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A composite system of three particles with different spin

  1. Dec 12, 2011 #1
    In general how does one find the basis of a composite system of particles with different spin. Say for example spin 1, spin 1/2, and spin 5/2.

    My first thought was to consider the uncoupled basis wherein the particles with spin 1/2 and spin 5/2 have similar state kets. Would the basis for the particle of spin 5/2 "envelope" the basis of the particle with spin 1/2?
     
  2. jcsd
  3. Dec 12, 2011 #2
    For starters, the uncoupled basis would be

    [itex]\left\{|\frac{1}{2},1,\frac{3}{2}\rangle ,|\frac{1}{2},1,\frac{1}{2}\rangle
    ,|\frac{1}{2},1,-\frac{1}{2}\rangle ,|\frac{1}{2},1,-\frac{3}{2}\rangle
    ,|\frac{1}{2},0,\frac{3}{2}\rangle ,|\frac{1}{2},0,\frac{1}{2}\rangle
    ,|\frac{1}{2},0,-\frac{1}{2}\rangle ,|\frac{1}{2},0,-\frac{3}{2}\rangle
    ,|\frac{1}{2},-1,\frac{3}{2}\rangle ,|\frac{1}{2},-1,\frac{1}{2}\rangle
    ,|\frac{1}{2},-1,-\frac{1}{2}\rangle ,|\frac{1}{2},-1,-\frac{3}{2}\rangle
    ,|-\frac{1}{2},1,\frac{3}{2}\rangle ,|-\frac{1}{2},1,\frac{1}{2}\rangle
    ,|-\frac{1}{2},1,-\frac{1}{2}\rangle ,|-\frac{1}{2},1,-\frac{3}{2}\rangle
    ,|-\frac{1}{2},0,\frac{3}{2}\rangle ,|-\frac{1}{2},0,\frac{1}{2}\rangle
    ,|-\frac{1}{2},0,-\frac{1}{2}\rangle ,|-\frac{1}{2},0,-\frac{3}{2}\rangle
    ,|-\frac{1}{2},-1,\frac{3}{2}\rangle ,|-\frac{1}{2},-1,\frac{1}{2}\rangle
    ,|-\frac{1}{2},-1,-\frac{1}{2}\rangle ,|-\frac{1}{2},-1,-\frac{3}{2}\rangle
    \right\}[/itex]

    The notation may be a little strange for some of you but it basically incorporates the spin states of each particle and the direction of orientation; this is mean't to save time.
     
  4. Dec 12, 2011 #3

    Vanadium 50

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    You can also use 6-J symbols and do it all in one shot. One painful shot, but one shot.
     
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