RELATIVISTIC QUANTUM DESCRIPTION OF MUONIC HYDROGEN ATOM: CASE OF FINITE SIZED NUCLEUS

Abstract

Muonic hydrogen is formed when electron is replaced by muon in an atom in which the mass of muon is 207 times larger than that of an electron, the radii of the muonic orbits are much smaller than that of electrons. The main purpose of this study is to theoretically describe muonic hydrogen with finite-sized proton in the framework of relativistic quantum mechanics. The main focus of this work are to determine the energy levels of muonic hydrogen with point-like proton at its center, to determine the energy eigenvalues of muonic hydrogen caused by the consideration of finite size nucleus. The derivation and solutions of the modified Dirac equations are performed in presence of central potential. The confluent hypergeometric function is used to find the exterior wave functions. In both analytical and numerical calculations we have used the experimental nuclear radius which is 0.8768fm. In the case of finite size nucleus, the matching equation is solved both analytically and numerically to obtain approximate energy eigenvalues of the muonic hydrogen atom. In the analytical calculations, the energy eigenvalue has been calculated for finite size nucleus and the result is 102.795345MeV which is lower than the numerical result 105.6545718MeV and this low result of energy is due to a number of bad approximations and the difficulties in manual calculation. Numerical results have generated using Matlab R2019a programming language. The numerical result of energy eigenvalue have been calculated and tabulated for different states. Probability density computed for the case of finite size nucleus shows that muon has a high probability to be found near the nucleus of an atom for 1𝑠1/2 state when compared to the 2𝑝1/2 excited state