Resolvendo a equa\c{c}\~ao de Schr\"odinger atrav\'es da substitui\c{c}\~ao direta da s\'erie de pot\^encias

TL;DR

This paper introduces a straightforward, highly accurate algebraic method using power series substitution to solve the Schrödinger equation, enabling precise energy and wavefunction calculations with minimal computational resources.

Contribution

It presents a novel direct algebraic approach using recurrence relations from power series to solve the Schrödinger equation efficiently and accurately.

Findings

High-accuracy solutions for the Schrödinger equation achieved

Energy eigenvalues estimated simply and refined iteratively

Wavefunctions computed for high quantum number states

Abstract

This work presents a direct and highly accurate method to solve ordinary differential equations, in particular the Schr\"odinger equation in one dimension, through the direct substitution of a power series solution to obtain a purely algebraical system containing the recurrence relations among the series coefficients. With these recurrence relations at hand it is possible to build an extremely simple routine using only basic arithmetic operations to find solutions of very high accuracy at a very low cost of machine resources. For the Schr\"odinger equation the energy eigenvalues may be estimated by a very simple method, an this estimate may be easily refined using the recurrence relations until the specified tolerance has been reached, which allows one to find high accuracy wavefunctions even for states with very high quantum numbers. ----- Este trabalho apresenta um m\'etodo direto e de alta acur\'acia para resolver equa\c{c}\~oes diferenciais ordin\'arias, em particular a equa\c{c}\~ao de Schr\"odinger em uma dimens\~ao, atrav\'es da substitui\c{c}\~ao direta de uma solu\c{c}\~ao em s\'erie de pot\^encias para obter um sistema puramente alg\'ebrico contendo as rela\c{c}\~oes de recorr\^encia entre os coeficientes da s\'erie. De posse dessas rela\c{c}\~oes de recorr\^encia \'e poss\'ivel construir uma rotina extremamente simples, usando somente opera\c{c}\~oes aritm\'eticas b\'asicas, para encontrar solu\c{c}\~oes de alt\'issima acur\'acia a um custo muito reduzido de recursos de m\'aquina. Para a equa\c{c}\~ao de Schr\"odinger, os autovalores de energia podem ser estimados por um m\'etodo simples, e a estimativa pode ser facilmente refinada com o uso das rela\c{c}\~oes de recorr\^encia at\'e que se alcance a toler\^ancia especificada, o que permite encontrar fun\c{c}\~oes de onda de alta acur\'acia mesmo para estados com n\'umero qu\^antico alto.