The potential acting on an electron near the surface of liquid helium can be approximately described by the sum of an infinitely high potential threshold (repulsive action of the He-surface) and an attractive potential due to the mirror charge (problem of a point charge in front of a dielectric half-space)
V(x)=⎩⎨⎧∞−xZe2for x≤0for x>0.
Here, x is the coordinate of the electron perpendicular to the surface, and Z=(ε−1)/4(ε+1), where for He ε=1.05723.
(a) Determine in analogy to the H-atom the energy eigenstates and eigenvalues.
(b) Give the value of the ground state energy.
Hint: Separate variables with respect to the coordinate x and the coordinates parallel to the surface.