Investigation of N-identical few-body bound systems in the relativistic and non-relativistic description

Accepted Manuscript

Investigation of N-Identical Few-Body Bound Systems in the Relativistic and Non-Relativistic Description Mohsen Mousavi , Mohammad Reza Shojaei , Azadeh Hejazi Juybari PII: DOI: Reference:

S0577-9073(16)30829-2 10.1016/j.cjph.2017.03.013 CJPH 211

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

8 December 2016 20 February 2017 2 March 2017

Please cite this article as: Mohsen Mousavi , Mohammad Reza Shojaei , Azadeh Hejazi Juybari , Investigation of N-Identical Few-Body Bound Systems in the Relativistic and Non-Relativistic Description, Chinese Journal of Physics (2017), doi: 10.1016/j.cjph.2017.03.013

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Highlights The Schrödinger and Klein–Gordon equations in D-dimensions are investigated.

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The PNU and SUSYQM methods are used for few-body bound systems.

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N-identical particles are investigated by using the Jacobi coordinate.

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Investigation of N-Identical Few-Body Bound Systems in the Relativistic and Non-Relativistic Description Mohsen Mousavi1, Mohammad Reza Shojaei1* and Azadeh Hejazi Juybari2 Physics Department, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran 2 Solid State Physics Department, University of Mazandaran, P.O. Box: 47416-95447, Babolsar, Iran *E-mail: [email protected]

1

Abstract: For the N-identical interacting particles that construct a bound system, by using the Jacobi

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coordinate transformation in the hyper-spherical approach, the non-relativistic Schrödinger and relativistic Klein– Gordon equations under spin symmetry in D-dimensions are investigated for the modified Hulthen plus quadratic Yukawa potentials. The parametric Nikiforov-Uvarov and super-symmetric quantum mechanics (SUSYQM) methods are used to obtain the energy eigen-values and wave functions for a few-body bound system. To show the accuracy of the present model, the dependence of the few-body binding energies on the potential parameters has been investigated.

Keywords: Quantum Field Theory; Few-body bound system; Jacobi coordinate; K-G equation;

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PACS: 03.65.Ge, 03.65.Pm, 02.30.Gp

1 Introduction

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Few-body systems appear in various branches of physics, from extremely tiny structures to largescale gravitational systems. In particular, our standard understanding of particle and nuclear physics includes two-body mesons and three-body baryons, which are present in almost every interaction. In investigations of such few-body systems, a variety of parallel experimental and theoretical approaches, with their associated achievements and failures, are being conducted. The physical world consists of interacting N-particle systems. In a real metal, in plasma physics, solid-state physics, statistical physics and chemical physics, we have many identical particles. An accurate description of such systems requires the inclusion of the interparticle potentials. The Schrödinger equation non-relativistically and the Klein-Gordon (K-G) and Dirac equation relativistically have long been recognized as essential tools for the study of atoms, nuclei, molecules and their spectral behaviors. Much effort has been spent on finding the exact bound state solution of these non-relativistic and relativistic equations for various potentials describing the nature of bonding or the nature of the vibrations of these quantum systems. However, a direct solution of the Schrödinger equation is impractical. Therefore, it is necessary to resort to other techniques, such as second quantization, quantum-field theory, and the use of Green’s functions. In a relativistic theory, the concept of second quantization is essential for describing the creation and destruction of particles. Even in a nonrelativistic theory, second quantization greatly simplifies the discussion of many identical interacting particles. The methods of quantum field theory also allow us to concentrate on the few matrix elements of interest. In most cases of interest, the first few orders of the perturbation theory cannot provide an adequate description for an interacting N-particle system. For this reason, it becomes essential to develop a systematic method for an analytical solution of the Schrödinger equation [1, 2, 3]. The studies of the 6nucleon bound-state properties for the case of few-body interactions, in particular, have received increasing attention theoretically and experimentally in recent years [4]. So far the analytical 2

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solution of the Schrödinger and Klein–Gordon equations in the presence of various potentials for two-, three-, and N-particle systems has been calculated [5, 6]. Many studies have been made in order to find the exact solutions of the Klein–Gordon equation. By assuming an equal magnitude scalar and vector potential, bound state solutions of the Klein–Gordon equation have been obtained in various potential fields [7, 8]. Although, the solution of the Klein–Gordon equation for one-particle is feasible and solvable numerically and analytically, studying a few-body Klein–Gordon equation is completely intractable with little relevant literature. This paper is organized as follows. In Sections 2, 3, 4, and 5 we derive the non-relativistic radial Schrödinger equation and the relativistic Klein– Gordon equation for the N-identical interacting particles that construct a bound system by using the Jacobi coordinate transformation in the hyper-spherical approach. Analytical solutions for the N-dimensional equations are discussed with the modified Hulthen plus quadratic Yukawa potentials by using the parametric Nikiforov-Uvarov (PNU) [9, 10] and super-symmetric quantum mechanics (SUSYQM) [11, 12] methods. To show the accuracy of the present model, some numerical values of the energy levels for a few-body bound system are shown. Finally, conclusions are given in Section 6.

