 Ettery Majorana's original work on non-adiabatic transitions between two quasi-crossing energy levels was revisited and significantly expanded upon. A new derivation of the Landau-Zenerstuckelberg Majorana formula was provided, along with a complete description of the wave function, including its phase. This wave function accurately describes the dynamics away from the avoided level crossing, but has limited accuracy in this region. This article was authored by P. O. Koffman, O. V. Ivekhanenko, S. M. Sivchenko, and others.