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Consider a finite potential barrier and the case where the energy of the particle is lower than that of finite potential. The figure is shown and we have three regions. As I solved the Schrödinger equation in the three regions. I can see that in the third region and first region, we have plane wave solution but in the third region the amplitude of the wave decreases as compare to the first region? Why it is happening so? Please elaborate it.

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    $\begingroup$ I'm skeptical of the function plot in the region of the barrier. If $E<V_0$, then the curvature sign should be the same as the function sign. That's not what I see. $\psi '' \propto -(E-V_0)\psi.$ $\endgroup$
    – Bill N
    Commented Aug 1, 2021 at 17:19
  • $\begingroup$ @Bill N Yes it usually looks like an exponential decay $\endgroup$ Commented Aug 1, 2021 at 17:48

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If it's about a particle moving from left to right, classically it wouldn't go past the barrier.

However in quantum mechanics it's possible that it can do 'tunnelling' and there is a small probability that it can get to the other side.

The amplitudes of the waves squared are proportional to the probability that the particle is at a given side of the barrier - so the amplitudes are showing that there is less chance that the particle is on the right than the left.

The amplitude reduces as the wave passes through the barrier, so there is more chance of a particle tunnelling through a thinner barrier.

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  • $\begingroup$ The total energy is greater than the height of the potential barrier, so the particle is allowed to move past the barrier, even classically. $\endgroup$
    – Bill N
    Commented Aug 1, 2021 at 12:26
  • $\begingroup$ @Bill N the question says: "Consider a finite potential barrier and the case where the energy of the particle is lower than that of finite potential" $\endgroup$ Commented Aug 1, 2021 at 13:15
  • $\begingroup$ Ok. My mistake for not reading. I was looking at the diagram. PLease forgive me. Trying to undo the downvote, but I can't unless you edit the answer. $\endgroup$
    – Bill N
    Commented Aug 1, 2021 at 17:09
  • $\begingroup$ @Bill N It's ok Bill, It'll be edited soon $\endgroup$ Commented Aug 1, 2021 at 17:32

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