2 Non-relativistic investigation for the N-particle system by the PNU method

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The many-body forces are more easily introduced and treated within the hyper-spherical harmonics formalism. Let us consider a system of fixed identical particles in the following Schrödinger equation [13, 14]:

i 

 i  1 i  ri 1   rj  , i 1  i j 1 

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(2) where

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 1 N 1 2   ri  V  ri , j   E    ri , j   0 ,   i , j 1  2m i 1  (1) where rij stands for the set of relative coordinates of the particle rij = rj-ri and V(rij) is a two-body potential of interaction. Let us assume that we intend to expand in the hyperspherical-harmonic a function V(rij) of the coordinates between the particles at a relative distance of rij = rj-ri. We define a set of Jacobi coordinates for which ζN=rij. The center of mass R can be eliminated by using the Jacobi coordinates:

N 1

x    2

i 1

2 i

N 1

   r i 1

i

2

R  

i  1,..., N  1

2  rk2 , N 1 k ;  k

R 

1 N

N

r

i

i

(3) For example, a three-particle system after eliminating the center-of-mass motion becomes a sixdimensional system (D=9-3=6). From Eqs. (2) and (3), the internal three-identical-particle motion is described by means of the Jacobi relative coordinates ζ1, ζ2 and R=R3 as: 3

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1 

r1  r2 , 2

2 

r1  r2  2r3 , 6

R3 

r1  r2  r3 3

.

(4) The hyper-spherical coordinates are defined in terms of the absolute values of ζ1, ζ2, and from Eq. (4) it follows that

  t  arctan  1  2 

.

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x   12   22 , (5)

The Laplace operator written in hyper-spherical coordinates in the D-dimensional space for N identical particles becomes [15, 16, 17]

 n m  x , D   R n  x Y

m

 D 

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N 1 N 1  d2 L2     D 1 d   2i     2x    2    x dx x2  i 1 i 1  dx (6) Applying the separation variable method by means of the solution

(7) Eq. (7) provides two separate equations, where the Y harmonics. m

 D   

 D  2 Y

m

 D 

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L2   Y

m

 D 

.

,

are known as the hyper-spherical .

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(8) The hyper-radial, or in short the ―radial‖ equation is

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 d 2 D 1 d   D  2   2m V x  E   R x  0  ,   n  n    2 2  x dx x2  dx  (9) where Ym    , R n  x  , E n and represent the hyper-spherical harmonics, the hyper-

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radial part, the energy eigen-values, and the orbital angular momentum, respectively. The Nbody problem in the center-of-mass frame is mathematically (3N-3) - dimensional. In the hyperspherical method, a point in the (D=3N-3)-dimensional configuration space is represented as lying on a (D-1)-dimensional hyper-sphere of radius x [18]. The modified Hulthen [19, 20] plus quadratic Yukawa [21, 22] potential is defined as: V(x)   v0

ex ex  v 1 (1  ex ) x2

,

(10) where the parameters v0 and v1 are real parameters, these are strength parameters, and the parameter α is related to the range of the potential.

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By considering the radial wave function as U  x   x 

D 1

2

R

n

x ,  



 D  3 2

, Eq. (9)

with a hyper-central potential derived from two-body potential, Eq. (10) results in: 2 d 2 U n  x  2m      1  v0 e x e x  E   v    Un  x   0 . n 1 2 2  x 2 dx (1  e ) x 2M x2  

(11)

x2



    1  2 ex

1  ex 

2

.

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    1

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Equation (11) is exactly solvable only for the case of λ= 0. Hence, we shall use an approximation in order to deal with the centrifugal-like terms. In order to obtain the analytical solutions of Eq. (11), we employ the improved Greene and Aldrich [23] approximation and replace the spin–orbit coupling term with an expression that is valid for α ≤ 1 [24]. The main characteristic of these solutions lies in the substitution of the centrifugal term by an approximation, so that one can obtain an equation, normally hyper-geometric, which is solvable [25]:

(12) Applying the approximations of (12) for the centrifugal term and the hyper-central potential along with introducing a new variable of the form s  exp(x) , we can write Eq. (11) as follows:

1  s  U s  1  s2   s    U s  0 n,   2 1 0  n,   2  s 1  s  s 2 1  s 

,

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Un,  s  

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(13) where the parameters χ2, χ1 and χ0 are considered as follows: 2m  v  2  v0   E n  , 2 2  1  2 2  2m   1   2 2  2E n  v0       1  ,   2M  2m En . 2 2 

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0  

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2 

(14) Applying the PNU method, we obtain the energy equation (by referring to references [9, 10]) as:

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  1 1 1 1 (2n  1)  2  1  0   0  (2n  1)   2 0 ( 2  1  0  )  20  1   0 4 4 4 4  

.

