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Homework Help: Commutation relation of R^2 with L

  1. Nov 2, 2016 #1
    1. The problem statement, all variables and given/known data
    Deduece the commutation relations of position operator (squared) [itex]\hat R^2[/itex] with angular momentum [itex]\hat L[/itex]

    2. Relevant equations
    [xi,xj]=0, Lj= εijkxjPk, [xi, Pl]=ih, [xi,Lj]=iℏϵijkxk

    3. The attempt at a solution
    The previous question related R and L and the result was [tex][\hat R,\hat L_j]=i \hbar \epsilon _{ijk}x_k[/tex] after setting up the commutator as [tex]\epsilon _{jkl}[x_i,x_kP_l][/tex] where I did not include the i in the epsilon.

    Now, I did the same with with [itex][\hat R^2,\hat L_j][/itex] and set it up as [tex][\hat R^2,\hat L_j]=[x_ix_i,L_j]=\epsilon_{jkl}[x_i,P_l]x_kx_i+x_i\epsilon_{jkl}[x_i,P_l]x_k[/tex], in which I simplified using the commutator property, and which is then equal to [tex]i\hbar\epsilon_{jkl}x_kx_i+i\hbar x_i\epsilon_{jkl}x_k[/tex]. I don't think I can reduce it any further.
    The solution has the i included in the epsilon in the setup and I don't know why that is.

    Any help will be appreciated
  2. jcsd
  3. Nov 2, 2016 #2


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    Science Advisor
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    Gold Member

    I can't follow your use of the epsilon symbol. Why not try calculating:

    ##[x^2, p_x]##

    And from there:

    ##[R^2, L_x]##

    Before you do the calculation, though, what do you think the answer will be?
    Last edited: Nov 2, 2016
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