Angular Momenta and Magnetic Dipole Moments — Question 5

Question 5

Adding orbital and spin angular momenta redistributes a fixed state count

An electron has $l=2$ and $s=1/2$. Angular-momentum addition permits $j=|l-s|,|l-s|+1,\ldots,l+s$, with $2j+1$ projections for each j.

1. Find the possible j values and their multiplicities.

2. Compare the total coupled-basis dimension with the uncoupled product-basis dimension.

Question 5 - Solution

Adding orbital and spin angular momenta redistributes a fixed state count

(a) Method and calculation

$$\boxed{j=3/2,\ 5/2}.$$$$\boxed{g_{3/2}=4,\quad g_{5/2}=6}.$$

(b) Interpretation and further calculation

$$d_{\mathrm{coupled}}=4+6=\boxed{10}.$$$$d_{\mathrm{product}}=(2l+1)(2s+1)=5(2)=\boxed{10}.$$

Physical check

Changing from separate orbital and spin projections to total angular momentum changes the basis, not the number of physical states.