Variational Calculus in Quantum Mechanics: A Computational Approach with Python
Alisson Ferreira Martins
ituto Federal de Educação, Ciência e Tecnologia de São Paulo – Campus Birigui
Abstract
The Variational Method is a powerful tool in Quantum Mechanics for solving minimization problems without the need to know exact analytical solutions. This work aims to analyze and attempt to implement the Variational Method through programming, numerical methods, and physical concepts. The approach considers the connection between theory and computational simulation, exploring the relationship between programming and physics, and examining the possibility of exploring these emerging areas.Initially, the necessary theoretical foundations were defined to deal with the matrix notation of quantum mechanics. Then, in the code implementation, boundary conditions and analytical solutions of classical problems, such as the hydrogen atom and the harmonic oscillator, were established, allowing comparisons with the obtained results. The code enabled the attempt to implement other problems in quantum mechanics. The implementation proved to be satisfactory, although there is a need for future improvements. This work not only attempts to implement the method but also validates it through comparisons with known analytical solutions. The implementation in Python, using libraries such as Scipy, Numpy, and Matplotlib, demonstrates the effectiveness of the method in different dimensions (1D, 2D, and 3D), highlighting the importance of choosing appropriate trial functions.
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