= Zeeman spectrum for orbital angular momentum one
{c}
For <orbital angular momentum> $\ell=1$ and <spin> $s=1/2$, consider
$$
H=\frac{a}{\hbar}(L_z+2S_z)+\frac{2K}{\hbar^2}\mathbf L\cdot\mathbf S,\qquad a=\mu_B B.
$$
The <quantum number> $m_j=m_\ell+m_s$ is conserved. The stretched states with $m_j=\pm3/2$ have energies $K\pm2a$. The remaining <eigenvalues> follow from two $2\times2$ blocks with off-diagonal entries $\sqrt2K$:
$$
E_{+}^{(\pm)}=\frac{a-K}{2}\pm\frac12\sqrt{(a+K)^2+8K^2},\qquad
E_{-}^{(\pm)}=\frac{-a-K}{2}\pm\frac12\sqrt{(a-K)^2+8K^2}.
$$
The subscripts denote $m_j=\pm1/2$. For $K>0$, the zero-field multiplets are $j=3/2$ at $K$ and $j=1/2$ at $-2K$. Their <weak-field Zeeman splitting> is $g_jam_j$, with <Landé g-factors> $4/3$ and $2/3$, respectively. When $a\gg|K|$, the <Paschen-Back effect> gives approximate product states $(m_\ell,m_s)$ and energies $a(m_\ell+2m_s)+2Km_\ell m_s$. Mixing within a fixed $m_j$ causes an <avoided crossing>; different $m_j$ sectors can cross because the <Hamiltonian> does not couple them.
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