Final answer:
The oscillations in the electron transmission probability curve in the presence of a numerical dispersion arise due to quantum mechanical tunneling and the wave-particle duality of electrons. As electrons tunnel through a potential barrier, the discrete energy levels and wave behavior of particles lead to variations in transmission probability.
Step-by-step explanation:
The oscillations observed in the Transmission probability (T(E)) curve when considering a numerical dispersion such as E=U0+2t(1−cos(ka)), where a is the mesh spacing, U0 is the barrier potential and t is the hopping parameter, can be understood through the concept of quantum mechanical tunneling and wave-particle duality. Electrons, much like other quantum particles, do not follow a classical path, but rather, their behavior is governed by wave equations. Through the time-independent Schrödinger equation for a particle encountering a potential barrier, it is demonstrated that the probability of an electron tunneling through the barrier is related to the energy of the particle relative to the barrier height and its width.
This wave-particle duality was experimentally confirmed by C. J. Davisson and L. H. Germer, who observed an interference pattern for electrons, illustrating that electrons have both wave and particle characteristics. Additionally, figures like Fig. 31.28 and 31.32 in the literature demonstrate how an alpha particle can quantum mechanically tunnel through a barrier, a process which the electron transmission through a potential barrier analogously follows. The decay in the wave function inside the barrier and its nonzero value outside implies a finite probability of tunneling, leading to a transmission probability curve that oscillates due to quantum mechanical effects and the discrete nature of energy levels in quantum systems.