Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section B / 21 / iii

Between two barriers, region III becomes a finite intermediate well containing both right- and left-moving oscillatory components, $P{e}^{ikx}+Q{e}^{-ikx}$. Repeated reflection produces standing-wave-like enhancement and quasi-bound levels with finite escape lifetimes. At resonant energies, the multiple transmitted amplitudes add constructively: an ideal symmetric lossless pair can have transmission approaching unity even when a single barrier transmits poorly. Away from resonance the pair strongly suppresses transmission.

A useful phase condition is $2k\ell +{\phi }_{r1}+{\phi }_{r2}=2\pi n$, where $\ell$ is the intermediate-well width and the reflection phases account for penetration into the barriers. The printed figure starts the second barrier at $a+L$, so its well width is $L$; if the prose's second-barrier position $L$ is used instead, the width is $L-a$. The resonance argument is unchanged once that geometrical convention is fixed.