International Conference on Mathematics, Science, and Education 2015 (ICMSE 2015)

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1 International Conference on Mathematics, Science, and Education 215 ICMSE 215 Solution of the Dirac equation for pseudospin symmetry with Eckart potential and trigonometric Manning Rosen potential using Asymptotic Iteration Method Resita Arum Sari, A Suparmi, and C Cari Physics Department, Post Graduate Study, Sebelas Maret University, Indonesia resitaarumsari@yahoocoid ABSTRACT Solution of the Dirac equation for Eckart potential and trigonometric Manning Rosen potential with pseudospin symmetry was obtained using asymptotic iteration method The combination of the two potential was substituted into the Dirac equation, then the separation of variables into radial and angular parts Solution of the Dirac equation using an asymptotic iteration method was done by reducing the second order differential equation into a differential equation with substitution variables of hypergeometry type The relativistic energy calculated using Matlab 211 This study is limited to the case of pseudospin symmetry With the asymptotic iteration method, the energy spectrum of the relativistic equations and equations of orbital quantum number l can be obtained, where both are interrelated between quantum numbers Energy spectrum also numerically solved using Matlab software, where the increase in the radial quantum number nr causes decrease in the energy spectrum The wave function of radial part and the angular are defined as hypergeometry functions and visualized with Matlab 211 The results show that the disturbance of combination Eckart potential and trigonometric Manning Rosen potential change in the wave function of the radial part and the angular part INTRODUCTION In particular, the Dirac equation, which describes the motion of a spin-12 particle, has been used in solving many problems of nuclear and high-energy physics Recently, there has been an increased in searching for analytic solution of the Dirac equation [1 7] One of the important tasks of relativistic quantum mechanics is to find an accurate and exact solution of the Dirac equation for a certain potential The spin symmetry appears when the magnitude of the scalar and vector potentials are nearly equal, in the nuclei ie, when the difference potential However, the pseudo-spin symmetry occurs when ie, when the sum potential [8] The bound states of nucleons seem to be sensitive to some mixtures of these potentials Dirac equations for the case of exact spin symmetric occur when the difference between the magnitude of the repulsive vector potential with the attractive scalar potential is zero and the sum of the magnitude of the repulsive vector and attractive scalar potentials is equal to the given potential The exact pseudo-spin symmetry occurs when the sum of the magnitude of the repulsive vector potential and the attractive scalar potential is zero and the difference between the vector with scalar potential is equal to the given potential, which is central or non-central [9] Solution of Dirac equation for some typical potential under special cases of spin symmetry and pseudospin P - 5 symmetry have been investigated For spin symmetry that gives for and for with is the eigenvalues of the spin orbit coupling operator, is quantum number orbital and is the total of angular momentum quantum number For pseudospin symmetry care, that gives for and for with is quantum number pseudospin orbital In nuclear physics, spin symmetry and pseudospin symmetry concepts have been used to study the aspect of deformed and super deformation nuclei The concept of spin symmetry has been applied to the spectrum of meson and antinucleon [1] Some researchers have completed the Dirac equation for a variety of potential and different methods, such as the approach to include the Hulthen potential and tensor type Coulomb potential with spin-orbit number k in a state of spin symmetry and p-spin symmetry [11], bound states of the Dirac equation with position-dependent mass for the Eckart potensial [12], the potential Deng-Fan and the Coulomb potential tensor using Asymptotic Iteration Method AIM [13], the Schrodinger equation with Hulthen plus Manning-Rosen potential [14], Scarf potensial with new tensor coupling potential for spin and pseudospin symmetries using Romanovski Polynomials [15], for q- deformed hyperbolic Poschl-Teller potential and trigonometric Scarf II non-central potensial by using

