The wave function of a physical system of particles specifies everything that can be known about the system. Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the time evolution of a known wave function can be deduced. The square of the absolute value of this wave function is directly related to the probability distribution of the particle positions, which describes the probability that the particles would be measured at those positions.
As shown in the animation, a wave packet impinges on the barrier, most of it is reflected and some is transmitted through the barrier. The wave packet becomes more de-localized: it is now on both sides of the barrier and lower in maximum amplitude, but equal in integrated square-magnitude, meaning that the probability the particle is somewhere remains unity. The wider the barrier and the higher the barrier energy, the lower the probability of tunneling.
Some models of a tunneling barrier, such as the rectangular barriers shown, can be analysed and solved algebraically. Most problems do not have an algebraic solution, so numerical solutions are used. "Semiclassical methods" offer approximate solutions that are easier to compute, such as the WKB approximation.