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Spin operators

  1. Apr 21, 2007 #1
    I need to show:

    [tex](\mathbf{\sigma} \cdot \mathbf{a})(\mathbf{\sigma} \cdot \mathbf{b})=\mathbf{a} \cdot \mathbf{b} I + i \mathbf{\sigma} \cdot (\mathbf{a} \times \mathbf{b})[/tex]

    where a and b are arbitrary vectors, sigma is the pauli spin operator.

    I was just wondering what the dot product and cross product were. Because a and b can be 2x1, 2x2, 2x3, etc... i'm not sure how to take a dot product of matricies much less a cross product. Since it specifies dot and cross, i assume that it is not just a regular matrix mulitpilication, however, i do not know how to take the dot and cross product of matrices. Any suggestions would be appreciated.
  2. jcsd
  3. Apr 21, 2007 #2


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    Just think of the [itex] {\vec \sigma} [/itex] as vectors. So, for example, [itex] {\vec a} \cdot {\vec \sigma} [/itex] is simply [itex] a_1 \sigma_1 + a_2 \sigma_2 + a_3 \sigma_3 [/itex] and so on. So you treat the [itex] {\vec \sigma} [/itex] as vectors while being careful to rememember that the components do not coommute. And then , you may write them as matrices to complete your calculation.
  4. Apr 21, 2007 #3
    thanks nrqed. are vectors a and b 2x1? or is it a general 2xn?

    Or is a=x1+x2+x3, where x1, x2, x3 are 2x1 vectors? I'm not too sure how general this should be
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