The chapter states

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Free particles, wave packets and tunnelling

Problem

The chapter states, without showing the algebra, that matching

\psi

and

\psi'

at both edges of a rectangular barrier gives the exact result

\mathcal T^{-1} = 1 + \dfrac{V_0^2\sinh^2\kappa a}{4E(V_0-E)}

. Carry out that matching. Write

\psi = Ae^{ikx} + Be^{-ikx}

for

x < 0

\psi = Ce^{\kappa x} + De^{-\kappa x}

for

0 < x < a

, and

\psi = Fe^{ikx}

for

x > a

, with

k = \sqrt{2mE}/\hbar

and

\kappa = \sqrt{2m(V_0-E)}/\hbar

. Show that

\mathcal T = |F/A|^2

(the two outer regions share the same

k

, so no current-weighting factor is needed) reduces to the stated exact formula.