Symbolic Solution for Generalized Quantum Cylindrical Wells using Computer Algebra

TL;DR

This paper introduces a computer algebra method to analytically solve generalized quantum cylindrical wells with non-zero potential and singularities, providing exact and approximate energy levels and wave functions.

Contribution

It presents a novel symbolic approach using computer algebra to solve complex quantum well problems involving special functions.

Findings

Exact energy levels and wave functions derived for specific cases

Approximate solutions obtained where integrals are intractable

Demonstrates the method's applicability to complex quantum problems

Abstract

This paper present how to solve the problem of cylindrical quantum wells with potential energy different from zero and with singularity of the energy on the axis of the cylinder. The solution to the problem was obtained using methods of computer algebra. The results depend of Bessel and Kummer functions. This paper present energy levels and wave functions in some of the cases with an exactly form and in other cases with an approximated form, this form depended on the possibility of integrating the special functions and calculating the zeros of these functions. Here we can see the power of the method in the applications concerning complex problems of quantum mechanics, and the possibility of being able to apply this method in order to solve other problems in science and also in engineering.