The semi-classical limit of the Aharonov-Bohm effect: The actualized approach

We suggest an approach, which formally allows us to describe the Aharonov-Bohm (AB) effect in the semi-classical language. In the framework of this approach, we keep the classical concepts of electromagnetic field and force. At the same time, instead of point-like classical charges, we introduce a finite-size elementary charge distribution, modelling the wave-like packet, associated with the motion of a given electron. In this case we derive the force on the wave-like packet on behalf of the solenoid via the minimization of action defined through the Lagrangian density (instead of the Lagrangian used in common classical electrodynamics of point-like charges). We show that this force due to the solenoid, being dependent on the vector potential, yields the common expression for the magnetic AB phase, when the original wave packet is splitted into a superposition of two packets encirling the solenoid. We also analyze in the classical language the implementation of total momentum conservation law for the isolated system “moving electrons plus elongated solenoid” and determine the properties of finite-size charge distribution, when this law is fulfilled. The results obtained are discussed.