A Computational Revisit of the Variational Principle: Estimating Ground State Energies of the 1D Harmonic Oscillator via Python

Purpose of the study:To estimate the ground state energy of the one-dimensional harmonic oscillator using the variational principle and Python-based numerical methods. Methodology:Python 3.11 was used with NumPy, SciPy, and Matplotlib libraries. The variational method was applied using multiple trial wavefunctions. Integrals were computed via Simpson’s rule, and optimization was done through parameter scanning. Main Findings:The Gaussian trial wavefunction produced a ground state energy of 0.5003 ℏω, showing 0.06% error. Other trial functions were less accurate. The results confirm that the choice of trial function critically affects the energy estimate, and Python effectively supports variational computations in quantum systems. Novelty/Originality of this study:This study integrates computational tools with the variational principle, presenting an accessible approach to energy estimation in quantum mechanics. It demonstrates how Python can facilitate variational analysis, making the method replicable and educationally useful for students and researchers.