Abstract:  This paper focuses on the relativistic hydrogen atom system. Starting from the Dirac equation, the relativistic wave function of the electron of the hydrogen atom is strictly solved. On this basis, the MonteCarlo simulation method is adopted to sample the probability cloud density of the electron, and the threedimensional visualization of the probability cloud of the electron with spin in the hydrogen atom under relativistic conditions is realized by using Mathematica software. The electron cloud distribution under different principal quantum numbers, orbital angular momentum quantum numbers and magnetic quantum numbers is analyzed, and the electron cloud morphology under relativistic and non-relativistic conditions is compared. In addition, this paper proposes a probability cloud sampling algorithm based on the acceptance rejection sampling method, which can be effectively applied to the visualization of the electron cloud of general atomic systems with spin.

Key words: relativistic quantum mechanics, Dirac equation, hydrogen atom model, Monte-Carlo simulation, visualization