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Addition of Angular momentum

  1. Sep 26, 2012 #1
    1. The problem statement, all variables and given/known data
    So I'm told I can't do it this way but I was wondering if anyone could clarify as to why? We're given [tex] |J=\frac{1}{2},M = \frac{1}{2}\!> [/tex] where [tex] j_1 = 1 \, and \, j_2 = \frac{1}{2} [/tex]

    2. Relevant equations



    3. The attempt at a solution
    So this can be composed as a linear combination:
    [tex] | \frac{1}{2} \frac{1}{2}\!> = C_1 |1 1\!>|\frac{1}{2} -\frac{1}{2}\!> + C_2 |10\!> \frac{1}{2}\frac{1}{2}\!> [/tex]
    Applying the raising operator to both sides [tex] J_+ [/tex] gives:
    [tex] 0 = C_1 |1 1\!>|\frac{1}{2} \frac{1}{2}\!> + \sqrt{2}C_2 |11\!> \frac{1}{2}\frac{1}{2}\!> [/tex] so that [tex] C_1 = -\sqrt{2}C_2 \, and \, C^2_1 + C^2_2 = 1 \, implies \, C_2 = \frac{1}{\sqrt3} \, and \, C_1 = \frac{\sqrt2}{\sqrt3} [/tex]
    But, I'm told this is wrong, why and thank you.
     
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  3. Sep 27, 2012 #2

    vela

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    Other than a sign mistake — one of the constants should be negative — it looks fine to me.
     
  4. Sep 27, 2012 #3
    I was told that this is true but that you cannot construct J=1/2 states directly, moreover that the coefficients are relative?
     
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