spectral equation, but nonetheless, yes.