Solution
= Solution
The states are defined by
$$
J^2|j,m\rangle=\hbar^2j(j+1)|j,m\rangle,
\qquad J_3|j,m\rangle=\hbar m|j,m\rangle,
$$
where $j=0,\tfrac12,1,\ldots$ and $m=-j,-j+1,\ldots,j$. From
$$
J_\mp J_\pm=J^2-J_3^2\mp\hbar J_3
$$
and positivity of the norm,
$$
C_\pm(j,m)=\hbar\sqrt{j(j+1)-m(m\pm1)}.
$$
\b[Thus $\lambda_1^+=\lambda_1^-=-1$, $\lambda_2^+=1$, and $\lambda_2^-=-1$.]