Research Article Scattering Study of Fermions Due to Double Dirac Delta Potential in Quaternionic Relativistic Quantum Mechanics

Hindawi Advances in High Energy Physics Volume 2018, Article ID 8124073, 7 pages https://doi.org/10.1155/2018/8124073

Research Article Scattering Study of Fermions due to Double Dirac Delta Potential in Quaternionic Relativistic Quantum Mechanics

Hassan Hassanabadi,1 Hadi Sobhani ,1 and Won Sang Chung2

1 Faculty of Physics, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran 2Department 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; [email protected]

Received 21 September 2017; Revised 15 November 2017; Accepted 23 November 2017; Published 3 January 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 Hassan Hassanabadi 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 SCOAP3.

We have studied the scattering problem of relativistic fermions from a quaternionic double Dirac delta potential. We have used Dirac equation in the presence of the scalar and vector potentials in the quaternionic formalism of relativistic quantum mechanics to study the problem. The wave functions of different regions have been derived. Then, using the reflection coefficient, transmission coefficient, and the continuity equation, the scattering problem has been investigated in detail. It has been shown that we have faced some fluctuations in the reflection and transmission coefficients.

1. Introduction The idea for making use of quaternions in quantum mechanics has a long shining story. Physicists and mathe- Among a lot of mathematical options in physics, quaternions matician have tried to found quantum mechanics on the qua- are a famous option. Quaternion was initiated by Hamilton ternions. Finkelstein et al. discussed a new kind of quantum for the first time [1–3]. Quaternions in general can be mechanics by using quaternions [4] or some works which are represented by about using real and complexified quaternions as underlying mathematical structure by adopting a complex geometry 𝜙=𝜙 +𝜙 𝑒 +𝜙 𝑒 +𝜙 𝑒 , 0 1 1 2 2 3 3 (1) to have ability to present a compatible face of quantum mechanics [5, 6]; tunneling phenomena for the relativistic 𝜙 (𝑙 = 0,1,2,3) where 푙 are real coefficients. There are three fermions have been studied in the presence of quaternionic imaginary units in quaternions which have the property step potential [7, 8] and there are a lot of valuable researches in this topic which are mentioned in [9–17]. All of them 𝑒 𝑒 =−𝛿 +𝜀 𝑒 , (𝑎, 𝑏, 𝑐 =) 1,2,3 . 푎 푏 푎푏 푎푏푐 푐 (2) tried to show that we can have a quantum mechanics based onthequaternions.Intherestofthepaper,weintendto 𝑖 𝑗 𝑘 If we set , , as the imaginary units, we conclude from (2) present quaternionic form of Dirac equation. We want to that extend our recent papers [18–21] to relativistic form as well as 𝑖𝑗=−𝑗𝑖=𝑘, having a study of scattering by double Dirac delta potential. In our recent paper [20], we derived the Dirac equation in 𝑗𝑘=−𝑘𝑗=𝑖, (3) quaternionic quantum mechanics, in general. Then scattering of