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EXACT SOLUTIONS IN QUANTUM MECHANICS: VIM RECONSTRUCTION AND BEYOND

ABSTRACT

The Schrödinger equation stands as a cornerstone in the realm of quantum mechanics, governing the behavior of quantum particles and waves. This work presents a comprehensive reconstruction of the Variational Iteration Method (VIM) for the precise determination of exact solutions to both linear and nonlinear Schrödinger equations. In the linear Schrödinger context, we revisit and extend the VIM approach, providing a systematic framework for obtaining closed-form solutions that capture the quantum behavior of non-interacting particles. This reconstruction not only enhances the efficiency and accuracy of existing methods but also unveils new insights into the mathematical structure of the linear Schrödinger equation. Furthermore, we explore the formidable domain of nonlinear Schrödinger equations, where the interaction between quantum particles gives rise to rich and complex phenomena. Through the reconstructed VIM, we offer a powerful tool to tackle these nonlinear challenges, enabling the discovery of exact solutions that were previously elusive. This advancement contributes to our understanding of the intricate interplay between nonlinearity, dispersion, and soliton dynamics in quantum systems. The reconstructed VIM methodology is demonstrated through a series of illustrative examples, showcasing its applicability across various physical scenarios. It offers a valuable resource for researchers in quantum mechanics, mathematical physics, and related fields, enabling them to navigate the intricacies of linear and nonlinear Schrödinger equations with precision and insight. This study not only refines existing techniques but also paves the way for further advancements in the exploration of quantum phenomena through mathematical modeling and analysis.

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