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

[kq6up]

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

 

[Dr Transport]

Define $\hat{T}$

 

[kq6up]

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

Chris

 

[CAF123]

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 $\hat{p}$.

 

[paranormal]

, you did an excellent job! Your solution is correct. The commutator of the position operator, $\hat{x}$, and the time-evolution operator, $\hat{T}$, is $\frac{i\hbar}{m}\hat{p}$. This result is significant in quantum mechanics, as it shows that the position and momentum operators do not commute, meaning that they cannot be measured simultaneously with complete precision. Keep up the good work!