Question #42823

A gas of Hydrogen atoms is placed in a uniform external electric field $\vec{E} = E_0 \hat{k}$. The first-order Stark effect perturbs the first excited state ($n=2$), splitting the degenerate manifold into three distinct energy levels: $E_2^{(0)}$, $E_2^{(0)} + 3ea_0E_0$, and $E_2^{(0)} - 3ea_0E_0$, where $E_2^{(0)}$ is the unperturbed energy, $e$ is the elementary charge, and $a_0$ is the Bohr radius. The ground state ($1s$) energy $E_1^{(0)}$ remains unshifted. The states corresponding to the shifted energies $E_2^{(0)} \pm 3ea_0E_0$ are linear combinations of the $2s$ and $2p_z$ orbitals, while the unshifted energy $E_2^{(0)}$ corresponds to the $2p_x$ and $2p_y$ orbitals. Let $\lambda_0$ be the wavelength of the Lyman-$\alpha$ line in the absence of the electric field. Assuming $3ea_0E_0 \ll (E_2^{(0)} - E_1^{(0)})$, what is the number of allowed distinct spectral lines observed for transitions from the $n=2$ manifold to the ground state, and what is the maximum difference in wavelength $\Delta \lambda$ between the highest and lowest energy photons emitted?