= Spin-orbit shifts of an isotropic harmonic oscillator
{title2=$\Delta E=\hbar^2\omega^2[j(j+1)-l(l+1)-3/4]/(4mc^2)$}
For an isotropic three-dimensional <quantum harmonic oscillator>, $V(r)=m\omega^2r^2/2$ makes $r^{-1}V'(r)=m\omega^2$. 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 $-\hbar^2\omega^2/(2mc^2)$ for $j=1/2$ and $+\hbar^2\omega^2/(4mc^2)$ 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.
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