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Solution of Schrödinger Equation Not Implementing Conventional Separation of Variables: Using the Trial and Error Brute Force Permutation Method

Abstract:

The Schrödinger equation is practically the base of quantum mechanics and the most use technique to solved this differential equation has been the separation of variable technique. From separation of variable process is possible obtain the time independent Schrödinger equation [TISE]; a useful equation for independent time potential. However, there are another situations in quantum mechanics that involves other mathematical ways to solve this equation. The methods implemented in this document are given by different potentials and avoiding the conventional separation of variables. The goal of this work is search wave functions that satisfy the Schrödinger equation with other unusual conditions. That makes easier reach more specific solution than in comparison are complicated to reach using the common separation of variables.