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Commutation relations between P and L

  1. Oct 27, 2013 #1

    CAF123

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    1. The problem statement, all variables and given/known data
    Compute the commutation relations of the momentum operator ##\underline{\hat{P}}## and the angular momentum operator ##\underline{\hat{L}}##

    2. Relevant equations
    $$\hat{L_i} = -i\hbar \epsilon_{ijk} x_j \frac{\partial}{\partial_k} = \epsilon_{ijk}x_j \hat{P_k}$$

    3. 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?
     
  2. jcsd
  3. Oct 27, 2013 #2

    vanhees71

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    This cannot work from the very beginning, because you used the index [itex]j[/itex] twice in your commutator, once as a free index at the momentum component and once when summing over the Levi-Civita-symbol. Relabel the free index, and you'll find out that the momentum components behave under rotations as a vector as it must be!
     
  4. Oct 27, 2013 #3

    CAF123

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    Thanks vanhees71, I got it. A similar question is to compute ##[\hat{L_i}, \hat{L_j}]## This can be rewritten as $$[\epsilon_{ipq} \hat{X_p} \hat{P_q}, \epsilon_{jrs} \hat{X_r}\hat{P_s}]$$Now using the relation [A,BC] in the previous post and then subsequently [AB,C] = A[B,C] + [A,C]B, I obtain after some cancellation of terms $$-\epsilon_{ipq} \epsilon_{jrs} \hat{X_p}i\hbar \delta_{rq} \hat{P_s} + \epsilon_{jrs}\epsilon_{ipq} \hat{X_r} i\hbar \delta_{ps}
    \hat{P_q}$$Now the first term is non zero only when r=q and the second term only non zero when p=s. So relabel the indices in the Levi-Civita tensors to give the equivalent$$-\epsilon_{ipr} \epsilon_{jrs} \hat{X_p} i\hbar \hat{P_s} + \epsilon_{jrp}\epsilon_{ipq} \hat{X_r} i\hbar\hat{P_q}$$ Rewrite in the following way: $$-\epsilon_{rip} \epsilon_{rsj} \hat{X_p} i\hbar \hat{P_s} + \epsilon_{pjr}\epsilon_{pqi} \hat{X_r} i\hbar\hat{P_q}$$

    Using the identity ##\epsilon_{ikl} \epsilon_{imn} = \delta_{km} \delta_{ln} - \delta_{kn}\delta_{lm}## I have that my expression is equal to: $$-\delta_{is} \delta_{pj} \hat{X_p}i \hbar \hat{P_s} + \delta_{ij} \delta_{ps} \hat{X_p} i\hbar \hat{P_s} + \delta_{jq}\delta_{ri} \hat{X_r} i \hbar \hat{P_q} - \delta_{ji} \delta_{rq} \hat{X_r} i\hbar \hat{P_q}$$
    First term nonzero when i=s and p=j, and similar analysis for the other terms. I end up with zero and this is not correct. Can you see where I went wrong?
     
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