Strong-field limit of the Zeeman effect

The strong-field limit of the Zeeman effect is the regime in which an atom or molecule in an external static magnetic field exhibits splitting of its spectral lines into multiple components, with the field strength large compared to the internal spin–orbit interaction.

To analyse the strong-field regime, we consider a hydrogen atom in an external magnetic field much stronger than the internal spin-orbit field $\boldsymbol{\mathit{B}}_{ext}\gg\boldsymbol{\mathit{B}}_{so}$ , so that the interaction of the electron with the external field dominates. In this regime, torques are generated separately on the orbital and spin magnetic moments, $\boldsymbol{\mathit{\mu}}_{l}$ and $\boldsymbol{\mathit{\mu}}_{s}$ , with the vectors $\boldsymbol{\mathit{L}}$ and $\boldsymbol{\mathit{S}}$ antiparallel to $\boldsymbol{\mathit{\mu}}_{l}$ and $\boldsymbol{\mathit{\mu}}_{s}$ respectively. Each torque, $\boldsymbol{\mathit{\tau}}=\frac{d\boldsymbol{\mathit{L}}}{dt}$ or $\boldsymbol{\mathit{\tau}}=\frac{d\boldsymbol{\mathit{S}}}{dt}$ , is perpendicular to the external magnetic field direction (taken as the lab $z$ -axis), so its component along the field axis is zero. Consequently, the projections $l_z$ and $s_z$ cannot change. The only way for $\boldsymbol{\mathit{L}}$ and $\boldsymbol{\mathit{S}}$ to evolve in time while maintaining constant projections onto the $z$ -axis is for them to precess independently about the direction of the magnetic field. Hence, $m_l$ and $m_s$ are good quantum numbers, with the state described by the uncoupled form $\vert n,l,m_l,m_s\rangle$ .

Using eq331 and taking $\boldsymbol{\mathit{B}}_{ext}=B\hat{\boldsymbol{\mathit{k}}}$ , where $\hat{\boldsymbol{\mathit{k}}}$ is a unit vector, the Zeeman Hamiltonian becomes

$\hat{H}_Z=\frac{eB}{2m}(\hat{L}_z+2\hat{S}_z)$

The corresponding expectation value is

$\langle\hat{H}_Z\rangle=\mu_BB(m_l+2m_s)\;\;\;\;\;\;\;\;335$

Furthermore, $\boldsymbol{\mathit{L}}\cdot\boldsymbol{\mathit{S}}=L_xS_x+L_yS_y+L_zS_z$ . Because $\boldsymbol{\mathit{L}}$ and $\boldsymbol{\mathit{S}}$ precess rapidly around the direction of the external field ( $z$ -axis), the time-average contributions from the $x$ and $y$ components vanish. Consequently, $\hat{H}_{so}\propto\hat{\boldsymbol{\mathit{L}}}\cdot\hat{\boldsymbol{\mathit{S}}}\approx \hat{L}_z\hat{S}_z$ and $\langle\hat{H}_{so}\rangle=Am_lm_s\hbar^2$ , where $A$ is a constant. From eq330,

$\langle\hat{H}\rangle=\langle\hat{H}_0\rangle+Am_lm_s\hbar^2+\mu_BB(m_l+2m_s)\;\;\;\;\;\;\;\;336$

where $\langle\hat{H}_0\rangle=-\frac{me^4}{32\pi^2\varepsilon_0^2\hbar^2}$ for the ground state of hydrogen (see eq297).

Eq336 shows that in a strong external magnetic field the energy levels are split according to the quantum numbers $m_l$ and $m_s$ , revealing the fine-structure of the system and largely lifting the degeneracy of the energy levels.

Question

Describe the strong-field Zeeman effect on the hydrogen atom configuration of 2p¹, with $l=1$ and $s=1/2$ .

Answer

The allowed $m_l$ and $m_s$ values for the 2p¹ configuration are $m_l=1,0,-1$ and $m_s=+1/2,-1/2$ , which according to eq335, results in six distinct energy sublevels. If we include the spin-orbit interactions, the energies corresponding to $\langle\hat{H}'\rangle=Am_lm_s\hbar^2+\mu_BB(m_l+2m_s)$ are given by the table below.