Question #43803

A hydrogen-like atom is placed in a rocket that starts from rest at $t=0$ and accelerates uniformly with acceleration $a$ in the $+z$ direction. An external time-varying electric field $\vec{E}(t) = E_0 \sin(\omega t) \hat{k}$ is applied within the rocket. In the presence of an effective electric field $E_{eff}$ (defined such that the total force on the electron is $-e E_{eff}$), the energy levels of this atom shift according to $\Delta E_n = -\beta n E_{eff}$, where $\beta$ is a positive constant and $n$ is the principal quantum number. The atom continuously undergoes transitions from the $n=2$ state to the $n=1$ state, emitting photons in the $+z$ direction. An observer at rest in the initial inertial frame (at $z \to \infty$) measures the frequency of the emitted photons. Let $\nu_0$ be the unperturbed transition frequency. Assuming $v \ll c$, what is the time-averaged shift in the observed frequency, $\langle \Delta \nu \rangle = \langle \nu_{obs} \rangle - \nu_0$, over the first period of the electric field $T = \frac{2\pi}{\omega}$? (Here $m_e$ and $-e$ are the mass and charge of the electron, $h$ is Planck's constant, and $c$ is the speed of light).