Approximate Analytical Solution of the Ground State Problem of He and He-like Ions using Symmetrized-Hydrogenic States

Abstract

The matrix method using three symmetrized-hydrogenic basis states has been applied to analytically obtain an approximate solution to the Schrödinger equation of the Helium atom and some He-like ions (2<Z<10). This study aims at obtaining more accurate ground state energies of the systems compared to our previous calculation using un-symmetrized basis states and some other simple calculations in the literature. The contribution of the symmetrized basis states on the ground state energies of the systems is also investigated. The time-independent Schrödinger equation involving a 3×3 Hamiltonian matrix, formed by hydrogenic s-states, was analytically solved to obtain three energy eigenvalues of the systems as well as their corresponding eigenvectors. Results showed that the 1s2 energies of the systems were more accurate than our previous unsymmetrized basis calculations, with significant error reduction observed for He and Li+. With the same matrix size, the ground state energies of He and He-like ions obtained from three symmetrized basis states in this study were found to be closer to the exact and experimental energies than those obtained from unsymmetrized basis states. It was also demonstrated that the |100;100> state made the largest contribution to the ground state energies of the systems, i.e. about 90.9% for He and around 99.9% for Ne8+, and consequently the smallest contribution came from the other two symmetrized states (less than 1%). To conclude, the calculated ground state energies were more accurate than some other simple calculations reported in the literature.