Slides: Barriers and tunneling

The discussion of the eigenstates of a particle in a barrier potential follows exactly the same method as the bound particle in the finite well, only the exponential form of the solutions is the most convenient in all cases, not the sine and cosine forms. (The case of the particle with energy greater than the depth of a finite potential energy well is also easy).

The important difference is that there are no boundary conditions at infinity, which removes two constraints. The result is that there is no quantization of the energy and the wave function is only quasi-normalizable.

Students are often anxious to throw out the decaying exponential in the barrier region. Although the decaying solution is, in fact, small, is is necessary to make the boundary conditions work and there is no physical reason for such a step.

Point out that one is mostly concerned with the ratio of the amplitude of the transmitted wave to the incident, or rather with the mod squared, which represents the particle intensity. It is safest to stick to the symmetric barrier; the asymmetric barrier leads to mathematical complexity that is best left to homework.

The interesting consequences of the barrier transmission are:

• The transmission is finite for the total energy lower than the height of the barrier. One should discuss the classical case, too.
• The transmission is <1 even when the energy reaches the height of the barrier (also violates classical intuition)
• The transmission oscillates for particle energy greater than the height of the barrier! It never exceeds 1, though (good!). This oscillation corresponds to quasi-resonances when the wavelength of the particle matches 1/4 of the width of the well. A comparison can be made to optics.