relativistic fermions due to a single quaternionic Dirac 𝑘𝑖 = −𝑖𝑘 =𝑗. delta function has been studied. It was seen that the reflection and transmission coefficients were treated smoothly and Equation (3) exhibits that in general multiplication of two without any fluctuation. But it will be shown that scattering quaternions has noncommutative nature; it means 𝑞𝑝 =𝑝𝑞̸ . due to the quaternionic double Dirac delta function results 2 Advances in High Energy Physics in some fluctuations in the reflection and transmission coef- calculations, we arrive at a coupled system of equations for ficients. We organized this article as presenting quaternionic the components Dirac equation in Section 2 and introducing the quaternionic 𝑑Φ− (𝑥) double Dirac delta potential and the discussion about it and Φ+ (𝑥) 𝑖𝐸 =𝜎 푥 𝑑𝑥 its effects in Section 3, and after bringing some information (10) + about probability current and conservation law in Section 4, + (𝑖𝑚 + 2푎 (𝑖𝑆 (𝑥) +𝑗𝑆푏 (𝑥))) Φ (𝑥) , the conclusions appeared at the end. 𝑑Φ+ (𝑥) Φ− (𝑥) 𝑖𝐸 =𝜎 −𝑖𝑚Φ− (𝑥) . (11) 2. Dirac Equation in Quaternionic Relativistic 푥 𝑑𝑥 Quantum Mechanics Quaternionic wave function components are in the form of ± ± ± ± ± Φ (𝑥) =푎 𝜙 (𝑥) + 푏𝑗𝜙 (𝑥),where𝜙푎 (𝑥) and 𝜙푏 (𝑥) are the As in the ordinary relativistic quantum mechanics, to inves- complex functions. Considering such a form of the wave tigate relativistic fermions, we should make use of Dirac function components in (11) yields equation. Quaternionic version of Dirac equation in the 𝜎 𝑑𝜙+ (𝑥) presence of vector and scalar potential can be written with 𝜙− (𝑥) = 푥 푎 , the help of [8] as 푎 𝑖 (𝐸+𝑚) 𝑑𝑥 (12) 𝜕Ψ (r,𝑡) + − 𝜎푥 𝑑𝜙푏 (𝑥) =−[𝛼 ⋅ ∇ + 𝛽 (𝑖 (𝑚푎 +𝑆 (r))+𝑗𝑆푏 (r)) 𝜙 𝑥 = . 𝜕𝑡 푏 ( ) (4) 𝑖 (𝐸−𝑚) 𝑑𝑥 +𝑖𝑉푎 (r) +𝑗𝑉푏 (r)]Ψ(r,𝑡) , Substitution of (12) into (10) provides a system of coupled differential equations where in (4) ℏ=𝑐=1and 𝑑2𝜙+ (𝑥) 푎 +(𝑝2 −2(𝐸+𝑚) 𝑆 (𝑥))𝜙+ (𝑥) 0 𝜎 2 푎 푎 𝛼 =( ) 𝑑𝑥 𝜎 0 −2𝑖(𝐸+𝑚) 𝑆∗ (𝑥) 𝜙+ (𝑥) =0, (5) 푏 푏 10 (13) 𝛽 =( ), 𝑑2𝜙+ (𝑥) 푏 +(−𝑝2 −2(𝐸−𝑚) 𝑆 (𝑥))𝜙+ (𝑥) 0 −1 𝑑𝑥2 푎 푏 𝜎 + where Pauli matrices are .Asisobviouslyclear,thepoten- +2𝑖(𝐸−𝑚) 𝑆푏 (𝑥) 𝜙푎 (𝑥) =0, tials have two parts. Real functions of them are shown by sub- 𝑝2 =𝐸2 −𝑚2 ∗ scribing 𝑎(𝑆푎(r), 𝑉푎(r)∈R) andforcomplexfunctions,they where and means the complex conjugation. have subscribed 𝑏(𝑆푏(r), 𝑉푏(r)∈C).Bysetting𝑆푏(r), 𝑉푏(r)→ Now, we are in a position to investigate scattering states due 0, we can get the well-known form of this equation as [22, 23] to quaternionic Dirac delta potential. 𝜕Ψ (r,𝑡) 3. Quaternionic Double Dirac 𝑖 = (𝛼 ⋅ P + 𝛽 (𝑚+𝑆푎 (r)) +𝑉푎 (r)) Ψ (r,𝑡) . (6) 𝜕𝑡 Delta and Scattering Since we are interested in the time-independent interactions, Here, we introduce a quaternionic form of double Dirac delta it is better to consider wave function as interaction [21] as −푖퐸푡 Ψ (r,𝑡) =Φ(r) 𝑒 . (7) 𝑆푎 (𝑥) =𝑉푎 (𝛿 (𝑥0 −𝑎 )+𝛿(𝑥+𝑎0)) (14) Inserting (7) into (4), the time-independent form of quater- 𝑆푏 (𝑥) =𝑖𝑉푏 (𝛿 (𝑥0 −𝑎 )+𝛿(𝑥+𝑎0)) . nionic Dirac equation derives as We assume that 𝑉푎 and 𝑉푏 are real constants. The well-