2 International Conference on Mathematics, Science, and Education 215 ICMSE 215 AIM [16], Eigensolutions of the Deformed Woods Saxon Potential via AIM [17], for Eckart potential and trigonometric Manning Rosen potential using Asymptotic Iteration Method [18], for spin symmetry with modified Poschl-Teller potential and trigonometric Scarf II noncentral potential using Asymptotic Iteration Method [19] This paper is organized as follows The asymptotic iteration method will be briefly reviewed in Section 2 In Section 3, we review the Eckart potential and trigonometric Manning Rosen potential, give a brief introduction to the Dirac equation with equal scalar and vector potentials, and apply the separation of variables in spherical coordinates In Section 4, we solve the radial and angular parts of the Dirac equation with modified Eckart potential combined with trigonometric Manning Rosen potential and obtain the relativistic energy spectrum and wavefunction via asymptotic iteration method In Section 5, we present graphically some wavefunctions of the Dirac equation, present several relativistic energy spectra, and discuss some consequences of the results obtained Finally, the conclusions obtained are given in Section 6 METHODS Asymptotic Iteration This method is used to solve diffrensial equation in terms:, 1 The solution of Eq1 can be obtained by using iteration of and, 2 Eigen values can be obtained using equation [2] : 3 While eigen functions of Eq1 can be obtained using [21] : 4 which can be solved by apply efect asymptotic of iteration k 5 Rewritten second-order differential of Eq1 take the following general form : 6 the following general formula for the exact solution of is given by : 7 where,,, is normalized constant and function [22] 9 is hipergeometry P - 6 The Dirac equation with equal scalar and vector Eckart potential combined with trigonometric Manning Rosen potential The Dirac equation with scalar potential and vector potential, { } * + 1 which is relativistic energy and is momentum operator, while and is matrix in term:, 11 with are the three-dimensional Pauli matrices and I is the 2 2 identity matrix The potential in Eq1 is spherically symmetric potential, ie it does not only depend on the radial coordinate, and we have taken, The Dirac spin may be written according to the upper and lower components as follows: : ; 12 Where is the Dirac spin components and is the Dirac pseudospin components and are the spin and pseudospin spherical harmonics, l is quantum numbers orbital and is quantum number pseudo orbital, m is the projection of the angular momentum on thez axis we have,, - 13, - 14 For pseudospin symmetry, equation 14 becomes, - 15, - Substituting Eq15 into Eq13 yields, -, - 16 In spherical coordinates, Eckart potential combined with trigonometric Manning Rosen potential is defined as 17 Putting Eq17 into Eq16 and simplifying the equation, and let we have,

3 International Conference on Mathematics, Science, and Education 215 ICMSE 215 Separating the variables in Eq19, we obtain Equation 22 is well known with its solution, 23 By substituting Eq12 to Eq1, we can differential equation for radial parts of Dirac eigen functions for pseudospin symmetry as follows:, - 24 where, and is quantum number spin orbital and is quantum number pseudospin orbital Analytical solution of radial and angular parts of the Dirac equation 11 Solution of the radial part In the case of exact pseudospin symmetry and, Eq24 becomes: 25 We assume that is Eckart potensial, which defined as 26 and we assume that with is used for the centrifugal term The radial Dirac equation take the following form: 27 which, By substituting variable to Eq 27, we have { by substituting, } If we insert this wave function into Eq32 to Eq31, we have the second-orde differential equation as in the following form: * + { } 33 where value and - - By comparing this equation with Eq1, we can write the and values and by means of Eq2, we may calculate and This gives * Combining these results with the quantization condition given by Eq3, yields where, 35 When the above expressions are generalized 36 so that eigenvalues turn our as 4 5, n r =,1,2,37 By substituting Eq28, Eq29, and Eq3 to Eq37, we obtain energy eigen values as follows: where is radial quantum number, is orbital quantum number which obtained from angular part solution And then, radial wavefunction can be obtain by Using Eq6, Eq7, Eq8 and Eq1, we have:,,, so, and P - 7

4 International Conference on Mathematics, Science, and Education 215 ICMSE 215 From Eq6, we have, 39 By substituting Eq39 to Eq32, we have radial wavefunction, 4 which, so 41 where is radial normalization constant, is hypergeometric function and is Pochamer symbol 12 Solution of the angular part For angular part in Eq21, can be obtain by using AIM to find orbital quantum number By multiplying Eq21 with, we have Eq42 must be simplified by using parameter, and by simplying it, yields, { } 43 by using 44 and simplying it, Eq43 can be transform to hypergeometri differential equation: * + * + 45 where, and is orbital quantum number Eq45 is hypergeometri differential equation, so we must transform it to AIM type by divide Eq45 with, yields * + { } 46 From Eq46, we have * + 47 { } 48 To have this eigen value of equation, further iterations and, which k is stated iteration By using Eq 3, energy eigen value can be obtain, by using software Matlab,, yield, yield, yield where, When the above expressions are generalized Energy eigen value can be obtain by generalized, which yields And then, angular wavefunction can be obtain by Using Eq6, Eq7, Eq8 and Eq1, we have,,,, so, and From Eq7, we have 5 By substituting Eq5 into Eq44, yields which, so the angular part wavefunction can be obtain, as follows is angular normalization constant 51 RESULT AND EXPLANATION In this section, we discuss several result obtained in the previous section From energy eigen value in Eq41 and orbital quantum number in Eq52 In Table 1 are energy eigen values with variation potential constant V and V 1 which plotted in Fig 1 for variation V and V 1 Fig1 shown that due to increase potential constant V and V 1 cause decrease energy In this section, we discuss several result obtained in the previous section From energy eigen value in Eq41 and orbital quantum number in Eq52, by using Matlab we have numeric solution of energy eigen value are listed in Table 1 with parameters C ps = -5; v = 5; alfa = 5; V P - 8

