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Operator Theory Problem on Momentum Operator (QM)

  • Thread starter LolWolf
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  • #1
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Homework Statement



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

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


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.
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

Homework Statement



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

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


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]##?
 
  • #3
6
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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.
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
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.
 

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