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

Abstract

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.

Appendix. Brief Review Functional Bethe Ansatz Method

Now, we briefly review the functional Bethe ansatz method. For details, please refer to literature [55,56,57,58,59,60,61,62]. For second-order ordinary differential equations of the form

$$\begin{aligned} \left[ {P\left( y \right) \frac{{{d^2}}}{{d{y^2}}} + Q\left( y \right) \frac{d}{dy} + W\left( y \right) } \right] S\left( y \right) = 0, \end{aligned}$$

(A.1)

in which P(y), Q(y) and W(y) are polynomials with the highest degrees of 4, 5, and 4 respectively, and they could be given as follows

$$\begin{aligned} P\left( y \right) = \sum \limits _{i = 0}^4 {{p_i}} {y^i},\;\;\;\;\;Q\left( y \right) = \sum \limits _{i = 0}^5 {{q_i}} {y^i},\;\;\;\;\;W\left( y \right) = \sum \limits _{i = 0}^4 {{w_i}} {y^i}, \end{aligned}$$

(A.2)

in which ${p_i}, {q_i}$ and ${w_i}$ are all constants. It is not difficult to find that the ordinary differential equation whose form satisfies (A.1) is quasi-exactly solvable, and whose corresponding exact solution can be given by n order polynomials of the variable y. Applying the functional Bethe ansatz method further, the polynomial solution of the ordinary differential equation with the form (A.1) can be written as

$$\begin{aligned} S\left( y \right) = \prod \limits _{i = 1}^n {\left( {y - {y_i}} \right) } ,\;\;\;\;\;S\left( y \right) \equiv 1,\;\;\;\;for\;\;\;\;n = 0, \end{aligned}$$

(A.3)

where the different roots $y_1, y_2, \ldots , y_n$ could be determined. So, according to the ordinary differential equation (A.1), we have

$$\begin{aligned} \begin{aligned} - {w_0}&= \left( {{p_4}{y^4} + {p_3}{y^3} + {p_2}{y^2} + {p_1}y + {p_0}} \right) \sum \limits _{i = 1}^n {\frac{1}{{y - {y_i}}}} \sum \limits _{i \ne j}^n {\frac{2}{{{y_i} - {y_j}}}} \\&\quad + \left( {{q_5}{y^5} + {q_4}{y^4} + {q_3}{y^3} + {q_2}{y^2} + {q_1}y + {q_0}} \right) \sum \limits _{i = 1}^n {\frac{1}{{y - {y_i}}}} + {w_4}{y^4} + {w_3}{y^3} + {w_2}{y^2} + {w_1}y. \end{aligned} \end{aligned}$$

(A.4)

It is not difficult to see that the left side of Eq. (A.4) is a constant, while its right is a meromorphic function which has simple poles $y={y_i}$ and singularity at $y=\infty$. Therefore, the precondition that the equation is always valid requires that its right side must also be constant, which is a sufficient and necessary condition that can be obtained from Liuville’s theorem. Thus, the constraints have

$$\begin{aligned} \begin{aligned} {w_4}&= - n{q_5},\;\;\;\;\; {w_3} = - {q_5}\sum \limits _{i = 1}^n {{y_i}} - n{q_4},\;\;\;\;\; {w_2} = - {q_5}\sum \limits _{i = 1}^n {y_i^2} - {q_4}\sum \limits _{i = 1}^n {{y_i}} - n\left( {n - 1} \right) {p_4} - n{q_3},\\ {w_1}&= - {q_5}\sum \limits _{i = 1}^n {y_i^3} - {q_4}\sum \limits _{i = 1}^n {y_i^2} - \left[ {2\left( {n - 1} \right) {p_4} + {q_3}} \right] \sum \limits _{i = 1}^n {{y_i}} - n\left( {n - 1} \right) {p_3} - n{q_2},\\ {w_0}&= - {q_5}\sum \limits _{i = 1}^n {y_i^4} - {q_4}\sum \limits _{i = 1}^n {y_i^3} - \left[ {2\left( {n - 1} \right) {p_4} + {q_3}} \right] \sum \limits _{i = 1}^n {y_i^2 - 2{p_4}} \sum \limits _{i < j}^n {{y_i}{y_j}} \\&\quad - \left[ {2\left( {n - 1} \right) {p_3} + {q_2}} \right] \sum \limits _{i = 1}^n {{y_i}} - n\left( {n - 1} \right) {p_2} - n{q_1}, \end{aligned} \end{aligned}$$

(A.5)

in which the different roots $y_1, y_2, \ldots , y_n$ satisfy the Bethe ansatz equations

$$\begin{aligned} \sum \limits _{i \ne j}^n {\frac{2}{{{y_i} - {y_j}}} + \frac{{{q_5}y_i^5 + {q_4}y_i^4 + {q_3}y_i^3 + {q_2}y_i^2 + {q_1}{y_i} + {q_0}}}{{{p_4}y_i^4 + {p_3}y_i^3 + {p_2}y_i^2 + {p_1}{y_i} + {p_0}}}} = 0,\;\;\;\;\;\;i = 1,2,\; \ldots ,\;n. \end{aligned}$$

(A.6)