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Spin and polarization

  1. Oct 1, 2011 #1
    Recently, I read an article about spin and polarization (http://www.mathpages.com/rr/s9-04/9-04.htm) , but I don't understand how the spin operator defined, can anyone give so help? any help would be great!!
     
    Last edited: Oct 1, 2011
  2. jcsd
  3. Oct 1, 2011 #2
    Consider if we have a complete set of vectors in 2D characterized experimentally by spin-up and spin-down in the z direction:
    [tex]
    \mathbb{I}=|\uparrow\rangle \langle \uparrow |+|\downarrow\rangle \langle \downarrow |
    [/tex]
    Then experimentally we find we can only observe two different numbers corresponding to two different physical situations, namely we measure something and it is spining up, or spining down with a number value for its angular momentum,
    [tex]
    S_{z}=\frac{\hbar}{2}|\uparrow\rangle \langle \uparrow |-\frac{\hbar}{2}|\downarrow\rangle \langle \downarrow |
    [/tex]
    Now lets consider the different possibilities for measuring these values along the z-axis, this is given by
    [tex]
    \sum_{n,m=1}^{2}\langle n|S_{z}|m\rangle
    [/tex]
    where by we examine the different situations
    [tex]
    \langle \uparrow |S_{z}|\uparrow\rangle
    [/tex]
    [tex]
    \langle \uparrow |S_{z}|\downarrow \rangle
    [/tex]
    [tex]
    \langle \downarrow |S_{z}|\uparrow \rangle
    [/tex]
    [tex]
    \langle \downarrow |S_{z}|\downarrow \rangle
    [/tex]
    These can be combined into a single object and the inner products can be evaluated explicitly to give
    [tex]
    \begin{pmatrix}
    \langle \uparrow |S_{z}|\uparrow\rangle & \langle \uparrow |S_{z}|\downarrow \rangle \\
    \langle \downarrow |S_{z}|\uparrow \rangle & \langle \downarrow |S_{z}|\downarrow \rangle
    \end{pmatrix}=\frac{\hbar}{2}
    \begin{pmatrix}
    1 & 0 \\
    0 & -1
    \end{pmatrix}=\frac{\hbar}{2}\sigma_{z}
    [/tex]
    This is one component of the spin matrices. a similar method yields the other two.
     
  4. Oct 2, 2011 #3
    thank you very much. Is that the operator you suggested in only focused on the electron in a hydrogen atom? And I am still having problems on the total orbital angular momentum and the total spin.
     
  5. Oct 2, 2011 #4
    It's for the z-component of the total spin. Once one has constructed all three components they can calculate [itex]S^2=\mathbf{S}\cdot \mathbf{S}[/itex] which is the total spin operator (squared). This operator commutes with all three components of the spin.
     
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