= Quarter-wave matching layer for a quantum potential step
{title2=$k_1a=\pi/2,\quad k_1^2=k_0k_2$}
For a constant-mass one-dimensional particle propagating through three regions of potentials $0,V_1,V_0$, let $k_j$ be the real positive wavevectors. A layer with $k_1a=\pi/2$ makes the two interface reflection amplitudes cancel exactly when $k_1^2=k_0k_2$. Equivalently $E-V_1=\sqrt{E(E-V_0)}$. For an increasing intermediate step this requires $E>V_0$ and $V_0/2<V_1<V_0$. The result follows from $r=(r_{01}+r_{12}e^{2ik_1a})/(1+r_{01}r_{12}e^{2ik_1a})$ and $r_{ij}=(k_i-k_j)/(k_i+k_j)$.
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