= Solution
Between two barriers, region III becomes a \b[finite intermediate well] containing both right- and left-moving oscillatory components, $Pe^{ikx}+Qe^{-ikx}$. 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.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2017/paper-2-resonant-tunnelling.png]
{title=Calculated single- and double-barrier transmission and a separate qualitative resonant-tunnelling current-voltage characteristic}
{source=paper-2-resonant-tunnelling.py}
A useful phase condition is $2k\ell+\varphi_{r1}+\varphi_{r2}=2\pi n$, 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.
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