Question

An electron moves above an impenetrable conducting surface. It is attracted toward this surface by its own image charge so that classically it bounces along the surface as shown in Fig.

(a) Write the Schrodinger equation for the energy eigenstates and energy eigenvalues of the electron. (Call y the distance above the surface.) Ignore inertial effects of the image.

(b) What is the x and z dependence of the eigenstates?

(c) What are the remaining boundary conditions?

(d) Find the ground state and its energy.

[Hint: they are closely related to those for the usual hydrogen atom).

(e) What is the complete set of discrete and/or continuous energy eigenvalues?

Answer 1

(a) Figure shows the electron and its image. Accordingly the electric energy for the system is

(b) Separating the variables by assuming solutions of the type

Note that since V(y) = \(- \frac{e^2 }{4y}\) depends on y only, p_x and p_z are constants

of the motion. Hence

(c) The remaining boundary condition is \(\psi\)(x, y, z) = 0 for y ≤ 0. 

(d) Now consider a hydrogen-like atom of nuclear charge Z. The Schrodinger equation in the radial direction is

which is identical with (1) with the replacements r → y, Z → a. Hence the solutions of (1) are simply y multiplied by the radial wave functions of the ground state of the atom. Thus

Note that the boundary condition in (c) is satisfied by this wave function. The ground-state energy due to y motion is similarly obtained:

(e) The complete energy eigenvalue for quantum state n is

where A is the normalization constant.