For a level with angular momentum $J$, the density operator is
\[\hat{\rho} = \sum_j p_j \lvert \psi_j \rangle \langle \psi_j \rvert,\]with $p_j \geq 0$ and
\[\sum_j p_j = 1.\]When the magnetic substates are chosen as an orthogonal basis, the density operator can be represented by the density matrix $\rho_{m,m’}$:
\[\rho = \begin{pmatrix} \rho_{J,J} & \rho_{J,J-1} & \cdots & \rho_{J,-J+1} & \rho_{J,-J} \\ \rho_{J-1,J} & \rho_{J-1,J-1} & \cdots & \rho_{J-1,-J+1} & \rho_{J-1,-J} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \rho_{-J,J} & \rho_{-J,J-1} & \cdots & \rho_{-J,-J+1} & \rho_{-J,-J} \end{pmatrix}.\]The requirements on $p_j$ imply that the density matrix is a positive-semidefinite Hermitian matrix. The diagonal elements $\rho_{m,m}$ describe the populations of the different magnetic substates, whereas the off-diagonal elements describe quantum coherences between different substates.
For example, the diagonal density matrix
\[\rho = \frac{1}{3} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\]describes a probabilistic mixture of an ensemble in the states $\lvert\psi_{-1}\rangle$, $\lvert\psi_0\rangle$, and $\lvert\psi_1\rangle$, with equal populations.
On the other hand, consider the coherent quantum superposition
\[\lvert\psi\rangle = \frac{ \lvert\psi_{-1}\rangle + \lvert\psi_0\rangle + \lvert\psi_1\rangle }{\sqrt{3}}.\]The corresponding density matrix is
\[\rho = \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix},\]where the non-zero off-diagonal elements represent the quantum coherence between the magnetic substates.
Now consider the temporal evolution of the density matrix. For a pure state, the state vector follows the Schrödinger equation,
\[i\hbar \frac{\mathrm{d}}{\mathrm{d}t} \lvert\psi\rangle = H\lvert\psi\rangle,\]and its Hermitian conjugate is
\[-i\hbar \frac{\mathrm{d}}{\mathrm{d}t} \langle\psi\rvert = \langle\psi\rvert H.\]Taking the time derivative of
\[\rho = \lvert\psi\rangle\langle\psi\rvert,\]we obtain
\[\frac{\mathrm{d}\rho}{\mathrm{d}t} = \frac{\mathrm{d}\lvert\psi\rangle}{\mathrm{d}t} \langle\psi\rvert + \lvert\psi\rangle \frac{\mathrm{d}\langle\psi\rvert}{\mathrm{d}t}.\]Substituting the Schrödinger equation gives
\[\frac{\mathrm{d}\rho}{\mathrm{d}t} = -\frac{i}{\hbar} H\lvert\psi\rangle\langle\psi\rvert + \frac{i}{\hbar} \lvert\psi\rangle\langle\psi\rvert H,\]or equivalently,
\[\boxed{ \frac{\mathrm{d}\rho}{\mathrm{d}t} = -\frac{i}{\hbar}[H,\rho] }.\]This is the Liouville–von Neumann equation.
For Zeeman splitting, the perturbation Hamiltonian is
\[\hat{H} = -\boldsymbol{\mu}\cdot\boldsymbol{B}.\]For a magnetic substate $\lvert m\rangle$,
\[\hat{H}\lvert m\rangle = m\hbar\omega_B\lvert m\rangle,\]where the Larmor frequency is
\[\omega_B \equiv \frac{g_J\mu_B B}{\hbar}.\]Since the magnetic substates are chosen as the basis of the density matrix, the matrix elements are
\[\rho_{m,m'} = \langle m\rvert\rho\lvert m'\rangle.\]Applying the Liouville–von Neumann equation,
\[\frac{\mathrm{d}\rho_{m,m'}}{\mathrm{d}t} = -\frac{i}{\hbar} \langle m\rvert[H,\rho]\lvert m'\rangle.\]Using
\[H\lvert m\rangle = m\hbar\omega_B\lvert m\rangle,\]we obtain
\[\frac{\mathrm{d}\rho_{m,m'}}{\mathrm{d}t} = -i(m-m')\omega_B \langle m\rvert\rho\lvert m'\rangle,\]and therefore
\[\boxed{ \frac{\mathrm{d}\rho_{m,m'}}{\mathrm{d}t} = -i(m-m')\omega_B\rho_{m,m'} }.\]A radiative decay term can also be included:
\[\frac{\mathrm{d}\rho_{m,m'}}{\mathrm{d}t} = -i(m-m')\omega_B\rho_{m,m'} - \frac{\rho_{m,m'}}{\tau},\]where
\[\tau = \frac{1}{A_{ul}}\]is the characteristic lifetime of the upper energy level.
For magnetic-dipole (M1) transitions in the solar corona,
\[\frac{1}{\tau} \ll \omega_B.\]The off-diagonal elements therefore undergo rapid oscillations, whereas the diagonal elements ($m=m’$) do not experience Larmor precession.
When averaged over timescales much longer than the Larmor period, the rapidly oscillating off-diagonal terms vanish:
\[\rho_{m,m'} \rightarrow 0, \qquad m\neq m'.\]Thus,
\[\rho_{m,m'} = \delta_{m,m'}\rho_{m,m}.\]This is why, for coronal M1 transitions in the strong-field or saturated-Hanle regime, the magnetic substates can be treated as non-coherent: the density matrix is effectively diagonal in the magnetic field reference frame.
Written on July 1st , 2026 by Yingjie Zhu