In Example 9.2, we introduced the Zeeman effect as a splitting of the energy levels of states in hydrogen due to the presence of a weak, external magnetic field. In that example, we just studied the Zeeman effect applied to the ground state of hydrogen and considering the different effects of an electron with spin-up vs. spindown in the magnetic field. As a charged particle orbiting the proton, the electron produces a magnetic moment and this will also introduce a splitting of states whose \(z\) components of orbital angular momentum, \(\hat{L}_{z}\), differ. We will study this effect in this exercise. Just as in Example 9.2, assume that the external magnetic field points along the \(z\) direction, as defined in Eq. (9.163).

(a) Consider first the electron as a classical, electrically charged particle orbiting the proton with angular momentum \(\vec{L}\). Show that the classical gyromagnetic ratio of this system is indeed

\[\begin{equation*}\gamma_{\text {class }}=-\frac{e}{2 m_{e}} \tag{9.175}\end{equation*}\]

Recall the Biot-Savart law applied to this orbiting electron.

(b) With this result, now consider the Zeeman effect on a general energy eigenstate of hydrogen, \(|n, \ell, mangle\). What is the change in the energy of this state due to the external magnetic field? Are the energies still independent of the eigenvalue of the \(z\)-component of angular momentum, \(m\) ?