03 - Spectrum of Angular Momentum Operators

Spectrum of Angular Momentum Operators

Consider Total Angular Momentum $J^2$ then

Eigenvalues of $J^2$ is in form

$$j(j+1){\hslash }^2\text{ where }j=0,\frac{1}{2},1,\frac{3}{2},2,\cdots$$

taking non-negative half-integer values

For each $j$ then
Eigenvalues of $J_3$ are $m\hslash$ where $m=-j,-j+1,-j+2,\cdots ,j-2,j$
where degeneracy of each eigenvalue is same for $j\hslash$ which is ($2j+1$)