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Pauli matrices

  1. Nov 24, 2011 #1
    Hey guys,

    I was wondering how to get the expression for pauli matrices. I know that for one electron:

    [tex] S_i = \frac{\hbar}{2} \sigma_i [/tex]

    But I also know that you can get to the above expression by explicitly calculating the matrix elements of the Sz, Sx and Sy operators (in the basis generated by Sz and S and composed of two vectors) by using a few rules about angular momentum operators, I just don't remember how exactly. Anyone can help?
     
    Last edited: Nov 24, 2011
  2. jcsd
  3. Nov 24, 2011 #2
  4. Nov 24, 2011 #3
    Thanks, but what equations should you use to find the x and y matrices? The z one is the easiest.
     
  5. Nov 24, 2011 #4
    You can write [itex]S_{x}[/itex] and [itex]S_{y}[/itex] just like [itex]S_{z}[/itex].
    [tex]
    S_{x}=\frac{\hbar}{2}|\uparrow\rangle \langle \uparrow |-\frac{\hbar}{2}|\downarrow \rangle \langle \downarrow |
    [/tex]
    Then if you hit this from both sides with z-basis eigenstates you can evaluate the inner products as [itex]1/\sqrt{2}[/itex] or [itex]i/\sqrt{2}[/itex] etc...
     
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