Asymptotic methods in quantum mechanics

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Quantum mechanics and the Schrodinger equation form the foundation for understanding the properties of atoms, molecules, and nuclei. Developing reliable solutions for the energy eigenfunctions of these systems is a complex challenge. The conventional method for obtaining these eigenfunctions relies on the variational extremum property of energy expectation values. However, the intricacies of these variational solutions hinder a clear, concise representation of the physical structure. Certain characteristics of wave functions in specific spatial domains are influenced by the general structure of the Schrodinger equation and the electromagnetic potential. These characteristics offer valuable insights for creating simple and accurate wave function solutions, yet they have not been sufficiently emphasized. This work focuses on the local properties of wave functions for a collection of particles, particularly their asymptotic properties when one particle is distanced from the others. The asymptotic behavior is primarily determined by the separation energy of the outermost particle. Recognizing the universal significance of this asymptotic behavior is crucial for both research and teaching. This presentation aims to highlight these aspects effectively.