Consider $V_1$, $V_2$ real potentials, $V1(x) \leq V_2(x) \leq 0$ for all x. $V_i(x) =0$ for $|x|>a$ . A particle obeying 1D shrodinger is input from left to each potential. Is it possible that probability of reflection from $V_1$ > probability reflection from $V_2$. Give proof or counterexample.
I’ve considered box potentials of varying width but the algebra is horrendous, I’m pretty sure this is a counterexample. Is there a nice way?
The formula achille hui mentions is on this OpenLearn page on Scattering from finite square wells and barriers
The transmission coefficient is $1$ for for depth $V=V_0$ and some values of $E$, so the reflection coefficient $R$ is $0$. For a different $V=V_1$ the zeros of $R$ will be displaced. Thus, you can find values of $E$ for which $R=0$ for $V_0$ but not $V_1$ or vice versa.