- #1
CAF123
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Homework Statement
Compute the commutation relations of the momentum operator ##\underline{\hat{P}}## and the angular momentum operator ##\underline{\hat{L}}##
Homework Equations
$$\hat{L_i} = -i\hbar \epsilon_{ijk} x_j \frac{\partial}{\partial_k} = \epsilon_{ijk}x_j \hat{P_k}$$
The Attempt at a Solution
$$[\hat{P_j}, \hat{L_i}] = [\hat{P_j}, \epsilon_{ijk} x_j \hat{P_k}] = [\hat{P_j}, \epsilon_{ijk} \hat{X_j}]\hat{P_k} + \epsilon_{ijk} \hat{X_j}[\hat{P_j}, \hat{P_k}],$$ where I used the relation ##[A,BC] = [A,B]C + B[A,C]## with A,B and C operators. The latter term is zero, and so this reduces to ##[\hat{P_j}, \epsilon_{ijk} \hat{X_j}]\hat{P_k}## which I think is the same as ##-i\hbar \epsilon_{ijk} \delta_{ij} \hat{P_k}##
I then said ##\delta_{ij}## is only nonzero (=1) when ##i=j##. But if ##i=j## then the tensor ##\epsilon_{ijk} = 0## hence my conclusion was that the expression is zero always.
I checked the answer online, and it appears this is not always the case. So where is my error?