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Commutators of vector operators

  1. Jan 19, 2012 #1
    I've been trying to work out some expressions involving commutators of vector operators, and I am hoping some of y'all might know some identities that might make my job a little easier.

    Specifically, what is [itex]\left[\mathbf{\hat A}\cdot\mathbf{\hat B}, \mathbf{\hat C}\right][/itex]? Are there any useful identities to express this in terms of simpler commutators?

    Any help is appreciated.
  2. jcsd
  3. Jan 22, 2012 #2
    Are you sure you mean a "vector operator", typically we talk about matrix operators when discussing the commutator relationships (or group elements in a more general setting).
    Vector operator: http://en.wikipedia.org/wiki/Vector_operator

    However, your question is straightforward,
    [S,T] = ST - TS (by definition)

    Start with [AB,C] = ABC - CAB (+ ACB - ACB )
    = ABC - ACB + ACB - CAB
    = A(BC - CB) + (AC - CA)B
    = A[B,C] + [A,C]B

    Therefore we conclude [AB,C] = A[B,C] + [A,C]B
    to be an identity.
    Does that answer your question.... you could have looked anywhere on the internet to get this.... so I'm guessing this isn't what you want.
  4. Jan 22, 2012 #3
  5. Jan 22, 2012 #4
    Thank you for the [STRIKE]reply[/STRIKE]replies.

    It seems to me that you're being pretty cavalier about vector multiplication, what with the way you're just putting vectors in a row next to each other without any dots or parentheses. For instance, what do you mean when you write “ABC”, when A, B and C are vector operators?

    I would think that you should define [itex][{\bf{\hat S}},{\bf{\hat T}}] = {\bf{\hat S}} \cdot {\bf{\hat T}} - {\bf{\hat T}} \cdot {\bf{\hat S}}[/itex], and therefore start your derivation with
    [tex][{\bf{\hat A}} \cdot {\bf{\hat B}},{\bf{\hat C}}] = ({\bf{\hat A}} \cdot {\bf{\hat B}}){\bf{\hat C}} - {\bf{\hat C}}({\bf{\hat A}} \cdot {\bf{\hat B}}).[/tex]
    But from there, I'm not sure how you can safely proceed, if you're being rigorous with your dots and parens. For instance — and correct me if I'm wrong on this — but I don't think [itex]({\bf{\hat A}} \cdot {\bf{\hat C}}){\bf{\hat B}}[/itex] is equal to [itex]{\bf{\hat A}}({\bf{\hat C}} \cdot {\bf{\hat B}})[/itex], so your next step seems iffy.
    Last edited: Jan 22, 2012
  6. Jan 23, 2012 #5
    Yeah, sorry I don't know. I thought you were intending for matrix operators. Good luck.
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