Question

Find the total angular momentum of an atom in the state with `S=3//2` and `L=2` if its magnetic moment is known to be equal to zero.

Answer 1

Since `mu=0` we must have either `J=0` or `g=0` is incomplete with `L=2` and `S=(3)/(2)`. Hence `g=0`. Thus
`0=1+(J(J+1)+(3)/(2)xx(5)/(2)-2xx3)/(2J(J+1))`
or `-3J(J+1)=(15)/(4)-6=-(9)/(4)`
Hence `J=(1)/(2)`
Thus `M= ħsqrt((1)/(2)xx(3)/(2))=( ħsqrt(3))/(2)`