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Comprehensive Study Notes

In 1896, Zeeman observed that application of an external magnetic field caused a splitting of atomic spectral lines. We shall consider this Zeeman effect for the hydrogen atom. We begin by reviewing magnetism.

Magnetic fields arise from moving electric charges. A charge $Q$ with velocity $\mathbf{v}$ gives rise to a magnetic field $\mathbf{B}$ at point $P$ in space, such that

\(
\begin{equation}
\mathbf{B}=\frac{\mu_{0}}{4 \pi} \frac{Q \mathbf{v} \times \mathbf{r}}{r^{3}} \tag{6.125}
\end{equation}
\)

where $\mathbf{r}$ is the vector from $Q$ to point P and where $\mu_{0}$ (called the permeability of vacuum or the magnetic constant) is defined as $4 \pi \times 10^{-7} \mathrm{~N} \mathrm{C}^{-2} \mathrm{~s}^{2}$. [Equation (6.125) is valid only for a nonaccelerated charge moving with a speed much less than the speed of light.] The vector $\mathbf{B}$ is called the magnetic induction or magnetic flux density. (It was formerly believed that the vector $\mathbf{H}$ was the fundamental magnetic field vector, so $\mathbf{H}$ was called the magnetic field strength. It is now known that $\mathbf{B}$ is the fundamental magnetic vector.) Equation (6.125) is in SI units with $Q$ in coulombs and $\mathbf{B}$ in teslas (T), where $1 \mathrm{~T}=1 \mathrm{NC}^{-1} \mathrm{~m}^{-1} \mathrm{~s}$.

Two electric charges $+Q$ and $-Q$ separated by a small distance $b$ constitute an electric dipole. The electric dipole moment is defined as a vector from $-Q$ to $+Q$ with magnitude $Q b$. For a small planar loop of electric current, it turns out that the magnetic field generated by the moving charges of the current is given by the same mathematical
expression as that giving the electric field due to an electric dipole, except that the electric dipole moment is replaced by the magnetic dipole moment $\mathbf{m} ; \mathbf{m}$ is a vector of magnitude $I A$, where $I$ is the current flowing in a loop of area $A$. The direction of $\mathbf{m}$ is perpendicular to the plane of the current loop.

Consider the magnetic (dipole) moment associated with a charge $Q$ moving in a circle of radius $r$ with speed $v$. The current is the charge flow per unit time. The circumference of the circle is $2 \pi r$, and the time for one revolution is $2 \pi r / v$. Hence $I=Q v / 2 \pi r$. The magnitude of $\mathbf{m}$ is

\(
\begin{equation}
|\mathbf{m}|=I A=(Q v / 2 \pi r) \pi r^{2}=Q v r / 2=Q r p / 2 m \tag{6.126}
\end{equation}
\)

where $m$ is the mass of the charged particle and $p$ is its linear momentum. Since the radius vector $\mathbf{r}$ is perpendicular to $\mathbf{p}$, we have

\(
\begin{equation}
\mathbf{m}_{L}=\frac{Q \mathbf{r} \times \mathbf{p}}{2 m}=\frac{Q}{2 m} \mathbf{L} \tag{6.127}
\end{equation}
\)

where the definition of orbital angular momentum $\mathbf{L}$ was used and the subscript on $\mathbf{m}$ indicates that it arises from the orbital motion of the particle. Although we derived (6.127) for the special case of circular motion, its validity is general. For an electron, $Q=-e$, and the magnetic moment due to its orbital motion is

\(
\begin{equation}
\mathbf{m}{L}=-\frac{e}{2 m{e}} \mathbf{L} \tag{6.128}
\end{equation}
\)

The magnitude of $\mathbf{L}$ is given by (5.95), and the magnitude of the orbital magnetic moment of an electron with orbital-angular-momentum quantum number $l$ is

\(
\begin{equation}
\left|\mathbf{m}{L}\right|=\frac{e \hbar}{2 m{e}}[l(l+1)]^{1 / 2}=\mu_{\mathrm{B}}[l(l+1)]^{1 / 2} \tag{6.129}
\end{equation}
\)

