Chapter 12, Problem 12.40E

The Stark effect is the change in energy of a system due to the presence of an electric field (discovered by German physicist Johannes Stark in 1913). Consider the hydrogen atom. Its normally spherical $1s$ orbital distorts slightly when exposed to an electric field. If the electric field is considered to be in the $z$ direction, then the field acts to introduce, or mix, some $2p_{\text{z}}$ character in with the $1s$ orbital. The atom is said to be polarizable, and the extent to which it changes is considered a measure of the atom’s polarizability (which is designated by the letter $\alpha$, not to be confused with the spin function $\alpha$!). The perturbation Hamiltonian is defined as $\widehat{H^{\prime }}=e\cdot E\cdot r\cdot \cos \theta$, where $e$ is the charge on the electron, $E$ is the strength of the electric field, and $r$ and $\theta$ represent coordinates of the electron. Evaluate the perturbation energy of the hydrogen atom. You will have to integrate all the three spherical polar coordinates in your evaluation of $\widehat{H^{\prime }}$. (A similar effect, the Zeeman effect, exists for magnetic fields. It too can be treated using perturbation theory.)