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Homework Help: Commutator, where have I gone wrong?

  1. Feb 24, 2010 #1
    This is for the Pauli Matrics 0 and 1 are different Hilbert Spaces
    [tex]\left[(I-Z)_{0}\otimes(I-Z)_{1} , Y_{0}\otimes Z_{1}\right][/tex]
    [tex]=\left((I-Z)_{0}\otimes(I-Z)_{1}\right)\left(Y_{0}\otimes Z_{1}\right)-\left(Y_{0}\otimes Z_{1}\right)\left((I-Z)_{0}\otimes(I-Z)_{1}\right) [/tex]
    [tex]=\left((I-Z)_{0}Y_{0}\otimes(I-Z)_{1} Z_{1} - Y_{0}(I-Z)_{0} \otimes Z_{1}(I-Z)_{1} [/tex]
    [tex]= (Y_{0} -Z_{0}Y_{0}) \otimes (Z-I)_{1} - ((Y_{0} + Y_{0}Z_{0})\otimes (Z-I)_{1} [/tex]
    [tex]= (Y_{0} -Z_{0}Y_{0}) \otimes (Z-I)_{1} - ((Y_{0} - Z_{0}Y_{0})\otimes (Z-I)_{1}
    =0
    [/tex]

    I think this is wrong because I have done it long hand (i.e multiplying the matrices) and also I really, really don't want zero for an answer :).

    Thanks
     
  2. jcsd
  3. Feb 24, 2010 #2
    Fourth line, second term, I think the plus should be turned to minus..
     
  4. Feb 24, 2010 #3
    Thanks for your reply, sorry I still can't see it though, could you explain why?
    Thanks again.
     
  5. Feb 24, 2010 #4

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    You really don't see why

    [tex]-Y_0(I-Z)_0\neq-(Y_0+Y_0Z_0)[/tex]

    ???
     
  6. Feb 25, 2010 #5
    :blushing: Oops, thanks!
     
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