What Are the Forbidden Regions in Finite Square Wells?

[Mike89]

Ok so I finished the 2nd year of my physics degree in july and have been looking at some of my notes so I don't forget everything before I start again but it struck me that I still don't understand square wells for quantum particles all too well.

I understand how for ψ(x) behaves inside and outside of the boundaries in a finite well, but I'm not really sure the idea of a classically forbidden region features in this kind of well. I get that in an infinite square well where were the particle to venture beyond the boundaries it would require an infinite potential and therefore cannot exist in a classical sense since V>E in this situation but in a finite well I can't see where the forbidden regions are unless we're saying that the particle has an E<V₀ where V₀ is the potential outside the well. I don't recall the idea of quantum tunneling being applied to a finite quare well but is this idea the same? If so I think I can understand where the forbidden regions come in, if the energy hasn't enough energy to truly escape a well it still has a small chance to tunnel into an adjacent well in a quantum system that couldn't occur in a classical system.

If I'm right a simple pat on the head would be appreciate but if not a point in the right direction should be all I need, thanks in advance :)

 

[jpreed]

Lets look at these two cases you are talking about.

1.) The infinite square well.

The infinite square well potential will completely prevent the particle from existing outside of the well. Inside of the well, the particle resides in sinusoidal wavefunctions that go to zero at the boundaries.

2.) The finite square well.

The finite square well on the other hand does allow the particle's existence to "leak" into the surrounding barrier.

Classically this is not possible, if you have a ball rolling up and down a valley, it can only go up the sides of the valley as high as its kinetic energy at the bottom. So the ball cannot go into this "classically forbidden region".

In the quantum case, the particles does exist *inside* of the barrier (the classically forbidden region). If you put two finite square wells next to each other with a thin barrier between them, then the particle can tunnel through the barrier from one well to the other and back.


A simple anology would be if had two types of cars, a regular car and a quantum car. Both sit on top of a 10 meter high hill and are going to try to roll up a 12 meter high hill and down the other side (see attachment). The classical car can only roll 10 meters up the 12 meter hill, but will NEVER make it to the other side. The quantum car can make it to the other side by tunneling through the barrier.

 

[maverick280857]

A simple anology would be if had two types of cars, a regular car and a quantum car. Both sit on top of a 10 meter high hill and are going to try to roll up a 12 meter high hill and down the other side (see attachment). The classical car can only roll 10 meters up the 12 meter hill, but will NEVER make it to the other side. The quantum car can make it to the other side by tunneling through the barrier.

I liked this

Mike89, you might want to try to take the limit of V0 going to infinity for a square well potential (assume the well is at V = 0 and the barrier outside is at V = V0) and see how the solution tends to the solution for an infinite square well. There are some technicalities involved here which you can look at later (specifically, the boundary condition on psi is automatically ensured for V0 = infinity, but it has to be explicitly enforced for V0 = some finite value). The tunneling (or leaking) feature of the wavefunction begins to disappear as V0 --> infinity.

Classically, an E < V0 region is forbidden because this would imply that the kinetic energy T = E - V0 < 0.