Φ (r) 𝑖𝐸 =(𝛼 ⋅ ∇ + 𝛽 (𝑖 (𝑚푎 +𝑆 (r))+𝑗𝑆푏 (r))+𝑖𝑉푎 (r) known property of Dirac delta interactions is producing (8) discontinuity condition for the derivative of wave function. +𝑗𝑉푏 (r))Φ(r) . These conditions can be derived by integrating (13) around 𝑥=𝑎0 and 𝑥=−𝑎0. So discontinuity condition at 𝑥=𝑎is Note the coordinate part of wave function is a quaternionic 󵄨 󵄨 𝑑𝜙+ 󵄨 𝑑𝜙+ 󵄨 function and has two components. Actually it is usually 푎 󵄨 푎 󵄨 󵄨 − 󵄨 𝑑𝑥 󵄨 + 𝑑𝑥 󵄨 − written like 󵄨푥=푎0 󵄨푥=푎0 + Φ (r) =2(𝐸+𝑚) (𝑉 𝜙+ (𝑎 )+𝑉𝜙+ (𝑎 )) , Φ (r) =( ), 푎 푎 0 푏 푏 0 − (9) Φ (r) 󵄨 󵄨 (15) 𝑑𝜙+ 󵄨 𝑑𝜙+ 󵄨 푏 󵄨 푏 󵄨 ± 󵄨 − 󵄨 where Φ (r) are quaternionic functions. This form of rep- 𝑑𝑥 󵄨 + 𝑑𝑥 󵄨 − 󵄨푥=푎0 󵄨푥=푎0 resentingiscalledspinorformofwavefunction.Ifweset + + 𝑖𝑆푎(r)+𝑗𝑆푏(r)=𝑖𝑉푎(r)+𝑗𝑉푏(r),bydoingsomealgebraic =2(𝐸−𝑚) (𝑉푎𝜙푎 (𝑎0)+𝑉푏𝜙푏 (𝑎0)) . Advances in High Energy Physics 3

Also for 𝑥=−𝑎,wehave where the coefficients are complex constants spinors in general. Therefore, according to our assumption about the parti- 󵄨 󵄨 𝑑𝜙+ 󵄨 𝑑𝜙+ 󵄨 cles, the physical wave functions can be written as 푎 󵄨 푎 󵄨 󵄨 − 󵄨 𝑑𝑥 󵄨푥=−푎+ 𝑑𝑥 󵄨푥=−푎− + 푖푝푥 −푖푝푥 푝푥 0 0 𝜙I (𝑥) =𝑒 +𝑟𝑒 +𝑗𝑒 , + + + 푖푝푥 −푖푝푥 푝푥 −푝푥 =2(𝐸+𝑚) (𝑉푎𝜙푎 (−𝑎0)+𝑉푏𝜙푏 (−𝑎0)) , 𝜙II (𝑥) =𝑐1𝑒 +𝑐2𝑒 +𝑗(𝑐3𝑒 +𝑐4𝑒 ), (19) + 󵄨 + 󵄨 (16) 𝑑𝜙 󵄨 𝑑𝜙 󵄨 + 푖푝푥 −푝푥 푏 󵄨 푏 󵄨 𝜙 (𝑥) =𝑡𝑒 +𝑗̃𝑡𝑒 . 󵄨 − 󵄨 III 𝑑𝑥 󵄨 + 𝑑𝑥 󵄨 − 󵄨푥=−푎0 󵄨푥=−푎0 + + It can be easily seen that if we remove the coefficient and =2(𝐸−𝑚) (𝑉푎𝜙푎 (−𝑎0)+𝑉푏𝜙푏 (−𝑎0)) . the imaginary unit 𝑗,weobtainthewavefunctionofthefree relativistic fermions in the ordinary quantum mechanics. In Bytheway,weshouldmakethreepartsinourproblem order to have explicit expression of the coefficients in (19), we should match the wave functions at 𝑥=𝑎0 and 𝑥=−𝑎0 which yields Region I 𝑥<−𝑎0, 푖푎 푝 −푖푎 푝 푖푎 푝 {𝑐 𝑒 0 +𝑐𝑒 0 =𝑡𝑒 0 , Region II −𝑎0 <𝑥<𝑎0, (17) 1 2 𝑥=𝑎0 { 푎0푝 −푎0푝 ̃ −푎0푝 𝑐3𝑒 +𝑐4𝑒 = 𝑡𝑒 , Region III 𝑥>𝑎0. { (20) 푖푎0푝 −푖푎0푝 −푖푎0푝 푖푎0푝 {𝑟𝑒 +𝑒 =𝑐1𝑒 +𝑐2𝑒 , For the free particles, we have 𝑥=−𝑎 0 { 푖푎 푝 −푖푎 푝 −푖푎 푝 푖푎 푝 𝑟𝑒 0 +𝑒 0 =𝑐𝑒 0 +𝑐𝑒 0 , { 1 2 + 푖푝푥 −푖푝푥 𝜙푎 (𝑥) =𝑐1𝑒 +𝑐2𝑒 , + 푝푥 −푝푥 (18) and four equations also are derived by applying the continuity 𝜙푏 (𝑥) =𝑐3𝑒 +𝑐4𝑒 , conditions

