Question #94941

Consider a hydrogen-like ion of atomic number $Z$ initially in its quantum mechanical ground state ($n=1, l=0$). It absorbs a photon of energy $E$ and transitions to the first excited state ($n=2$). A student incorrectly models this process classically, assuming the photon imparts its entire momentum $p = E/c$ as an instantaneous tangential impulse to the electron in its initial Bohr orbit of radius $r_1 = a_0/Z$ (where $a_0$ is the Bohr radius). The student uses this to calculate a classical change in the orbital magnetic moment, $\Delta \mu_{\text{class}}$. However, according to quantum mechanics, the transition obeys the electric dipole selection rule $\Delta l = \pm 1$. Let $\Delta \mu_{z, \text{QM}}$ be the actual change in the maximum possible $z$-component of the electron's orbital magnetic moment. Given the fine structure constant $\alpha = \frac{e^2}{4\pi\epsilon_0 \hbar c}$, what is the ratio $\frac{\Delta \mu_{\text{class}}}{\Delta \mu_{z, \text{QM}}}$? (Assume the gyromagnetic ratio $\gamma = \frac{e}{2m_e}$ is invariant in both models).