Discussion on Exact Solution of Dirac Equation with Generalized Exponential Potential in the Presence of Generalized Uncertainty Principle

In this work, the relativistic particle with the action of the generalized exponential potential is studied in the Dirac equation in the context of minimum length, subsequently finding a suitable variable substitution and giving its wave function and explicit energy spectrum by using the Bethe ansatz method. Further, we will see that this research could be further extended to various special exponential potential, and the explicit energy spectrum and wave functions of various special exponential potential could be reproduced by selecting and adjusting the potential parameters. By observing the expressions of the exact solution of the generalized exponential potential, it can be found that for finite β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}, its energy spectrum is not only related to the principal quantum number n, but also related to the square of n. Moreover, we will also see that this paper not only extends the study of Dirac equation with generalized exponential potential effect to the background of minimal length, but also provides an easier, alternative and valid method for solving the Dirac equation with exponential-type potential.