Introduction to Quantum Mechanics · Problem 2.34

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · step_potential

Problem

Consider the "step" potential:

V(x) = \begin{cases} 0, & x \leq 0, \\ V_0, & x > 0. \end{cases}

(a) Calculate the reflection coefficient, for the case

E < V_0

, and comment on the answer. (b) Calculate the reflection coefficient for the case

E > V_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

T = \sqrt{\frac{E - V_0}{E}}\frac{|F|^2}{|A|^2},

for

E > V_0

. Hint: 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 < V_0

? (d) For

E > V_0

, calculate the transmission coefficient for the step potential, and check that

T + R = 1