The calculation of the probability for quantum tunneling of an electron is derived from the mathematical formalism of quantum mechanics. Let me explain the basic steps involved in this derivation.
In quantum mechanics, particles such as electrons are described by wave functions, which are mathematical functions that contain information about the particle's position and momentum. The time evolution of a wave function is governed by the Schrödinger equation, a fundamental equation in quantum mechanics.
When a particle encounters a potential barrier, such as an energy barrier or a potential energy well, there is a possibility for the particle to tunnel through the barrier, even though classically it would not have enough energy to overcome it. This phenomenon is known as quantum tunneling.
To calculate the probability of quantum tunneling, physicists typically use a technique called the "WKB approximation" (Wentzel-Kramers-Brillouin approximation), which is an approximation method that allows for the calculation of wave functions in the presence of a potential barrier.
In the WKB approximation, the wave function is approximated as a combination of two solutions: one for the region before the barrier, and another for the region after the barrier. The wave function is assumed to decay exponentially within the barrier region. By matching the wave functions and their derivatives at the boundaries between these regions, physicists can derive an expression for the transmission coefficient, which represents the probability that the particle will tunnel through the barrier.
The transmission coefficient depends on various factors, such as the height and width of the barrier, as well as the energy of the particle. It is important to note that the WKB approximation is an approximation method and may not provide an exact solution for all cases. In some situations, more advanced mathematical techniques, such as quantum mechanical scattering theory, may be required to calculate the precise tunneling probability.
Overall, the calculation of the probability for quantum tunneling involves applying the principles of quantum mechanics, specifically the Schrödinger equation and the WKB approximation, to determine the likelihood of a particle tunneling through a potential barrier.