= Quantum quarter-wave matching layer
{title2=$k_1^2=k_0k_2,\quad k_1a=\pi/2$}
For equal-mass one-dimensional scattering between regions with real wavenumbers $k_0>k_2>0$, insert a layer of width $a$ and wavenumber $k_1$. For a purely right-moving incident wave of amplitude $A$ and no reflection, continuity at the first edge gives interior amplitudes $C=A(1+k_0/k_1)/2$ and $D=A(1-k_0/k_1)/2$. If $k_1a=\pi/2$, continuity at the second edge gives a transmitted edge amplitude $F=i(k_0/k_1)A$ and the matching condition $k_1^2=k_0k_2$. Thus $F/A=i\sqrt{k_0/k_2}$ and the transmitted <probability current> equals the incident current, despite their unequal amplitudes.
For potentials $0,V_1,V_2$ with $E>V_2>V_1>0$, the condition is $(E-V_1)^2=E(E-V_2)$, or $E=V_1^2/(2V_1-V_2)$, requiring $V_2/2<V_1<V_2$. It is the quantum-wave counterpart of a <quarter-wave impedance transformer>: the intermediate wavenumber is the geometric mean and the two reflected amplitudes cancel.
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