1.II.15B

Let $V_{1}(x)$ and $V_{2}(x)$ be two real potential functions of one space dimension, and let $a$ be a positive constant. Suppose also that $V_{1}(x) \leqslant V_{2}(x) \leqslant 0$ for all $x$ and that $V_{1}(x)=V_{2}(x)=0$ for all $x$ such that $|x| \geqslant a$. Consider an incoming beam of particles described by the plane wave $\exp (i k x)$, 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 $p_{1} \leqslant p_{2}$ ? Justify your answer carefully, either by giving a rigorous proof or by presenting a counterexample with explicit calculations of $p_{1}$ and $p_{2}$.

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