O método numérico de Numerov aplicado à equação de Schrödinger

Abstract

ResumoNeste artigo mostra-se como resolver numericamente problemas de autovalor associados a equações diferenciais ordinárias lineares de segunda ordem, contendo também termos que dependem da derivada primeira da variável incógnita. Nesse sentido, faz-se uma apresentação didática do método de Numerov e, em seguida, ele é aplicado a dois problemas clássicos da mecânica quântica não relativística cujas soluções analíticas são bem conhecidas: o oscilador harmônico simples e o átomo de hidrogênio. Os resultados numéricos são confrontados com os obtidos analiticamente. Palavras-chave: método de Numerov, oscilador harmônico, átomo de hidrogênio. AbstractIn this paper it is shown how to solve numerically eigenvalue problems associated to second order linear ordinary differential equations, containing also terms which depend on the variable. A didactic presentation of the Numerov Method is given and, in the sequel, it is applied to two quantum non-relativistic problems with well known analytical solutions: the simple harmonic oscillator and the hydrogen atom. The numerical results are compared to those obtained analytically.

How the inter-electronic potential Ansätze affect the bound state solutions of a planar two-electron quantum dot model

“…This take us to solve numerically the radial Schrödinger equation, Eq. (3), by using the Numerov method [35,36,37,38,39,40]. The algorithm was implemented in a program developed by the authors using C ++ language.…”

Solution of the 1d Schrödinger Equation for a Symmetric Well

“…The value of 0.500 2 k 2 /m is exactly the ground state value obtained analytically in section 3. This method produces good results in solving 1d-Schrödinger equation like pointed out by F. Caruso and V. Oguri [8]. The energy results are summarized in the last column and will serve to compare with other approximated methods.…”

“…To this end we can approximate the potential to O(x 2 ) with the aid of Eq. 's (7) and (8), as plotted in figure 2. The boundary condition for the potential well in Eq.…”

“…We have already encountered from equations (7) and (8) that the expansion of potential contains terms like x 2 , x 4 , and so on. Thus, it is useful to make an approach with the Stationary Perturbation Theory to study the effect of the perturbed potential with x 4 and x 6 on the energy levels into the one-dimensional harmonic oscillator Hamiltonian [2,[4][5][6].…”

“…Its algorithm is very efficient and converges very fast with at least O(h 6 ) of precision. Under this perspective, it is a good exercise to solve the 1d-Schrödinger equation using this technique to gain good insights about the Schrödinger equation during the classes [8].…”