Angular momentum III Eigenfunctions Using Operator Algebra

In Chapter 13 we had shown that we can have a set of simultaneous eigenkets of the operators J2 and Jz: $$ J^2 \left| {\left. {j,m} \right\rangle = j\left( {j + 1} \right)\hbar ^2 } \right|\left. {j,m} \right\rangle $$ (1) $$ J_z \left| {\left. {j,m} \right\rangle = m\hbar ^2 } \right|\left. {j,m} \right\rangle $$ (2) where $$ j = 0,\quad \frac{1} {2},\quad 1,\quad \frac{3} {2},\quad 2,\quad \frac{5} {2}, \ldots $$ (3) and $$ m = - j, - j + 1, \ldots ,j $$ (4)