Advances in High Energy Physics

Volume 2016, Article ID 4502312, 8 pages

http://dx.doi.org/10.1155/2016/4502312

## Quantum Tunneling in Deformed Quantum Mechanics with Minimal Length

^{1}School of Science, Southwest University of Science and Technology, Mianyang 621010, China^{2}Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu 610064, China

Received 14 September 2016; Revised 16 November 2016; Accepted 5 December 2016

Academic Editor: Elias C. Vagenas

Copyright © 2016 Xiaobo Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In the deformed quantum mechanics with a minimal length, one WKB connection formula through a turning point is derived. We then use it to calculate tunneling rates through potential barriers under the WKB approximation. Finally, the minimal length effects on two examples of quantum tunneling in nuclear and atomic physics are discussed.

#### 1. Introduction

Various theories of quantum gravity, such as string theory, loop quantum gravity, and quantum geometry, predict the existence of a minimal length [1–3]. For a review of a minimal length in quantum gravity, see [4]. Some realizations of the minimal length from various scenarios have been proposed. Specifically, one of the most popular models is the Generalized Uncertainty Principle (GUP) [5, 6], derived from the modified fundamental commutation relation:where is the Planck mass, is the Planck length, and is a dimensionless parameter. With this modified commutation relation, one can easily findwhich leads to the minimal measurable length:The GUP has been extensively studied recently; see, for example, [7–14]. For a review of the GUP, see [15].

To study 1D quantum mechanics with the deformed commutators (1), one can exploit the following representation for and :where . It can easily show that such representation fulfills the relation (1) to . Furthermore, we can adopt the position representation:Therefore for a quantum system described bythe deformed stationary Schrodinger equation in the position representation iswhere and terms of order are neglected.

If (7) with can be solved exactly, one could use the perturbation method to solve (7) by treating the term with as a small correction. However for the general , one might need other methods to solve (7). In fact, the WKB approximation in deformed space has been considered [16]. In [16], the authors considered the deformed commutation relationwhere is some function. For , one could solve the differential equation:for , and denotes the inverse function of . It is interesting to note that there might be more than one inverse function for . However, one usually finds that there is only one inverse function which vanishes at . The remaining ones are called “runaway” solutions, which are not physical and should be discarded [17, 18]. Then, they used the WKB approximation to show that the solution to the deformed Schrodinger equation:waswhere . Moreover, it also showed that condition,had to be satisfied to make the WKB approximation valid. However, condition (12) fails near a turning point, where .

For the case with , we derived one WKB connection formula through turning points and Bohr-Sommerfeld quantization rule in [19]. In this paper, we continue to consider other WKB connection formulas and calculate tunneling rates through potential barriers. The remainder of our paper is organized as follows. In Section 2, we derive one WKB connection formula and use it to find the formula for the tunneling rate through a potential barrier. Then two examples of quantum tunneling in nuclear and atomic physics are discussed in Section 3. Section 4 is devoted to our conclusions.

#### 2. Tunneling through Potential Barriers

We now consider WKB description of tunneling through a potential barrier , which vanishes as and rises monotonically to its maximum at as approaches from either the left or the right side of . In Figure 1, we plot the potential . For a particle of energy , there are two turning points and , at which . There are two classical allowed regions, Region I with and Region III with . To describe tunneling, we need to choose appropriate boundary conditions in the classical allowed regions. We postulate an incident right-moving wave in Region I, where the WKB approximation solution to (10) includes a wave incident, the barrier and a reflected wave:In Region III, there is only a transmitted wave:In the classically forbidden Region II, there are exponentially growing and decaying solutions:whereTo calculate the tunneling rate, we need to use connection formulas to relate , and to . In [19], we derived one WKB connection formula around in the case with . If , we found thatwhich givesIn what follows, we will derive a WKB connection formula around to relate and to and then calculate the tunneling rate through the potential barrier.