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# 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, $$$$\epsilon=\mu_{B} B$$$$. Not to scale.
