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Spin problem

  1. Apr 24, 2005 #1
    Hi,

    I have to find out the possible total spins for a three-particle system composed of spin-1/2-particles. My guess is that there are two possible spins; 1/2 (one up, the others down or vice versa) and 3/2 (all up or all down), but I'm not sure.

    In my book they show how to find the total spin of a system composed of two spin-1/2-particles, but I don't understand the derivation. He talks about triplets and singlets (what is that!?) and apparently the state,

    [tex]\tfrac1{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle[/tex],​

    represents a system of total spin 1. How come? I don't get it.

    Also, another question: Is the total spin of a spin-1/2 particle s = 1/2 or is it slightly bigger (like for orbital angular momentum, where the total is always bigger than the z-component). I would think that it is, since if it is 1/2 you would know the direction of the spin vector completely (Sx = 0, Sy = 0, Sz = +/-1/2), which would violate the uncertainty principle.
     
    Last edited: Apr 24, 2005
  2. jcsd
  3. Apr 24, 2005 #2

    dextercioby

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    What did u make of Clebsch-Gordan theorem and the C-G coefficients...?

    There's one way to do it.Use the 2 1/2 spins case & compose it with a spin 1/2 case.Instead of 4,u'll have 8 states...

    Daniel.
     
  4. Apr 24, 2005 #3
    Huh? We aren't suppose to use the Clebsch-Gordon coefficients (we skipped that part).
     
  5. Apr 24, 2005 #4

    dextercioby

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    Composing spins (and angular momenta in general) is done starting with the theorem of Clebsch-Gordan...Read it and compute

    [tex] \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}}\otimes \mathcal{E}_{\frac{1}{2}} [/tex]



    Daniel.
     
  6. Apr 24, 2005 #5
    Can anyone give a more intuitive explanation? Am I right in my initial guess?

    And what about my last question?

    I'm not familiar with that notation or the Clebsch-Gordan theorem. We're not supposed to use that (trust me).
     
  7. Apr 24, 2005 #6

    dextercioby

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    There are 3 irreducible representations (3 irreducible spaces) spanned by the vectors given by the C-G theorem...

    Daniel.
     
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