The particle INSIDE the barrier

1. Oct 11, 2009

iLIKEstuff

So I know this topic of tunneling has been discussed hundreds of times on this forum, but I simply cannot find nor deduce an answer to a burning question I have about what happens exactly INSIDE the barrier and how to describe it (preferably in layman's terms.)

The problem is basic: A particle with energy E that is impinging on a finite barrier with height V where E < V.

There are valid solutions to the wave function which describes the particle before it hits the barrier, inside the barrier, and after the barrier. Now, the solutions before and after the barrier are "wave-like" i.e. they are oscillating sinusoidal functions. However, the solution inside the barrier is not... it is exponentially decreasing.

So my multi-part question:

What is this "thing" inside the barrier?

Is it a particle? If so, why? Does it have single particle properties?

Is it a wave? If so, why? Also, why doesn't the solution show oscillating behavior? Isn't QM supposed to describe the simultaneous wave- and particle-like behavior?

Any response is appreciated. Thanks for your input guys.

Last edited: Oct 11, 2009
2. Oct 12, 2009

Pythagorean

The wave function is used to determine the probability that the particle will be in a particular location*: (I'm just showing this equation to point out that the wave function itself is not the probability. It has to be operated on, along with it's conjugate. The operator that you use is based on what you want to know about it. In this case, we want to know the 'expected' position of the particle, denoted by <x>)

So when you see the exponential curve in the potential barrier, or the wave-shape where the potential is zero, it's not the shape of the particle itself. It's related to the probability that it will be in a particular location, x, in that region.

*note: position determination isn't the only use of the wavefunciton. You can use a different operator to measure a different observable (such as the momentum or the kinetic energy of the particle).

3. Oct 12, 2009

Staff: Mentor

Actually, this is a little incorrect. There is a single valid solution, and that solution is non-zero on one side of the barrier, inside the barrier, and on the other side of the barrier. The part inside the barrier is not a separate solution, nor are the parts on either side.
The entire wavefunction is the state of the particle, and represents everything that can be measured about the particle. When the wavefunction is displayed as a function of space (in the position basis) then its value at some point is related to the probability of measuring it to be at that point. The fact that some parts of the wavefunction are inside the barrier represents the fact that there is a finite probability of measuring it to be within the barrier.