Advances in High Energy Physics

Volume 2017 (2017), Article ID 9530874, 9 pages

https://doi.org/10.1155/2017/9530874

## -Deformed Relativistic Fermion Scattering

^{1}Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrud, Iran^{2}Department of Physics and Research Institute of Natural Science, College of Natural Science, Gyeongsang National University, Jinju 52828, Republic of Korea

Correspondence should be addressed to Hadi Sobhani

Received 28 November 2016; Revised 24 December 2016; Accepted 27 December 2016; Published 19 January 2017

Academic Editor: Chun-Sheng Jia

Copyright © 2017 Hadi Sobhani 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 this article, after introducing a kind of -deformation in quantum mechanics, first, -deformed form of Dirac equation in relativistic quantum mechanics is derived. Then, three important scattering problems in physics are studied. All results have satisfied what we had expected before. Furthermore, effects of all parameters in the problems on the reflection and transmission coefficients are calculated and shown graphically.

#### 1. Introduction

-Deformation for quantum group and physical system has been one of the remarkable and interesting issues of studies such as conformal quantum mechanics [1], nuclear and high energy physics [2–4], cosmic string and black holes [5], and fractional quantum Hall effect [6]. Applications of -deformation emerged in physics and chemistry after introducing -deformed harmonic oscillator [7, 8] such as investigation of electronic conductance in disordered metals and doped semiconductors [9], analyzing of the phonon spectrum in [10], and expressing of the oscillatory-rotational spectra of diatomic and multiatomic molecules [11, 12]. Basically, -calculus was established for the first time by Jackson [13] and then Arik and Coon used it. Arik and Coon studied generalized coherent states that are associated with generalization of the harmonic oscillator commutation relation [14]. They utilized where the relation between number operator and step operators is given by where a -number is defined as

Another -deformation exists that has been introduced by Tsallis [15] and has a different algebraic structure from Jackson’s. For Tsallis’s case, the -derivative and -integral were given by Borges [16].

In what follows, Section 2 is devoted to the introduction to the kind of -deformation of quantum mechanics which will be used in the next sections. In Section 3, -deformed version of Dirac equation is derived. As first relativistic scattering problem in -deformed version of relativistic quantum mechanics, scattering from a Dirac delta potential is done in Section 4. Section 5 is devoted to the extended form of problem in Section 4, scattering problem from a double Dirac delta potential. At last, Ramsauer-Townsend effect is studied in considered formalism of quantum mechanics.

#### 2. -Deformed Quantum Mechanics

In this section, we want to introduce postulates of -deformed quantum mechanics to use for the next sections. In this formalism of quantum mechanics, we deal with the following:(1)In this formalism of quantum mechanics like the ordinary one, time-dependent form of Schrödinger equation in -deformed quantum mechanics is written in form of in which we deal with the operators as where is a positive constant and the wave function is .(2)Inner product of Hilbert space in one-dimensional -deformed quantum mechanics can be written as (3)Expectation value of an operator regarding the wave function is given by and also we have Hermitian definition for the operator if we get It should be noted that the deformation is considered only for the coordinate part; then, the time part has no deformation. This point can be checked in the first postulate.

In this formalism of quantum mechanics, commutation relation between coordinate and its momentum should be deformed in form of Considering operator form of coordinate and momentum we can rewrite (4) in terms of the operators and to obtain time-independent form of Schrödinger equation in this formalism, we set ; then, we have

Using (11), we can easily find continuity relation in this formalism of quantum mechanics as where

By these considerations, we are in a position to study relativistic scattering of fermions in -deformed relativistic quantum mechanics.

#### 3. Scattering of Relativistic Fermions in -Deformed Quantum Mechanics

In this section, we want to study scattering of fermions in -deformed formalism of relativistic quantum mechanics. Study of fermions can be done by Dirac equation. This can be written as [17]: in which the matrices are where stands for Pauli matrices. We have considered direction as interaction direction. To obtain stationary states, we choose the wave function as also, we would like to consider for simplicity. These assumptions give us a system of equation likeFrom (19), we find that If one substitutes (20) into (18), one easily can derive

In the next sections, we will study three important and famous types of scattering.

#### 4. Scattering due to Single Dirac Delta Potential

As first scattering study, we want to consider single Dirac delta potential as where and are real constants. This point can be derived that this potential produces a discontinuity for the first derivative of wave function as We assume that particles come from ; then, because of Dirac delta existence, they scatter. Consequently, some of them are reflected to region I and the others are transmitted to region II . According to this assumption, we can find wave functions of the regions as The coefficients and can be determined by using boundary condition of continuity and discontinuity of wave functions at . These are whereby, solving them, we can find that

On the other hand, current density of fermions can be derived by . Because there is no sink or source, we have from current density that We plot this equation considering , , , and in Figure 1.