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Suppose we have a potential$$ V(x) = \begin{cases} 0, & \text{if $ x \le 0$} \\ V_0 & \text{if $ x \gt 0$ } \end{cases}$$ The reflection coefficient for the case $E \lt V_0$ is $1$, which means all the waves are reflected back.

But I've got a question: we know from the calculation, there is some waves which can pass through the barrier, though decaying exponentially, then how is the reflection coefficient be $1$. Does that mean the waves which pass through the barrier will eventually reflect back?

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From the definition of the reflection and transmission coefficients, for the case $E<V$, $R=1$ means that there is NO CURRENT, or NO FLUX of particles at all. You can find particles behind the potential step: yes, it's true. But it does not mean that you can find a FLUX of particles. So, it is ok here.

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First off, a derivation of the transmission and reflection coefficients can be found here

Second, unless I'm grossly mistaken, the reflection coefficient at a step potential is not 1.0, even for the case that $E < V_{0}$.

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  • $\begingroup$ The reflection coefficient is exactly 1 if the potential has infinite width. See the graph on p. 8 of your link. $\endgroup$ – Michael Seifert Jul 12 '17 at 13:42

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