Graphical Solution of Eigenstate of an Electron in a Finite Quantum Well

Abstract

This study explores the eigenstates of an electron in a finite quantum well using the Schrödinger wave equation. Quantum mechanics, a fundamental theory in physics, describes the properties of molecules, atoms, and subatomic particles through quantization of energy and wave-particle duality. A quantum well, a nanometer-thin layer in semiconductor materials, confines electrons to a two-dimensional layer, resulting in quantized energy spectra essential for various electronic and optoelectronic devices. Unlike the infinite potential well, the finite potential well allows for the probability of finding particles outside the well, necessitating accurate calculations of bound states. This research employs a graphical method using MATLAB to solve for the eigenstates and eigenenergies of electrons in a finite quantum well. By deriving the time-independent Schrödinger equation, applying boundary conditions, and utilizing transcendental equations, we determine the energy levels and eigenfunctions of the system. The study highlights the practical applications of quantum wells in modern electronic devices and underscores the importance of understanding quantum confinement in developing advanced technologies.