Métodos numéricos para a solução da equação de Dirac potenciais de oscilador harmônico

Abstract

The Harmonic Oscillator (HO) is a system addressed in various areas of physics and with many applications in vasts fields of science. We present the equations of the HOs for the classical, quantum and relativistic cases and solve them analytically and numerically. To do so, we apply the Euler, Runge-Kutta and Numerov methods to solve the second-order differential equations (DOEs) of the oscillators, developing algorithms in C and Python programming languages for the numerical solutions, and utilizing the MATHEMATICA software for analytical solutions. We compare both results to test the effectiveness of these methods, concluding that the Numerov method is the most efficient. We use the relativistic harmonic oscillator (RHO) as a physical-mathematical model to study energy level degeneracy and wave functions for the limits of exact nuclear spin and pseudospin symmetries. Regarding the RHO energy eigenvalues, the maximum relative difference between the numerical and analytical calculations was approximately $0.1 \%$ for the ground state $n = 0$ and $ 2.0\%$ for the excited state $n = 5$. In addition, the numerically calculated wave functions were compatible with those obtained from the analytical functions, allowing us to reveal the exact degeneracy expected for the pairs of spin and pseudospin energy levels.