Question #43655

A beam of atoms of mass $m$ is initially in the first excited state ($n=2$). In this state, the atoms possess an effective permanent electric dipole moment $\vec{p} = p_0 \hat{i}$ ($p_0 > 0$) whose orientation remains fixed. In the ground state ($n=1$), the dipole moment is zero. The atoms are released from rest at $t=0$ from the position $x = L$ into a region ($x > 0$) containing a non-uniform electric field $\vec{E} = -E_0 \left( \frac{x^2}{L^2} \right) \hat{i}$, where $E_0 > 0$. At time $t = \frac{\pi}{3} \sqrt{\frac{m L^2}{2 p_0 E_0}}$, an atom undergoes a transition to the ground state, emitting a photon. If the energy difference between the $n=2$ and $n=1$ states in the absence of an electric field is $\Delta E_0$, what is the frequency $\nu$ of the emitted photon? (Assume the classical velocity of the atom does not change during the emission process).