“Application of Quantum Mechanics at The Non-Relativistic Regime for Obtaining the Energy Levels of an Electron at the Aharonov-Bohm Effect.”

Abstract

 In this research work, the dynamics of the electron were developed under gauge fields or also called electromagnetic potentials. The importance of potentials in quantum theory was studied, and how these are considered as more fundamental physical quantities, than the fields themselves. The modification of the electron dynamics, caused by the potentials in quantum theory, is known as the Aharonov-Bohm effect. The effect for the magnetic case was raised, that is, in the presence of the potential vector , considering the motion of the electron around a small, very long radius solenoid, with a uniform magnetic field  inside. For the description of the effect and the importance of the potentials, the electron orbits a region of the solenoid where the magnetic field is zero, but the potential vector is not. At the same time, the bound energy states of the electron were calculated for this case, and thus they are modified by the value of the magnetic flux, inside the solenoid, flux to which the electron is not exposed. It explained the meaning of the potential vector appearing as a phase factor in the wave function that characterizes an electron, which describes the change in the interference pattern between two electron beams, which pass through the exterior of the solenoid, in a region excluded from fields; however, in the presence of the potential vector, which implies that the interference pattern moves on the screen, because the beams arrive on different phases, that difference is represented as ∆Φ. The potential vector is an auxiliary field in classical electrodynamics; however, in quantum mechanics it has physical implications on charged particles. It is recommended for experimental evidence, instruments that generate a magnetic field without leaks, that is, a region excluded from fields.