Wigner 3j symbol recursion relation

[pstq]

Hi all!

Homework Statement

I have to show:

\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 >

Homework Equations

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

\begin{pmatrix}<br /> j_1 &amp; j_2 &amp; j_3\\<br /> m_1 &amp; m_2 &amp; m_3<br /> \end{pmatrix}<br /> \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

j_3=j, m_3=m

The Attempt at a Solution

I've tried to put each term &lt;j_1 j_2 m_1 \pm1, m_2 | j_1 j_2 j m &gt; and &lt;j_1 j_2 m_1 , m_2 \pm1 | j_1 j_2 j m &gt; on the matrix form , but I don't know how i can get the square roots, any idea?

thanks in advance

 

[fzero]

Do you even need the 3j symbol to do the problem? Hint: try to compute \langle j_1 j_2 m_1 m_2 | J_\mp | j_1 j_2 j m \rangle.

 

[pstq]

Thanks!