In the present article we propose a simple equality involving the Dirac operator and the Maxwell operators from a chiral approach. This equality establishes a direct connection between solutions of the two systems. Moreover, we show that the connection is valid when a fairly natural relationship between the frequency of the electromagnetic wave and the energy of the Dirac particle is fulfilled, if the electric field <img border=0 width=15 height=19 src="/fbpe/img/ingeniare/v16nespecial/image06-1.gif">is parallel to the magnetic field <img border=0 width=17 height=19 src="/fbpe/img/ingeniare/v16nespecial/image06-2.gif">. Our analysis is based on the quaternionic form of the Dirac equation and on the quaternionic form of the Maxwell equations. In both cases the quaternionic reformulations are completely equivalent to the traditional form of the Dirac and Maxwell systems. This theory is a new quantum mechanics (QM) interpretation. The research below shows that the QM represents the electrodynamics of the curvilinear closed chiral waves. This concords entirely with the modern interpretation and results of the quantum field theory.