Tunneling through a finite potential barrier

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BearY
I have 3 questions. After thinking about it I feel these questions may indicate that I have some misunderstanding in basic knowledge or some missed parts.
1. Why is the (time independent) wavefunction an exponential decay inside the potential barrier? I know the mathematical derivation, but I am confused why traveling in a region with constant potential would cause the probability density to decrease.
2. When crossing from the finite barrier region to zero potential region, reflection still happens. Why is that happening? I think if I just take it for granted that it would happen, I can derive the wavefunction, but I am confused why it is happening.
3. When the initial kinetic energy is larger than the potential energy the particle would acquire in the barrier, neither reflection nor the exponential decay happens. At first, it is natural to me so I didn't think about it too much about it. But after thinking about the first 2 questions, I am starting to question why is this happening as well.

A. Neumaier

I am confused why traveling in a region with constant potential would cause the probability density to decrease.
Imagine a random walk inside the barrier. If the barrier is wide it is far more likely to return to the region one started than to cross the barrier. This likelihood decreases exponentially with the width of the barrier.

BearY
BearY
Imagine a random walk inside the barrier. If the barrier is wide it is far more likely to return to the region one started than to cross the barrier. This likelihood decreases exponentially with the width of the barrier.
Thank you for the reply. That does explain why it is exponential if I imagine the displacement inside the barrier as a Bernoulli process about going backward or forward. But what is causing it to go back to start with? When the particle went into the barrier, it seems it acquired potential energy that it does not have. Does it have anything to do with this "random walk"?

Mentor
When the particle went into the barrier, it seems it acquired potential energy that it does not have.
The particle does not "go into" the barrier, and if it were to tunnel from one side of the barrier to the other it has not moved through the barrier or ever been "in" it, so there's no question of it ever "going back" The wave function isn't telling you anything about where the particle is or is not, it is telling you the probability of finding it at any given location when and if you measure its position.

Any measurement involves a interaction with the particle, and that interaction will necessarily transfer energy and momentum between the particle and the measuring device. If a measurement finds the particle in the classically forbidden region, then the measuring device has transferred energy to the particle so that the total energy of the particle+device system is conserved. If we don't measure the particle and find it in the classically forbidden region, then it was never there (even if successive measurements find it on opposite sides of the barrier) so there's no potential energy to acquire.