Write the spectral designations of the terms of the hydrogen atom whose electron is in the state with principle quantum number `n=3`

Explanation

We shall ignore hyperfine interaction. The state with principal quantum number `n=3` has orbital angular momentum number
`l=0,1,2`
The levels these terms are `3S,3P,3D`. The total angular momentum is obtained by combining spin and angular momentum. For a single electron this leads to
`J=(1)/(2)` if `L=0`
`J=L-(1)/(2)` and `L+(1)/(2)` if `L!=0`
We then get the final designations
`3S_(1/2), 3p_(1/(2))3p_(3//2),3D_(5//2)`