Probability difference scattering potentials

Question

Let $V_1(x)$ and $V_2(x)$ be two real potential functions of one space dimension, and let $m$ be a positive constant. Suppose $V_1(x)\le V_2(x) \le 0$ for all $x$ and that $V_1(x) = V_2(x) = 0$ for all $x$ such that $|x| > a.$

Consider an incoming beam of particles described by the plane wave $\exp(ikx),$ for some $k > 0,$ scattering off one of the potentials $V_1(x)$ or $V_2(x).$ Let $p_i$ be the probability that a particle in the beam is reflected by the potential $V_i(x).$ Is it necessarily the case that $p_1$ is greater than $p_2?$

It looks like you could choose shape factors so that the Resonance Curves cross. — Keith McClary, May 30 '19 at 21:41
For wells of different widths and same depth, the transmission spikes are space differently. — Keith McClary, May 31 '19 at 14:45
could you possibly provide an example? Im still very confused — MathematicianP, May 31 '19 at 15:27

score 0 · Answer 1 · answered Jun 01 '19 at 04:18

I assume you are talking about the reflection coefficient $R(k) = 1 - T(k)$ where $T(k)$ is the transmission coefficient discussed in the wiki. There, $R(k) = 0$ for some values of $k$ depending on the length $L$ of the well $V_2$. If we take $V_1$ to be a well with greater length $M$ (but same depth) then the zeros will not generally coincide, so for some values of $k$, $p_2 = 0$ but $p_1 \ge 0$.

Probability difference scattering potentials

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