 Spin-orbit shifts of an isotropic harmonic oscillator

For an isotropic three-dimensional quantum harmonic oscillator, $V(r)=m{\omega }^2{r}^2/2$ makes $r^{-1}{V}^{\prime }(r)=m{\omega }^2$. The spin-orbit energy shift in a central potential is therefore independent of the radial excitation. For spin one-half, the $l=0,j=1/2$ shift is zero, while the two $l=1$ shifts are $-{\hslash }^2{\omega }^2/(2m{c}^2)$ for $j=1/2$ and $+{\hslash }^2{\omega }^2/(4m{c}^2)$ for $j=3/2$.

These shifts are quadratic in trap frequency. Their degeneracy-weighted sum in the $l=1$ manifold is zero, since the multiplicities are two and four. They are perturbative corrections rather than full oscillator energies; a nonrelativistic treatment requires them to be small compared with the orbital spacings.