Operator Theory Problem on Momentum Operator (QM)

[LolWolf]

Homework Statement

Given the operators $\hat{x}=x\cdot$ and $\hat{p}=-i\hbar \frac{d}{dx}$, prove that:

$[\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}$

Homework Equations

None.

The Attempt at a Solution

I have very little idea on how to begin this problem, but I don't want a solution, I simply want a hint in the right direction.

Thanks, mates.

 

[Dick]

Homework Statement

Given the operators $\hat{x}=x\cdot$ and $\hat{p}=-i\hbar \frac{d}{dx}$, prove that:

$[\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}$

Homework Equations

None.

The Attempt at a Solution

I have very little idea on how to begin this problem, but I don't want a solution, I simply want a hint in the right direction.

Thanks, mates.

Expand $g(p)$ in a power series. What's $[x,p^n]$?

 

[LolWolf]

Actually, I realized it was even easier than that, but thank you!

Consider the case in momentum-space rather than position-space, and this reduces nicely using elementary operations.

 

[Dick]

Actually, I realized it was even easier than that, but thank you!

Consider the case in momentum-space rather than position-space, and this reduces nicely using elementary operations.

Sure, that works also. x is a differentiation operator in p space.