Electron on Helium Surface

An electron close to the surface of liquid helium experiences an attractive force due to the electrostatic polarization of the helium and a repulsive force due to the exclusion principle(hard core). To a reasonable approximation for the potential when helium fills the space where $z<0$:

$V(z)={\begin{cases}-{\frac {Q^{2}e^{2}}{z}}&{\mbox{if}}\qquad z>0\\infinity&{\mbox{if}}\qquad else\end{cases}}$

(a) Solve the Schrodinger equation. Find the Eingenenergies and Eigenvalues.

(b) An electric field is turned on at t=0 which produces the perturbation:

$V(z)={\begin{cases}eE_{0}ze^{t/\tau }&{\mbox{if}}\qquad t>=0\\0&{\mbox{if}}\qquad else\end{cases}}$

If the electron is initially in its ground state, find the probability makes a transition to its first excited state for times $t>>\tau$.