What Is the Result of the Commutator [x,T] in Quantum Mechanics?

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Homework Help Overview

The discussion revolves around the commutator of the position operator and the kinetic energy operator in quantum mechanics, specifically finding the result of ##[\hat{x},\hat{T}]##.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the definition of the kinetic energy operator and its relation to the position operator. One participant attempts to compute the commutator and shares their result, while another seeks clarification on the definition of the kinetic energy operator.

Discussion Status

The conversation includes a verification of the initial attempt, with one participant confirming the correctness of the result. There is also a mention of a method involving canonical commutation relations, suggesting a potential approach to the problem.

Contextual Notes

There is a focus on the definitions of operators and the application of commutation relations, with some participants questioning the setup and assumptions related to the operators involved.

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Homework Statement



Find ##[\hat{x},\hat{T}]##.

Homework Equations



##[\hat{x},\hat{T}]=\hat{x}\hat{T}-\hat{T}\hat{x}##

The Attempt at a Solution


I wind up with ##\frac{i\hbar}{m}\hat{p}##. Did I do good, boss?

Chris
 
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It is the kinetic energy operator. p^2/2m.

Chris
 
Your answer is correct.

I am not sure if this is the way you did it, but using the fact that ##[A,BC] = B[A,C] + [A,B]C## for operators ##A, B## and ##C##, the result follows in one line using the canonical commutation relations for ##\hat{x}## and ##\hat{p}##.
 
Last edited:

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