Approximate Solution to the Schrödinger Equation of Exotic Doubly Muonic Helium-like Systems Using Hydrogenic-based Matrix Mechanics

Abstract

The approximate solution to the Schrödinger equation of exotic doubly muonic helium-like systems has been obtained using a simple matrix method based on hydrogenic s-states. Each system considered consists of a positively charged nucleus surrounded by 2 negatively charged muons X^Z+_μ^-_μ^- (2 Z 36). The present work aims to obtain approximate ground-state energies of the systems and to decompose the energies in terms of the basis states used. Here, the wave function was expressed as a linear combination of 15 eigenfunctions, each written as the product of two hydrogenic s-states. The elements of the Hamiltonian matrix were calculated and finally, the energy eigenvalue equation was numerically solved to obtain the ground-state energies of the systems with their corresponding eigenvectors. From the results, ground-state energies of all systems were in agreement with others from the literature, with percentage differences between 0.03% and 2.05%. The analysis of the probability amplitudes from the eigenvectors showed that the 1s1s state made the largest contribution to the ground state energies of the systems, approaching 90.99%, 96.98% and  98.54% for He²⁺_μ^-_μ^{- ,} Li³⁺_μ^-_μ^- , and Be⁴⁺_μ^-_μ^- , respectively.

Keywords: Doubly muonic helium-like systems, ground state energy, hydrogenic basis states, matrix mechanics, Schrödinger equation