If an ensemble of quantum partcles, with energy E, traveling in x direction encounter a very wide potential barrier V0 > E, the ensemble wavefunction will exponentially decay within the barrier. I thought that meant that there was a small probability of detecting an electron within the barrier. But the reflection coefficient of the probability current is unity. So whatâ€™s going on here?
For a very wide barrier, the inverse if the transmission coefficient (T) goes to infinity, so T --> 0 and R --> 1, like you said. The probabilty density of the wavefunction does decay exponentially within the barrier, but the amplitude decreases as the width of the barrier increases. So, for a very wide barrier, the probability amplitude inside the barrier would be extrememly small. If we take the extreme case where R = 1, then the probability amplitude would basically be zero. Here is a Shockwave movie that might help. Modify the width, and watch the behavior of the probability density. http://phys.educ.ksu.edu/vqm/html/qtunneling.html