The mathematical model used to predict the probability of quantum tunneling can indeed take into account various additional factors beyond a simple transmission and reflection from a low potential barrier. In more realistic scenarios, the tunneling process can be influenced by a range of factors, including the presence of other potentials and interactions.
One approach to incorporating these effects is to use a more comprehensive potential energy landscape that includes multiple barriers and wells. This can be achieved by solving the Schrödinger equation for the given potential energy profile, taking into account the interactions between particles and the associated quantum states.
In some cases, quantum tunneling calculations involve considering tunneling through complex potential landscapes or barriers that are time-dependent or spatially varying. Numerical techniques, such as the finite difference method or the finite element method, can be employed to solve the time-dependent Schrödinger equation and obtain the tunneling probabilities under such conditions.
Moreover, interactions between particles can be taken into account through perturbation theory or more sophisticated methods like the Feynman diagram approach. These interactions can influence the tunneling probability by modifying the potential energy profile or introducing additional scattering processes that affect the tunneling process.
Additionally, quantum tunneling can be affected by environmental factors, such as the presence of electromagnetic fields or thermal effects. These influences can be included in the calculations through various theoretical frameworks, such as the WKB approximation, density functional theory, or path integral methods, which can provide a more accurate description of the tunneling behavior in the presence of these external influences.
It's important to note that the level of complexity in incorporating these additional factors depends on the specific system and the desired level of accuracy. In some cases, simplified models or approximations may be used to obtain an approximate description of the tunneling process, while in other cases, more sophisticated computational methods and numerical simulations may be necessary.
Overall, the mathematical model used to predict the probability of quantum tunneling can be extended to account for various additional potentials, interactions, and environmental effects. The level of complexity and the techniques employed will depend on the specific system under consideration and the desired level of accuracy.