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

  • Redi Kristian Pingak Department of Physics, Faculty of Science and Engineering, University of Nusa Cendana
  • Johanes Eliaser Jacob Department of Physics, Faculty of Science and Engineering, University of Nusa Cendana
  • Bernandus Bernandus Department of Physics, Faculty of Science and Engineering, University of Nusa Cendana
  • Albert Zicko Johannes Department of Physics, Faculty of Science and Engineering, University of Nusa Cendana
  • Ely Setiawan Department of Chemistry, Faculty of Mathematics and Natural Sciences, Universitas Jenderal Soedirman, Indonesia

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 XZ+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

Published
2024-03-14
How to Cite
PINGAK, Redi Kristian et al. Approximate Solution to the Schrödinger Equation of Exotic Doubly Muonic Helium-like Systems Using Hydrogenic-based Matrix Mechanics. Molekul, [S.l.], v. 19, n. 1, p. 189-197, mar. 2024. ISSN 2503-0310. Available at: <http://jos.unsoed.ac.id/index.php/jm/article/view/11224>. Date accessed: 17 apr. 2024. doi: https://doi.org/10.20884/1.jm.2024.19.1.11224.
Section
Articles