Eigenfunctions and eigenvalues of modified double-center Woods-Saxon potential

This study explores the quantum mechanical dynamics of nuclear fission of uranium-235 using a modified double-center Woods-Saxon potential. The Schrödinger equation is solved via the Nikiforov-Uvarov method to derive eigenfunctions and eigenvalues characterizing the fission process. The research models binary fission to reveal subatomic phenomena such as energy transitions and particle probabilities during nucleus splitting process. By solving the radial component of the three-dimensional Schrödinger equation, the study computes eigenfunctions, eigenvalues, and probability densities, providing insights into the nuclear system's quantum states. The key aspects examined include energy level transitions, the influence of eccentricity, and double-fold degeneracies. Results illustrate the dynamic evolution of fragment separation, transitioning from overlap at equilibrium to distinct potential wells as fragments become independent. The eigenfunctions exhibit even parity π⁺ for eigenvalues n_z = 0, 2, 4, 6 and odd parity π⁻ for n_z = 1, 3, 5, 7, with the number of nodes reflecting their states. Energy eigenvalues are determined by applying continuity conditions at the boundary (z = 0) and solving transcendental equations across various eccentricity values. The eigenvalue spectrum reveals double-fold degeneracy, indicating a transition from two independent schemes to a single configuration at infinite separation, describing the complete asymmetric fission of ²³⁵U into two fragments. These findings of the research underscore the model's capability to depict fission dynamics, offering significant insights into nuclear interactions, energy transitions, and the mechanics of heavy nuclei splitting. The study highlights the potential of modified double-center Woods-Saxon potentials in advancing our understanding of nuclear binary fission processes.