= Zeeman populations from an unpolarized angular-momentum multiplet
{c}
{title2=$p(m_L,m_S)=\langle m_L,m_S|P_J|m_L,m_S\rangle/(2J+1)$}
An equal incoherent mixture within a fixed total-$J$ multiplet has <density operator> $\rho=P_J/(2J+1)$, where $P_J$ is its <orthogonal projection>. If the applied-field <Hamiltonian operator> is proportional to $L_z+2S_z$ and no <spin-orbit interaction> is retained, its <energy eigenstates> are the uncoupled states $|m_L,m_S\rangle$. The <Born rule> gives their populations by the formula above, or equivalently by squared <Clebsch-Gordan decomposition> coefficients. Equal populations of coupled states need not give equal populations of magnetic <energy eigenvalues>.
For $L=1,S=1/2,J=3/2$, the nonextremal coupled states have weights $2/3$ and $1/3$ in their two uncoupled components. The five dimensionless magnetic energies $-2,-1,0,1,2$ then have populations $1/4,1/6,1/6,1/6,1/4$; the zero energy contains two independent states, each with population $1/12$. In a different, weak-field approximation with a dominant <spin-orbit interaction>, projecting into the isolated multiplet gives the <Landé g-factor> spectrum instead. The full operator and its projection must not be interchanged.
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