Introduction to Quantum Mechanics · Problem 2.33

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · rectangular_barrier

Problem

Determine the transmission coefficient for a rectangular barrier (same as Equation 2.148, only with

V(x) = +V_0 > 0

in the region

-a < x < a

). Treat separately the three cases

E < V_0

E = V_0

, and

E > V_0

(note that the wave function inside the barrier is different in the three cases). Partial answer: for

E < V_0

T^{-1} = 1 + \frac{V_0^2}{4E(V_0 - E)}\sinh^2\!\left(\frac{2a}{\hbar}\sqrt{2m(V_0 - E)}\right).