ABSTRACT
When we dealt in the last chapter with the finite potential well, it was observed that the wave function penetrated into the barriers. This process does not occur in classical mechanics, since a particle will in all cases bounce off the barrier. However, when we treat the particle as a wave, then the wave nature of barrier penetration can occur. This is familiar in electromagnetic waves, where the decaying wave (as opposed to a propagating wave) is termed an evanescent wave. For energies below the top of the barrier, the wave is attenuated, and it decays exponentially. Yet, it takes a significant distance for this decay to eliminate the wave completely. If the barrier is thinner than this critical distance, the evanescent wave can excite a propagating wave in the region beyond the barrier. In optics, this effect is termed frustrated total internal reflection. We are familiar with total internal reflection just from a swimming pool. Water has a higher dielectric constant than free space, so when we are under water, there is critical angle beyond which we can no longer see out of the pool. In optics, we can use a prism, and there will be reflection from the back surface for angles greater than this critical angle. The wave in back of the prism is attenuated and decaying. But, if we place another high dielectric constant material close to the prism, the decaying wave can excite a wave in this new material. Thus, the wave can penetrate the barrier, and continue to propagate, with an attenuated amplitude, in the trans-barrier region. For particles, this process is termed tunneling, with analogy to the miners who burrow through a mountain in order to get to the other side! This process is quite important in modern semiconductor devices, and Leo Esaki received the Nobel prize for first recognizing that tunneling was important in degenerately doped p–n junction diodes. Nevertheless, tunneling arises when the particle is treated as a wave. The behavior is fully consistent with optics in that regard.
