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I Question about commuting operators

  1. Aug 7, 2016 #1
    Hi everybody. I have a (I gess rather silly) question.

    If I define [Jk,Ll]=iħΣmεklmLm, what would happen if I made [J, L]?. I gess it would be iħΣjεiijLi=0.

    Can someone please confirm this?

    Thanks for reading.
     
  2. jcsd
  3. Aug 7, 2016 #2

    blue_leaf77

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    If ##\mathbf J## and ##\mathbf L## are both vector operators, how are we supposed to define ##\mathbf J \mathbf L## and ##\mathbf L \mathbf J##?
     
  4. Aug 7, 2016 #3

    stevendaryl

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    By [itex][\boldsymbol{J}, \boldsymbol{L}[/itex], do you mean the commutator of the vectors? The definition of commutator is this:
    [itex][A,B] = AB - BA[/itex], so to make sense of a commutator, you must first have a notion of multiplication. What notion of multiplication of vectors do you mean?

    I think the most straight-forward is the tensor product. If you have two vectors [itex]\boldsymbol{A}[/itex] and [itex]\boldsymbol{B}[/itex], then you can define [itex]\boldsymbol{A}\boldsymbol{B}[/itex] to the be the tensor [itex]\boldsymbol{T}[/itex] with 9 components:

    [itex]T_{ij} = A_i B_j[/itex]

    where [itex]i[/itex] and [itex]j[/itex] range from 1 to 3. In that case, the commutator [itex][\boldsymbol{A},\boldsymbol{B}][/itex] would be just the tensor [itex]\boldsymbol{T}'[/itex] with components [itex]T'_{ij} = T_{ij} - T_{ji}[/itex]
     
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