This project uses the procedure of root-finding to resolve the eigenvalue problem of a rectangular quantum well. The procedure is applied to determine the first two to three energy levels of a simplified model of a hydrogen atom, represented by the rectangular quantum well. The project also explores the eigenvalue problem within various physics fields. While the project involves simple mathematical operations, it is rooted in complex physical concepts like quantum mechanics, often unfamiliar to first-year students. The fundamentals of quantum mechanics are introduced, providing enough understanding for successful project execution. The project initially focuses on the central object, the quantum state, and its probabilistic nature in quantum mechanics. The Schrödinger equation, an eigenvalue problem, is used to find state functions. This project explores eigenenergies and eigenfunctions within a rectangular finite quantum well, treating the well as a simplistic 1D model of the hydrogen atom.