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Vector Operator

  1. May 24, 2015 #1

    blue_leaf77

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    A vector operator ##\mathbf{V}## is defined as one satisfying the following property:

    ## [V_i,J_j] = i\hbar \epsilon_{ijk}V_k##

    where ##\mathbf{J}## is an angular momentum operator. My question is what is the role of ##\mathbf{J}##, does it have to be the total angular momentum from all angular momenta appearing in the system, i.e. if we don't consider spin then ##\mathbf{J}=\mathbf{L}##, if we consider it then ##\mathbf{J} = \mathbf{L}+\mathbf{S}##?
     
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  3. May 24, 2015 #2

    DrDu

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    Yes, exactly.
     
  4. May 24, 2015 #3

    blue_leaf77

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    But using L instead of the total angular momentum J even in the presence of spin doesn't prevent the relation ##[x_i,L_j] = i\hbar \epsilon_{ijk}x_k## to prevail.
     
  5. May 24, 2015 #4

    DrDu

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    Of course, but if you take V=S or V=J, you will need J=L+S to rotate also the spin part.
     
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