Aplicación de la mecánica cuántica en el régimen no-relativista para obtener los niveles de Landau de un electrón sometido a un campo magnético homogéneo

Abstract

This thesis is aimed at studying Landau levels which appear from the quantization of electrons in the presence of magnetic fields, to study the behavior of electrons with two-dimensional motion within a homogeneous magnetic field. The impressive thing about the study of this system was the motion equations were reduced to the shape of a one-dimensional quantum harmonic oscillator, where the frequency is the cyclotron frequency and its energy spectrum looks the same as the oscillator. To begin with the classic treatment was carried out based on the canonical formalism with which the motion equations were raised in order to arrive to Lagrangian and Hamiltonian that contains the system information of an electron in a homogeneous magnetic field based on the electromagnetic potentials. The second part of the work corresponds to the non-relativistic quantum treatment to represent the electron with two-dimensional motion in a homogeneous magnetic field; before starting the quantum theory for calculations simplicity the Landau gauge was chosen, condition which it is ensured the Magnetic field is in the z-direction, which does not affect the observable physical properties. Finally, the Hamiltonian and the Landau gauge were introduced into the independent time Schrödinger equation to be solved, which was solved through power series, which the eigenfunction solution was obtained in an elegant way, it was rewritten according to Hermite polynomials and it gets the Landau levels that are degenerated and therefore the set of states for each energy level represents a Landau level, in addition this degeneration was also appreciated in the probability densities.