Rectangular Potential Barrier

Rectangular potential barriers, also called square potential barriers, are formed by energy potentials which create wall-like barricades for particles. Essentially, a potential barrier is a potential step except the energy potential returns to zero at some finite positive $x$-position, $a$, and remains zero beyond that point. Here, we'll derive the wave function of a particle facing a rectangular potential barrier, then find the transmission and reflection coefficients of the particle upon encountering the barrier.

Hyperbolic Functions

Hyperbolic functions are related to the unit hyperbola, given by $x^2 - y^2 = 1$, analogous to the way trigonometric functions are related to the unit circle. Both trigonometric and hyperbolic functions can be used to parameterize their respective unit conics. However, while the unit circle's central angle, the argument taken by trigonometric functions, is indeed what one might consider to be an "angle" in the usual meaning of the word, the hyperbolic angle taken by hyperbolic functions is perhaps less intuitively defined.