5 International Conference on Mathematics, Science, and Education 215 ICMSE 215 = 25; V 1 = 25 and, the positive value is In Table 2 are energy eigenvalues with variation taken due to the spin symmetric limit By inspecting potential constant V and V 1 which plotted in Fig 1 for Table 1, show that increase of value and in the same variation V and V 1 By inspecting Fig1 shown that due to quantum state cause decrease energy eigenvalue increase potential constant V and V 1 cause decrease energy Table 1 Energy eigenvalues corresponding to several sates of a particle under the influence of Eckart potential and trigonometric Manning Rosen potential 1-2,8531-2,9531-2, ,8878-2,9568-2, ,8998-2,9689-2, ,9362-2,9816-2, ,9148-2,9753-2, ,945-2,9831-2, ,9491-2,9845-2, ,9733-2,9921-2,9999 Table 2 Energy eigenvalues in with M = 5; Cps = -5; v = 5; q =,2; = 5, for particle under the influence of Eckart potential and trigonometric Manning Rosen potential variation V andv 1 M V V 1 E E 1 E 2 E 3 E ,9328-3,129-3,3129-3,5231-3, ,8339-2,9231-3,635-3,2282-3, ,6339-2,7231-2,8635-3,282-3,2394 Fig 1 Energy eigenvalues for variation V and V 1 By varying parameter which corresponding value and, some of the radial wavefunctions are listed in Table 3 Radial wavefunctions for particle under the influence of Eckart potential and trigonometric Manning Rosen potential be affected by potential constant V, V 1 and by The parameter q has a dimension inverse of distance in space that describes the range of trigonometric Manning Rosen potential effect in space If q is enlarged, physically could mean that the potential range smaller in a space By inspecting Table 3 and Fig2, the variations of for the radial wavefunctions influences distribution particle to move away or approach from the core of atom From Fig2, amplitude <, which means that at the distribution of particles move away from nucleus when compared with smaller This shows that the higher the atomic shell causes the greater distance of particle from the core P - 9

6 International Conference on Mathematics, Science, and Education 215 ICMSE 215 Table 3 Energy eigenvalues in with M = 5; Cps = -5; nl = 2; m = 2; v = 25; V = 2; V2 = 2; q =,2; =,5 for particle under the influence of Eckart potential and trigonometric Manning Rosen potential variation nr nr Enr -2, , ,8635 Fig2 Radial wavefunction follow Table 3 variations n r a b c d e Fig3 a-c Polar diagram and d-f three-dimensional anguler wavefunctions with variations n l a and d,c and f CONCLUSION In this paper, we study the Dirac equation for particle spin 12 in the Eckart potential combined with the trigonometric Manning Rosen potential under the condition that the scalar potential equals to the vector potential The radial part of the spinor wavefunction is obtained approximately from Eq44 and the angular part in Eq54 The results show that the disturbance of Eckart and trigonometric Manning Rosen potential change in the wavefunction of the radial part and the angular part Energy eigenvalue can be obtain via Asymptotic Iteration Method in Eq41 and equations of orbital quantum number l in Eq52, where both are interrelated between quantum numbers Energy spectrum also numerically solved using Matlab software, where the increase in the radial quantum number n r causes a decrease in the energy spectrum ACKNOWLEDGEMENT This research was partly supported by Higher Education Project Grant with contract No 351UN2711PN214 P f, b and e BIBLIOGRAPHY [1] SM Ikhdair and R Sever, Appl Math Comp [2] GF Wei and SH Dong, Phys Lett B [3] JY Guo and ZQ Sheng, Phys Lett A [4] M Eshghi and H Mehraban, Few-Body Syst in press [5] BF Samsonov and AA Pecheritsyn, Russ Phys J [6] CS Jia, JY Liu, PQ Wang, and X Lin, Int J Theor Phys [7] AD Alhaidari, Phys Rev Lett [8] JN Ginocchio, Phys Rev Lett [9] Suparmi and Cari 214 ITB Journal Publisher, 46, 31 [1] Suparmi A, Cari C and Angraini L M 214 AIP Conf Proc [11] Zhou S G, Meng J and Ring P 23 Physical Review Letters, 91, 26 [12] Bahar M K and Yasuk F 212 Chin Phys B 22 1

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