Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    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

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2017 Award

    You can also use 6-J symbols and do it all in one shot. One painful shot, but one shot.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook