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Physics LibreTexts

11.9: Zeeman Effect

( \newcommand{\kernel}{\mathrm{null}\,}\)

Consider a hydrogen atom placed in a uniform z-directed external magnetic field of magnitude |{\bf B}|. The modification to the Hamiltonian of the system is H_1 = -\mu\cdot{\bf B}, where \mu = - \frac{e}{2\,m_e}\,({\bf L} + 2\,{\bf S}) is the total electron magnetic moment, including both orbital and spin contributions. [See Equations ([e10.57])–([e10.59]).] Thus, H_1 = \frac{e\,B}{2\,m_e}\,(L_z+ 2\,S_z).

Suppose that the applied magnetic field is much weaker than the atom’s internal magnetic field, ([e12.124]). Because the magnitude of the internal field is about 25 tesla, this is a fairly reasonable assumption. In this situation, we can treat H_1 as a small perturbation acting on the simultaneous eigenstates of the unperturbed Hamiltonian and the fine structure Hamiltonian. Of course, these states are the simultaneous eigenstates of L^2, S^2, J^{\,2}, and J_z. (See the previous section.) Hence, from standard perturbation theory, the first-order energy-shift induced by a weak external magnetic field is \begin{aligned} {\mit\Delta} E_{l,1/2;j,m_j} &= \langle l,1/2;j,m_j|H_1|l,1/2;j,m_j\rangle\nonumber\\[0.5ex] &= \frac{e\,B}{2\,m_e}\,\left(m_j\,\hbar + \langle l,1/2;j,m_j|S_z|l,1/2;j,m_j\rangle\right),\end{aligned} because J_z=L_z+S_z. Now, according to Equations ([e11.47]) and ([e11.48]),

\label{e12.143} \psi^{(2)}_{j,m_j} = \left(\frac{j+m_j}{2\,l+1}\right)^{1/2}\psi^{(1)}_{m_j-1/2,1/2} + \left(\frac{j-m_j}{2\,l+1}\right)^{1/2}\,\psi^{(1)}_{m_j+1/2,-1/2} when j=l+1/2, and \psi^{(2)}_{j,m_j} = \left(\frac{j+1-m_j}{2\,l+1}\right)^{1/2}\psi^{(1)}_{m_j-1/2,1/2} - \left(\frac{j+1+m_j}{2\,l+1}\right)^{1/2}\,\psi^{(1)}_{m_j+1/2,-1/2} when j=l-1/2. Here, the \psi^{(1)}_{m,m_s} are the simultaneous eigenstates of L^2, S^2, L_z, and S_z, whereas the \psi^{(2)}_{j,m_j} are the simultaneous eigenstates of L^2, S^2, J^{\,2}, and J_z. In particular, \label{e12.145} S_z\,\psi^{(1)}_{m,\pm 1/2} = \pm \frac{\hbar}{2}\,\psi^{(1)}_{m,\pm 1/2}. It follows from Equations ([e12.143])–([e12.145]), and the orthormality of the \psi^{(1)}, that \langle l,1/2;j,m_j|S_z|l,1/2;j,m_j\rangle = \pm \frac{m_j\,\hbar}{2\,l+1} when j=l\pm 1/2. Thus, the induced energy-shift when a hydrogen atom is placed in an external magnetic field—which is known as the Zeeman effect —becomes \label{e12.147} {\mit\Delta} E_{l,1/2;j,m_j} = \mu_B\,B\,m_j\left(1\pm \frac{1}{2\,l+1}\right) where the \pm signs correspond to j=l\pm 1/2. Here, \mu_B = \frac{e\,\hbar}{2\,m_e} = 5.788\times 10^{-5}\,{\rm eV/T} is known as the Bohr magnetron. Of course, the quantum number m_j takes values differing by unity in the range -j to j. It, thus, follows from Equation ([e12.147]) that the Zeeman effect splits degenerate states characterized by j=l+1/2 into 2\,j+1 equally spaced states of interstate spacing \label{e12.149} {\mit\Delta} E_{j=l+1/2} = \mu_B\,B\left(\frac{2\,l+2}{2\,l+1}\right). Likewise, the Zeeman effect splits degenerate states characterized by j=l-1/2 into 2\,j+1 equally spaced states of interstate spacing \label{e12.150} {\mit\Delta} E_{j=l-1/2} = \mu_B\,B\left(\frac{2\,l}{2\,l+1}\right).

In conclusion, in the presence of a weak external magnetic field, the two degenerate 1S_{1/2} states of the hydrogen atom are split by 2\,\mu_B\,B. Likewise, the four degenerate 2S_{1/2} and 2P_{1/2} states are split by (2/3)\,\mu_B\,B, whereas the four degenerate 2P_{3/2} states are split by (4/3)\,\mu_B\,B. This is illustrated in Figure [fzee]. Note, finally, that because the \psi^{(2)}_{l,m_j} are not simultaneous eigenstates of the unperturbed and perturbing Hamiltonians, Equations ([e12.149]) and ([e12.150]) can only be regarded as the expectation values of the magnetic-field induced energy-shifts. However, as long as the external magnetic field is much weaker than the internal magnetic field, these expectation values are almost identical to the actual measured values of the energy-shifts.

clipboard_eaad045acf4f250237cbbb152fde7efc7.png

Figure 24: The Zeeman effect for the $n=1$ and $2$ states of a hydrogen atom. Here, \begin{equation}\epsilon=\mu_{B} B\end{equation}. Not to scale.

Contributors and Attributions

  • Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

    \newcommand {\ltapp} {\stackrel {_{\normalsize<}}{_{\normalsize \sim}}} \newcommand {\gtapp} {\stackrel {_{\normalsize>}}{_{\normalsize \sim}}} \newcommand {\btau}{\mbox{\boldmath$\tau$}} \newcommand {\bmu}{\mbox{\boldmath$\mu$}} \newcommand {\bsigma}{\mbox{\boldmath$\sigma$}} \newcommand {\bOmega}{\mbox{\boldmath$\Omega$}} \newcommand {\bomega}{\mbox{\boldmath$\omega$}} \newcommand {\bepsilon}{\mbox{\boldmath$\epsilon$}}

This page titled 11.9: Zeeman Effect is shared under a not declared license and was authored, remixed, and/or curated by Richard Fitzpatrick.

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