Question

Consider a one-dimensional square-well potential (see Fig.):

(a) For E < 0, find the wave function of a particle bound in this potential. Write an equation which determines the allowed values of E.

(b) Suppose a particle with energy E > 0 is incident upon this potential. Find the phase relation between the incident and the outgoing wave.

Answer 1

The SchrGdinger equations for the different regions are

(a) E < 0.

(i) Consider first the case of KJ < -E, for which the wave function is

The continuity conditions of the wave function give

As cot h x > 0 for x > 0, there is no solution for this case.

(ii) For V₀ > -E, ik → k, k = 2m(V₀ + E)/h, and the equation determining the energy becomes k cot (ka) = -k' The wave function is

From the continuity and normalization of the wave function we get

(b) E > 0. The wave function is

Hence the phase shift of the outgoing wave in relation to the incident wave is