Consider the step potential (a) Calculate the reflection coefficient, for the case E < V 0 , and comment on the answer. (b) Calculate the

Consider the “step” potential

(a) Calculate the reflection coefficient, for the case  E 0, and comment on the answer.

(b) Calculate the reflection coefficient for the case  E 0.

(c) For a potential (such as this one) that does not go back to zero to the right of the barrier, the transmission coefficient is not simply |F|² / |A|² (with A the incident amplitude and F the transmitted amplitude), because the transmitted wave travels at a different speed. Show that

for E > V₀, You can figure it out using Equation 2.99, or—more elegantly, but less informatively—from the probability current (Problem 2.18). What is T, for E 0?
(d) For E > V₀, calculate the transmission coefficient for the step potential, and check that T+R = 1.

Equation 2.99

Problem 2.18

Find the probability current, J (Problem 1.14) for the free particle wave function Equation 2.95. Which direction does the probability flow?

- 18. V (x) = x 0, X Vo, x > 0.