Integrable conditions for Dirac Equation and Schr\"odinger equation

TL;DR

This paper derives integrable conditions for Dirac and Schrödinger equations using commutative operator chains, linking symmetry to solvability, and discusses solution methods including separation of variables and approximations.

Contribution

It introduces a method to determine integrability conditions for these equations based on symmetry and commutative relations, highlighting cases solvable by separation of variables.

Findings

Few cases are fully solvable by separation of variables.

Most cases require perturbation or approximation methods.

The approach links symmetry to integrability of quantum equations.

Abstract

By constructing the commutative operators chain, we derive the integrable conditions for solving the eigenfunctions of Dirac equation and Schr\"odinger equation. These commutative relations correspond to the intrinsic symmetry of the physical system, which are equivalent to the original partial differential equation can be solved by separation of variables. Detailed calculation shows that, only a few cases can be completely solved by separation of variables. In general cases, we have to solve the Dirac equation and Schr\"odinger equation by effective perturbation or approximation methods, especially in the cases including nonlinear potential or self interactive potentials.