Implementación computacional de la teoría espectral. Escuela Superior Politécnica de Chimborazo

Abstract

This study delves into the computational implementation of spectral theory, emphasizing particularly in the Schrödinger equation within quantum mechanics. The main challenge encountered was the complexity of applying spectral theory in computational contexts, especially in those concerning quantum systems. The objective was to clarify and implement current computational methods for spectral analysis, focusing on the Schrödinger equation. A qualitative and descriptive methodology was adopted, centered on the review and implementation of algorithms in the Python programming language. Key components included the Lanczos method for eigenvalue problems and the use of Chebyshev polynomials for spatial discretization. The results highlighted the effectiveness of the Lanczos method in handling large dispersed matrices, a critical aspect of quantum mechanics. The Chebyshev node implementation proved to be accurate for the radial space discretization, essential to analyzing the properties of the hydrogen atom. The combination of these methods facilitated a precise representation of quantum states and energy levels. In conclusion, this research significantly advances the understanding and practical application of spectral theory in quantum physics, offering valuable insights and tools for future research and computational analysis of quantum systems.