푖푎0푝 −푖푎0푝 푖푎0푝 ̃ −푎0푝 푖푎0푝 {−𝑖𝑐1𝑝𝑒 +𝑖𝑐2𝑝𝑒 + 𝑖𝑝𝑡𝑒 =2(𝐸+𝑚) (𝑉푏𝑡𝑒 +𝑡𝑉푎𝑒 ) 𝑥=𝑎 0 { −푎 푝 푎 푝 −푎 푝 −푎 푝 푖푎 푝 𝑝̃𝑡(−𝑒 0 )−𝑐𝑝𝑒 0 +𝑐𝑝𝑒 0 =2(𝐸−𝑚) (𝑉 ̃𝑡𝑒 0 +𝑡𝑉𝑒 0 ) { 3 4 푏 푎 (21) −푖푎0푝 푖푎0푝 푖푎0푝 −푖푎0푝 −푎0푝 푖푎0푝 −푖푎0푝 {𝑖𝑐1𝑝𝑒 −𝑖𝑐2𝑝𝑒 +𝑖𝑝𝑟𝑒 −𝑖𝑝𝑒 =2(𝐸+𝑚) (𝑉푏𝑟𝑒̃ +𝑉푎 (𝑟𝑒 +𝑒 )) 𝑥=−𝑎 0 { −푎 푝 −푎 푝 푎 푝 −푎 푝 푖푎 푝 −푖푎 푝 𝑝𝑟(−𝑒̃ 0 )+𝑐𝑝𝑒 0 −𝑐𝑝𝑒 0 =2(𝐸−𝑚) (𝑉 𝑟𝑒̃ 0 +𝑉 (𝑟𝑒 0 +𝑒 0 )) . { 3 4 푏 푎

So we have eight equations and undetermined coefficients. By each region. This form of the wave functions can be obtained solving these equations, explicit form of each coefficient can using(12)as be determined. 𝑒푖푝푥 +𝑟𝑒−푖푝푥 +𝑗𝑟𝑒̃ 푝푥 Φ (𝑥) =( 𝑒푖푝푥 −𝑟𝑒−푖푝푥 𝑟𝑒̃ 푝푥 ), 4. Probability Current and Conservation Law I 𝜎 𝑝( +𝑗 ) 푥 𝐸+𝑚 𝑖 (𝐸−𝑚) In the quaternionic quantum mechanics, similar to the (24) complex version, we have the continuity equation as [20] 𝑡𝑒푖푝푥 +𝑗̃𝑡𝑒−푝푥 푖푝푥 −푝푥 𝜕𝜌 ΦII (𝑥) =( 𝑡𝑒 ̃𝑡𝑒 ). + ∇ ⋅ J =0, (22) 𝜎푥𝑝( −𝑗 ) 𝜕𝑡 𝐸+𝑚 𝑖 (𝐸−𝑚) where As is indicated in the Appendix, it is straightforward to prove † that, by using the definition 𝐽푥 =Ψ𝛼푥Ψ, we derive the 𝜌=ΨΨ, constraint (23) † 2 2 J =Ψ𝛼Ψ. |𝑟| + |𝑡| =1. (25)

