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

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

$j_3=j, m_3=m$

The Attempt at a Solution

I've tried to put each term $<j_1 j_2 m_1 \pm1, m_2 | j_1 j_2 j m >$ and $<j_1 j_2 m_1 , m_2 \pm1 | j_1 j_2 j m >$ 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!