Charge and Plane

A particle moving in one dimension feels the potential

(This potential would be appropriate for an electron moving in the presence of a uniformly charged sheet where C is the transparency of the sheet.)

a) Using the WKB approximation, find the energy spectrum, E_n, for this one-dimensional problem for all n for C = 0.

b) Find the energy spectrum, E_n, for C → ∞.

c) Derive an equation that describes the energies E_n for even wave functions for an arbitrary value of C. What can you say about the energies E_n for odd wave functions?

SOLUTION

a) Since V(x) = γ|x|, we may write

(1)

In the WKB approximation

or, between turning points,

(2)

Substituting (1) into (2) and using the symmetry of the motion

about x = 0, we obtain

(3)

Thus,

(4)

(5)

b) For the potential where C → ∞,

the quantization condition gives

(6)

(7)

(8)

c) Using the boundary conditions at x = 0, we obtain

(9)

It implies that the odd states, for which ѱ(0) = 0, are not affected by Cδ(x), while even states should satisfy the condition

(10)

Since

(11)

where

(12)

this condition takes the following form:

(13)