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I'm trying to understand quantum mechanics and I have a problem with the first application. The solution of Schrödinger equation for a step potential https://en.wikipedia.org/wiki/Solution_of_Schr%C3%B6dinger_equation_for_a_step_potential

Considering the case $E<V_0$, we have a transmission coefficient $T=0$, so I would interpret it as no particle can enter. But looking at the wave function, we see that inside, it decreases exponentially but it is not zero. So I should have a non-null probability to have a particle inside. I don't understand this form of apparent contradiction, a non-null wave function but a transmission coefficient null.

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I think I got it, just have to process it. According to the Cohen-Tannoudji, p.282. Inside we have a positive current corresponding to the entrance of the wave, so there is entrance, there is no contradiction. But at the same time, there is a negative current corresponding to the return of the wave packet. They are exactly equal. So the total current is zero. It seems to be a "coincidence" in 1D, while we don't have this situation in a 2D problem.

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    $\begingroup$ The key here is to realize that the wave-function is not normalized. Imagine that the particle could only be found (for some reason) up to a distance L to the left and right of the barrier. You could then 'normalize' the wave-function. The integral from 0 to L of this 'normalized' wave function would then go to zero as L approaches infinity. That's how I wrapped my head around the problem. $\endgroup$ – chuckstables Sep 12 '17 at 22:53

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