Question

A particle of mass m and momentum p is incident from the left on the potential step shown in Fig.

Calculate the probability that the particle is scattered backward by the potential if

(a) p²/2m < V_0,

(b) p²/2m > V₀.

Answer 1

The Schrodinger equations are

The condition that \(\psi\)(x) is finite for x \(\to \infty\) having been made use of. The continuity conditions give 1 + r = t, ik - ikr = -k't, whence T = (k' + ik)/(ik - k'). The probability of reflection is R = j_r/j_i = |rI² = 1.

(b) If E > V₀. We have

Noting that there is only outgoing wave for x > x₀. The continuity conditions give 1 + r = t, ik - ikr = ik't, and hence r = (k - k')/(k + k'). The probability of reflection is then R = |rI² = [(k - k')/(k + k')|².