To check the conservation law of probability, we need to So form of conservation law of probability is (25). This equa- calculate the currents of each region. To derive the currents of tion has been plotted in Figure 1 considering 𝑚=𝑎0 =𝑉푎 = each region, we need the spinor form of the wave function of 𝑉푏 =1and 𝐸∈[1,4]. 4 Advances in High Energy Physics

thedoubleDiracdeltafunctions.Itcanbeseenthatby 1 increasing the distance, the number of fluctuations in the reflection and transmissions increases. 0.8 5. Conclusions 0.6 In this paper, we presented quaternionic version of Dirac equation in the presence of vector and scalar potentials. To 0.4 ensure the correctness of this type of equation, we checked it inspecialcasewherewedonothavequaternionicpotential, 0.2 and the result was what we expected. After introducing the scattering potential, we investigated effects of it. It causes the discontinuity conditions for the derivative of the wave 0 functions. The probability current for each region of our 1 1.5 2 2.5 3 3.5 4 considered problem was determined. At the end conservation law of probability was derived. How different parameters E of the scattering potential can affect the reflection and |r|2 transmission coefficients was shown; for instance, increasing |t|2 the first part of the potential causes to have sharper reflection |r|2 + |t|2 and transmission coefficients but the second part of the Figure 1: Treatments of the coefficients in terms of energy. For this potential was treated vice versa. It means that by decreasing plot, we have set 𝑚=𝑎0 =𝑉푎 =𝑉푏 =1and energy ∈[1,4]. thesecondpartofthepotentialwehavethesharpreflection and transmission coefficients. For the last case, effect of the distance between the double Dirac delta functions was AsisshowninFigure1,(25)isvalid.Itisconstructiveifwe shown. It was seen that the number of fluctuations of check effects of the potential coefficients and the distance of the reflection and transmission increases by enlarging the the Dirac delta functions on the reflection and transmission distance of the double Dirac delta functions. coefficients. In Figures 2 and 3, the effects of potential coefficients on the reflection and transmission coefficients Appendix were shown. It is seen that in Figure 1 by increasing 𝑉푎,the reflection and transmission coefficient appear sharper but In this section, details of derivation of (25) are mentioned. in Figure 2 we face a different case. When 𝑉푏 grows up, the In order to brief the calculation, we will indicate them in a reflection and transmission coefficients arise smoother than compact form. thoseinFigure2. At the first step, we are going to derive the current of In Figure 4, by considering fix values for the potential probability of region I. Using the definition of the probability coefficients and the energy, we change the distance between current, we have [20]

† 𝐽I =Ψ𝛼푥Ψ 𝑒푖푝푥 +𝑟𝑒−푖푝푥 +𝑗𝑟𝑒̃ 푝푥 −푖푝푥 ∗ 푖푝푥 ∗ 푝푥 0𝜎 −푖푝푥 ∗ 푖푝푥 푝푥 ∗ 𝑒 −𝑟 𝑒 𝑟̃ 𝑒 푥 (A.1) =(𝑒 +𝑟 𝑒 −𝑒 𝑟̃ 𝑗, 𝜎 𝑝( + 𝑗))( )( 𝑒푖푝푥 −𝑟𝑒−푖푝푥 𝑟𝑒̃ 푝푥 ). 푥 𝐸+𝑚 𝑖 (𝐸−𝑚) 𝜎 0 𝜎 𝑝( +𝑗 ) 푥 푥 𝐸+𝑚 𝑖 (𝐸−𝑚)

This point should be noted as the spinor of wave function is a in daggered form of the spinor, we face a reversed order quaternion and since the coefficients are complex constants, in (A.1). Proceeding more in the matrix multiplication, we their order and the imaginary unit 𝑗 are important. Hence have

퐵 퐵 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞2 𝑒푖푝푥 −𝑟𝑒−푖푝푥 𝑟𝑒̃ 푝푥 퐴 퐴 𝑝( + 𝑗 ) ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1 2 −푖푝푥 ∗ 푖푝푥 ∗ 푝푥 𝐸+𝑚 𝑖 𝐸−𝑚 −푖푝푥 ∗ 푖푝푥 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞푝푥 ∗ 𝑒 −𝑟 𝑒 𝑟̃ 𝑒 ( ( ) ) 𝐽I =(𝑒 +𝑟 𝑒 − 𝑒 𝑟̃ 𝑗,𝜎푥𝑝 ( + 𝑗)) ( ) , (A.2) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝐸+𝑚 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑖 (𝐸−𝑚) 퐴 3 퐴4 푖푝푥 −푖푝푥 푝푥 𝜎푥 (𝑒⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟+𝑟𝑒 +𝑗⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑟𝑒̃ ) 퐵 ( 3 퐵4 )

