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Wigner 3j symbol recursion relation

  1. Nov 24, 2011 #1
    Hi all!

    1. The problem statement, all variables and given/known data
    I have to show:

    [itex]\sqrt{(j \pm m ) (j \mp m+1} <j_1 j_2 m_1 m_2 | j_1 j_2 j m\mp 1 > = \sqrt{(j_1 \mp m_1 ) (j_1 \pm m_1+1} <j_1 j_2 m_1 \pm1, m_2 | j_1 j_2 j m > +\sqrt{(j_2 \mp m_2 ) (j_2 \pm m_2+1} <j_1 j_2 m_1 , m_2 \pm1 | j_1 j_2 j m > [/itex]


    2. Relevant equations

    Wigner 3-j symbols are related to Clebsch–Gordan coefficients through

    [itex]\begin{pmatrix}
    j_1 & j_2 & j_3\\
    m_1 & m_2 & m_3
    \end{pmatrix}
    \equiv \frac{(-1)^{j_1-j_2-m_3}}{\sqrt{2j_3+1}} \langle j_1 m_1 j_2 m_2 | j_3 \, {-m_3} \rangle [/itex]

    [itex]j_3=j, m_3=m [/itex]

    3. The attempt at a solution
    I've tried to put each term [itex] <j_1 j_2 m_1 \pm1, m_2 | j_1 j_2 j m > [/itex] and [itex] <j_1 j_2 m_1 , m_2 \pm1 | j_1 j_2 j m > [/itex] on the matrix form , but I don't know how i can get the square roots, any idea?

    thanks in advance
     
  2. jcsd
  3. Nov 24, 2011 #2

    fzero

    User Avatar
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    Gold Member

    Do you even need the 3j symbol to do the problem? Hint: try to compute [itex]\langle j_1 j_2 m_1 m_2 | J_\mp | j_1 j_2 j m \rangle[/itex].
     
  4. Nov 25, 2011 #3
    Thanks!!
     
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