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 \(\beta \), 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.

本文在狄拉克方程中研究了具有广义指数势作用的相对论粒子，在最小长度的背景下，找到了一个合适的变量代换，并使用Bethe ansatz方法给出了它的波函数和显式能谱。进一步，我们将看到，这项研究可以进一步扩展到各种特殊指数势，通过选择和调整势参数，可以再现各种特殊指数势的显能谱和波函数。通过观察广义指数势的精确解的表达式，可以发现对于有限的\(\beta\)，其能谱不仅与主量子数n有关，但也与n的平方有关。此外，我们还将看到，本文不仅将广义指数势狄拉克方程的研究扩展到了最小长度的背景，而且为求解指数型势狄拉克方程提供了一种更简单、替代和有效的方法。