𝐽I =𝐴1𝐵1 +𝐴1𝐵2 +𝐴2𝐵1 +𝐴2𝐵2 +𝐴3𝐵3 +𝐴3𝐵4 +𝐴4𝐵3 +𝐴4𝐵4. (A.3) Advances in High Energy Physics 5

1 1

0.8 0.8

0.6 0.6 2 2 | | t r | | 0.4 0.4

0.2 0.2

0 0

1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4 E E

Va =1 Va =1 Va =2 Va =2 Va =3 Va =3

Figure 2: Effects of different values of 𝑉푎 on the coefficients.

1 1

0.8 0.8

0.6 0.6 2 2 | | t r | | 0.4 0.4

0.2 0.2

0 0

1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4 E E

Vb =1 Vb =1 Vb =2 Vb =2 Vb =3 Vb =3

Figure 3: Effects of different values of 𝑉푏 on the coefficients.

𝑝 To avoid complicity of multiplication of (A.2), we have 𝐴 𝐵 = (1 + 𝑟𝑒−2푖푝푥 −𝑟∗𝑒2푖푝푥 − |𝑟|2), 3 3 𝐸+𝑚 𝑝 −2푖푝푥 ∗ 2푖푝푥 2 𝐴 𝐵 = (1 − 𝑟𝑒 +𝑟 𝑒 − |𝑟| ), 𝑝 1 1 𝐸+𝑚 𝐴 𝐵 =𝑗 (𝑟𝑒̃ 푝푥(1+푖) −𝑟𝑟𝑒̃ 푝푥(1−푖)), 3 4 (𝐸+𝑚) 𝑝 ̃ 푝푥(1+푖) ̃ 푝푥(1−푖) 𝐴1𝐵2 =𝑗 (𝑟𝑒 +𝑟𝑟𝑒 ), 𝑝 𝑖 (𝐸−𝑚) 𝐴 𝐵 =−𝑗 (𝑟𝑒̃ 푝푥(1+푖) +𝑟𝑟𝑒̃ 푝푥(1−푖)), 4 3 𝑖 (𝐸−𝑚) 𝑝 푝푥(1+푖) 푝푥(1−푖) 𝐴2𝐵1 =𝑗 (−𝑟𝑒̃ +𝑟𝑟𝑒̃ ), (𝐸+𝑚) 𝑝 2 2푝푥 𝐴4𝐵4 =− (|𝑟̃| 𝑒 ). 𝑝 𝑖 (𝐸−𝑚) 𝐴 𝐵 = (|𝑟̃|2 𝑒2푝푥), 2 2 𝑖 (𝐸−𝑚) (A.4) 6 Advances in High Energy Physics

1 1

0.8 0.8

0.6 0.6 2 2 | | t r | | 0.4 0.4

0.2 0.2

0 0

1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4 E E

a0 = 0.5 a0 = 0.5 a0 =1 a0 =1 a0 = 1.5 a0 = 1.5

Figure 4: Effects of different values of 𝑎0 on the coefficients.

