(untitled) (Universal International Scientific Journal, 2026)
The transmission of electron wave functions in a material whose physical parameters vary along a specific spatial direction has been investigated from a rigorous theoretical perspective. In such systems, spatial inhomogeneity plays a crucial role in determining the quantum-mechanical behavior of charge carriers, since variations in potential energy, effective mass, or compositional profile significantly influence wave propagation. When an electron travels through a non-uniform medium, its wave function adapts to the changing environment, leading to reflection, transmission, and interference phenomena that must be described consistently within the framework of quantum mechanics. The present analysis is based on the single-particle, time-independent Schrödinger equation, which provides a fundamental description of stationary states in quantum systems. This equation is formulated under the assumption of total energy conservation, meaning that the sum of kinetic and potential energies remains constant throughout the motion of the particle. By applying appropriate boundary conditions at interfaces where material properties change, one can determine the amplitudes of reflected and transmitted waves, as well as calculate physically measurable quantities such as transmission and reflection coefficients. Within this formalism, elastic scattering processes of non-interacting, spinless particles are examined in detail. Elastic scattering implies that the particle’s total energy is conserved during interaction with the potential landscape, although its momentum and direction of motion may change. Particular attention is given to quantum tunneling phenomena, in which particles penetrate potential barriers even when their kinetic energy is lower than the barrier height. Such processes arise purely from the wave nature of matter and have no classical analog. Altogether, the approach provides a unified, internally consistent framework for describing wave transmission, reflection, and tunneling in spatially non-uniform quantum systems.