Editing Matrix Elements and the Wigner Eckart Theorem Example

Let ${\vec {J}}_{1}$ and ${\vec {J}}_{2}$ be two angular momentum operators, ${\vec {J}}={\vec {J}}_{1}+{\vec {J}}_{2}$ is the sum of these two vectors, and $|JM\rangle$ denotes the eigen states of ${\vec {J}}^{2}$ and ${\mathbf {J}}_{z}$.

${\mathbf {(}}a)$ Show that the matrix elements of $J_{1}^{-},\langle JM|J_{1}^{-}|J'(M+1)\rangle$, vanish, unless ${\mathbf {J}}'={\mathbf {J}}$ or ${\mathbf {J}}'={\mathbf {J}}\pm 1$.

${\mathbf {(}}b)$ Show also that the following expressions are independent of ${\mathbf {M}}$:

$i){\frac {\langle JM|J_{1}^{-}|J(M+1)\rangle }{\sqrt {(J+M+1)(J-M)}}}$

$ii){\frac {\langle JM|J_{1}^{-}|(J-1)(M+1)\rangle }{\sqrt {(J-M+1)(J-M)}}}$

$iii){\frac {\langle JM|J_{1}^{-}|(J+1)(M+1)\rangle }{\sqrt {(J+M+2)(J+M+1)}}}$

<math> \bold (a) <\math> According to the Wigner-Eckart Theorem