The constant $e \hbar / 2 m{e}$ is called the Bohr magneton $\mu{\mathrm{B}}$ :

\(
\begin{equation}
\mu{\mathrm{B}} \equiv e \hbar / 2 m{e}=9.2740 \times 10^{-24} \mathrm{~J} / \mathrm{T} \tag{6.130}
\end{equation}
\)

Now consider applying an external magnetic field to the hydrogen atom. The energy of interaction between a magnetic dipole $\mathbf{m}$ and an external magnetic field $\mathbf{B}$ can be shown to be

\(
\begin{equation}
E_{B}=-\mathbf{m} \cdot \mathbf{B} \tag{6.131}
\end{equation}
\)

Using Eq. (6.128), we have

\(
\begin{equation}
E{B}=\frac{e}{2 m{e}} \mathbf{L} \cdot \mathbf{B} \tag{6.132}
\end{equation}
\)

We take the $z$ axis along the direction of the applied field: $\mathbf{B}=B \mathbf{k}$, where $\mathbf{k}$ is a unit vector in the $z$ direction. We have

\(
E{B}=\frac{e}{2 m{e}} B\left(L{x} \mathbf{i}+L{y} \mathbf{j}+L{z} \mathbf{k}\right) \cdot \mathbf{k}=\frac{e}{2 m{e}} B L{z}=\frac{\mu{\mathrm{B}}}{\hbar} B L_{z}
\)

where $L{z}$ is the $z$ component of orbital angular momentum. We now replace $L{z}$ by the operator $\hat{L}_{z}$ to give the following additional term in the Hamiltonian operator, resulting from the external magnetic field:

\(
\begin{equation}
\hat{H}{B}=\mu{\mathrm{B}} B \hbar^{-1} \hat{L}_{z} \tag{6.133}
\end{equation}
\)

The Schrödinger equation for the hydrogen atom in a magnetic field is

\(
\begin{equation}
\left(\hat{H}+\hat{H}_{B}\right) \psi=E \psi \tag{6.134}
\end{equation}
\)

where $\hat{H}$ is the hydrogen-atom Hamiltonian in the absence of an external field. We readily verify that the solutions of Eq. (6.134) are the complex hydrogenlike wave functions (6.61):

\(
\begin{equation}
\left(\hat{H}+\hat{H}{B}\right) R(r) Y{l}^{m}(\theta, \phi)=\hat{H} R Y{l}^{m}+\mu{\mathrm{B}} \hbar^{-1} B \hat{L}{z} R Y{l}^{m}=\left(-\frac{Z^{2}}{n^{2}} \frac{e^{2}}{8 \pi \varepsilon{0} a}+\mu{\mathrm{B}} B m\right) R Y_{l}^{m} \tag{6.135}
\end{equation}
\)

where Eqs. (6.94) and (5.105) were used. Thus there is an additional term $\mu_{\mathrm{B}} B m$ in the energy, and the external magnetic field removes the $m$ degeneracy. For obvious reasons, $m$ is often called the magnetic quantum number. Actually, the observed energy shifts do not match the predictions of Eq. (6.135) because of the existence of electron spin magnetic moment (Chapter 10 and Section 11.7).

In Chapter 5 we found that in quantum mechanics $\mathbf{L}$ lies on the surface of a cone. A classical-mechanical treatment of the motion of $\mathbf{L}$ in an applied magnetic field shows that the field exerts a torque on $\mathbf{m}{L}$, causing $\mathbf{L}$ to revolve about the direction of $\mathbf{B}$ at a constant frequency given by $\left|\mathbf{m}{L}\right| B / 2 \pi|\mathrm{~L}|$, while maintaining a constant angle with $\mathbf{B}$. This gyroscopic motion is called precession. In quantum mechanics, a complete specification of $\mathbf{L}$ is impossible. However, one finds that $\langle\mathbf{L}\rangle$ precesses about the field direction (Dicke and Wittke, Section 12-3).