potential barrier

In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schr?dinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.
Although a particle hypothetically behaving as a point mass would be reflected, a particle actually behaving as a matter wave has a finite probability that it will penetrate the barrier and continue its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by thetransmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schr?dinger s wave-equation allows these coefficients to be calculated.



Transmission and reflection


At this point, it is instructive to compare the situation to the classical case. In both cases, the particle behaves as a free particle outside of the barrier region. A classical particle with energy larger than the barrier height would always pass the barrier, and a classical particle with incident on the barrier would always get reflected.
To study the quantum case, consider the following situation: a particle incident on the barrier from the left side ( ). It may be reflected ( ) or transmitted ( ).
To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations (incoming particle), (reflection), =0 (no incoming particle from the right), and (transmission). We then eliminate the coefficients from the equation and solve for and .