Withthehelpof(A.3)and(A.4),wecanfindtheprobability [6] L. P. Horwitz and L. C. Biedenharn, “Quaternion quantum current of region I as mechanics: second quantization and gauge fields,” Annals of Physics,vol.157,no.2,pp.432–488,1984. 2𝑝 2 [7] S. De Leo and S. Giardino, “Dirac solutions for quaternionic po- 𝐽I = (1 − |𝑟| ). (A.5) 𝐸+𝑚 tentials,” Journal of Mathematical Physics,vol.55,no.2,022301, 10 pages, 2014. In the same manner, we can find the probability current of [8] S. De Leo, G. Ducati, and S. Giardino, “Quaternionic dirac scat- region II as tering,” Journal of Physical Mathematics,vol.6,article1000130, 2𝑝 2015. 𝐽 = |𝑡|2 . II 𝐸+𝑚 (A.6) [9] S. De Leo and G. Ducati, “Quaternionic differential operators,” Journal of Mathematical Physics,vol.42,no.5,pp.2236–2265, Since there is no sink or source for the particles, we have 2001. [10] A. J. Davies and B. H. J. McKellar, “Nonrelativistic quaternionic 𝐽I =𝐽II 󳨐⇒ quantum mechanics in one dimension,” Physical Review A: Atomic, Molecular and Optical Physics,vol.40,no.8,pp.4209– 2 2 (A.7) |𝑟| + |𝑡| =1. 4214, 1989. [11] A. J. Davies and B. H. J. McKellar, “Observability of quaternionic Conflicts of Interest quantum mechanics,” Physical Review A: Atomic, Molecular and Optical Physics,vol.46,no.7,pp.3671–3675,1992. The authors declare that they have no conflicts of interest. [12] S. De Leo, G. C. Ducati, and C. C. Nishi, “Quateraionic potentials in non-relativistic quantum mechanics,” Journal of Physics A: Mathematical and General,vol.35,no.26,pp.5411–5426, References 2002. [1] W. R. Hamilton, Elements of Quaternions,NewYork,NY,USA, [13] A. Peres, “Proposed test for complex versus quaternion quan- 1969. tum theory,” Physical Review Letters,vol.42,no.11,pp.683–686, [2] W. R. Hamilton, The mathematical papers of Sir William Rowan 1979. Hamilton, Cambridge University Press, Cambridge, UK, 1967. [14] H. Kaiser, E. A. George, and S. A. Werner, “Neutron interfero- [3]B.A.Rosenfeld,A history of Non-Euclidean Geometry,vol.12, metric search for quaternions in quantum mechanics,” Physical Springer, New York, NY, USA, 1988. Review A: Atomic, Molecular and Optical Physics,vol.29,no.4, [4] D. Finkelstein, J. M. Jauch, and D. Speiser, “Quaternionic repre- pp. 2276–2279, 1984. sentations of compact groups,” Journal of Mathematical Physics, [15] K. Shoemake, “Animating rotation with quaternion curves,” vol.4,pp.136–140,1963. ACM SIGGRAPH Computer Graphics,vol.19,no.3,pp.245– [5] J. Rembielinski, “Tensor product of the octonionic Hilbert spa- 254, 1985. ces and colour confinement,” Journal of Physics A: Mathematical [16] S. L. Altmann, Rotations, Quaternions, and Double Groups, and General, vol. 11, p. 2323, 1978. Oxford, UK, Claredon, 1986. Advances in High Energy Physics 7

[17] M. Gogberashvili, “Split quaternions and particles in (2+1)- space,” The European Physical Journal C,vol.74,article3200, 2014. [18] H. Sobhani and H. Hassanabadi, “Scattering in quantum mechanics under quaternionic Dirac delta potential,” Canadian Journal of Physics, vol. 94, no. 3, pp. 262–266, 2016. [19] H. Sobhani, H. Hassanabadi, and W. S. Chung, “Observations of the Ramsauer–Townsend effect in quaternionic quantum mechanics,” The European Physical Journal C,vol.77,article425, 2017. [20] H. Hassanabadi, H. Sobhani, and A. Banerjee, “Relativistic scattering of fermions in quaternionic quantum mechanics,” The European Physical Journal C,vol.77,article581,2017. [21] H. Sobhani and H. Hassanabadi, “Investigation of spin-zero bosons in q-deformed relativistic quantum mechanics,” Indian Journal of Physics,vol.91,no.1205,2017. [22] H. Sobhani and H. Hassanabadi, “Two-dimensional linear dependencies on the coordinate time-dependent interaction in relativistic non-commutative phase space,” Communications in Theoretical Physics,vol.64,no.3,pp.263–268,2015. [23] H. Sobhani and H. Hassanabadi, “Rashba effect in presence of time-dependent interaction,” Communications in Theoretical Physics, vol. 65, no. 5, pp. 543–545, 2016. Journal of Journal of The Scientific Journal of Advances in Gravity Photonics World Journal Soft Matter Condensed Matter Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Aerodynamics

Journal of Fluids Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Submit your manuscripts at https://www.hindawi.com

International Journal of International Journal of Statistical Mechanics Optics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Thermodynamics

Journal of Computational Advances in Physics Advances in Methods in Physics High Energy Physics Research International Astronomy Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

Journal of Journal of Journal of International Journal of Journal of Atomic and Solid State Physics Astrophysics Superconductivity Biophysics Molecular Physics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014