Derivation of advanced Python code for solving the
Dirac-Coulomb-Breit equation in muonic atoms
Poster-Athanasios Gkrepis
ABSTRACT
In this work, initially the Dirac-Coulomb equation for exotic muonic atoms is formulated in the relative coordinate r of the muon-nucleus system by separating out the center of mass motion. Then, an advanced algorithm (in Python language) is derived on the basis of the neural networks techniques for solving numerically this equation and obtain the large (upper) f(r) as well as the small (bottom) g(r) components of the muon orbiting around a nucleus.
For the required optimization process, an error function of a concrete muonic system is defined and the BFGS or trust-constr methods (as provided from the SciPy submodule optimize) are being utilized. Finally, in order to asses the confidential level of the designed algorithm, the numerical solutions are compared with the corresponding analytic wave functions of the Dirac-Coulomb equation.
One of our main goals of this work is to test previous muon capture predictions obtained by solving the Schrödinger equation for several muon-nucleus systems and, then, to perform systematic studies on exotic purely leptonic atoms that are promising probes to test the quantum electrodynamics and beyond the standard model theories. Our numerical method provides fast and accurate numerical solutions of the Dirac-Coulomb equation a fact that encourages us to apply the same scheme to solve the corresponding Dirac-Coulomb-Breit equation which includes higher order relativistic corrections for the description of the aforementioned exotic atomic systems.