11.9: Zeeman Effect

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.

Figure 24: The Zeeman effect for the and 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$}}$