Question

There is no tunneling in the case of infinite potential barrier, but there is when we have a finite well. In the classical analog, in the first case we have a particle bouncing between to infinitely rigid impenetrable walls and there is no tunneling, same as the quantum case. But if we have a finite barrier, means we have walls of finite rigidity, say made of cork or something. Then the particle would just break through some of the cork and it's probability of being found further in the cork wall will decay steadily.

I can understand discrete energy levels being a new thing, because they behave like a wave that's confined and not like particles confined, but why tunneling?

Answer 1

Not a good analogy. Here's why.

Your cork wall system is very complicated. That wall isn't just a potential with finite height. Rather, it contains holes where the potential is zero (or near zero), so that the particle can just propagate further and it contains full woody parts where the potential is infinite and the particle bounces back. So it is a very complicated many particle system which you can reduce to one parameter (like penetration depth, possibly depending on energy). But the resulting system will be probabilistic and very different from just a single particle with finite height potential. Because in this simple system the particle will bounce back with 100% probability if its energy is below the potential and pass with 100% probability if its energy is bigger.

Now, if you take this simple system (with just one degree of freedom) and investigate it from the point of view of quantum mechanics you'll discover that there's non-zero probability that particle will pass through the potential even if it doesn't have sufficient energy to pass it.

Why is this strange? Well, the particle has a total energy . This means that its kinetic energy is . As you can see, this equation makes no sense classically if the quantity on the r.h.s. is negative because you'll obtain imaginary velocity. But in quantum mechanics this still makes sense because the particle is also a wave and that imaginary factor means that this wave will exponentially decay, rather than just propagate freely.