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Write Sz in the J angular momentum basis?

  1. Apr 28, 2013 #1
    I'm working on a problem where I want to write the operator [itex] S_z [/itex] down in terms of some operator(s) in the [itex] \vec{J} = \vec{L} + \vec{S} [/itex] basis so that I can operate [itex] S_z [/itex] on the states [itex]\mid \ell, s=1/2, j= \ell\pm1/2, m\rangle[/itex] but I'm having trouble finding the correct combination of operators to do so.

    Is this possible, and if so could anyone point me in the right direction? Thanks.
     
  2. jcsd
  3. May 15, 2013 #2
    If you're using z as the preferred orientation for your problem(Jz), and your orbital and spin AM are in the same direction, you are already working in a basis with elements that are tensor products [itex]\|j,m_j\rangle_z \otimes |s,m_s \rangle_z[/itex], which are still eigenvectors of Sz with the same eigenvalues. Sz in this "wider" basis constructed from Lz and Sz basis vectors simply acts like the unit matrix on the j-part.
     
    Last edited: May 15, 2013
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