(15) And the hyper-radial wave function can be written in the form, by referring to the PNU method in references [26, 27], as: R n, (x)  Nx

 D 1     2 





 1 1 x  2 1 0  4  2   

 e  1  e  x

0

 1  2 0 ,2 2 1 0  4    n

P

1  2e  x

,

(16) 5

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where N is the normalization constant and the functions Pn ,  x  are the Jacobi polynomials.

3 Non-relativistic investigation for the N-particle system by SUSYQM method Applying Eq. (12), Eq. (11) is as summarized below: 2

dx



2m 2

E

n

V eff  x U n  x   0

,

(17) where Veff  x    v0

e x e2 x 2  v   1 2 1  e x  1  e x 

 2     1

2

2m

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d 2U n  x 

e2 x

1  e   x

2

.

(18) Through super-algebra we make the following ansatz for the super-potential as follows:  e  x A  B 2m  1  e  x

   

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W x   

.

(19) This satisfies the associated Riccati equation, and substituting this expression in the SUSYQM [11, 12] method, we obtain the following identity:

x  

2m

W   x  V eff  x   E n

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2

W

(20)  2 x A 2   B 2   B  e 2m  1  e  x  2

2

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

  B  2AB 

(21)

 e  x  V  x   E n 1  e  x   eff

.

A2 

2m 2

E0,

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By comparing each side of Eq. (21) we obtain

2

B    ,  

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B

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2m  B  2AB     ,   2 v 0 , 2m  2 2  v 1  

 2     1 

2

2m

 . 

(22) Solving Eq. (22) yields

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E0  

2

2m

A 2,

   2  4 , 2      B . A 2B 2 B 

 2 x A 2   B 2   B  e 2m  1  e  x  2  2 x A 2   B 2   B  e V  2m  1  e  x  V 

2

 

2

2

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(23) Consequently, these super-symmetric partner potentials are obtained as:  e  x  , 1  e  x    e  x     B  2AB  . 1  e  x     B  2AB 

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(24) And

S  ai  V   B  i  , x  V   B   i  1 , x  ,

2 2      B   i  1    B  i          S  ai       ,    2m   2  B  i   2 2   2  B   i  1      2

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(25) where the remainder U(ai) is independent of x. We can obtain the energy levels as follows with the reference to the SUSYQM [15,17] method as: n

E n ,   S  ai   E 0 i 1

.

2 2     B  n         B                   2m   2  B  n  2 2   2    2  4   2B 

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2

En,

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(26) Finally, the energy relation is written as:

   

2

    

.

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(27) In super-symmetric quantum mechanics the ground state eigen-function U0(x) can be written as  2m U 0  x   N 0 exp   



 W  y dy  . x



(28)

Using Eq. (19) and (28), the eigen-function for the ground state in terms of x will be obtained as: U 0,  x   N 0e Ax 1  e  x





B



.

(29) So, we have

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R 0, (x )  N 0 x

 D 1    2  

e Ax 1  e  x





B



,

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(30) where N0 is a normalization constant. According to Eq. (15) and Eq. (27), the obtained energy eigen-values of the PNU and SUSYQM methods depend on the hyper-central potential parameters. We carried out some calculations for the ground state using Eq. (15) for two, three and four-body bound systems. 0

0 -20

-20

E3-SH (fm-1) 0,0

-60 -80 -100

v 1=1

-120

v 1=1.5

-140

v 1=2

-160

v 1=2.5

-180

5

(a)

-40

v 1=1

-60

v 1=2

(b)

v 1=4

-80

v 1=6

10

15

v0

20

25

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E2-SH (fm-1) 0,0

-40

-100 10

30

0

-10

-15

v 1=1 v 1=5

-20

v 1=10

-25

30

40

v0

50

60

70

80

(c)

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E4-SH (fm-1) 0,0

-5

20

v 1=15

-30 20

30

40

50

60

70

80

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v0

Fig. 1 The variation of the ground state binding energy for the Schrödinger equation on the parameter v0 with different values of v1 for the fixed value of α = 0.08 fm−1, for two-nucleon (a), three-nucleon (b) and four-nucleon (c) bound systems.

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We have investigated the dependence of the two-, three- and four-nucleon binding energies on the parameters v0 and v1, in Figs. 1 and 2, respectively. For a particular v1, the few-nucleon binding energies are found to increase with increasing v0, as it should be. Also, it can be seen that for a particular v0, the few-nucleon binding energies decrease with increasing v1. 0

0

-20

AC

-10

-60

E3-SH (fm-1) 0,0

E2-SH (fm-1) 0,0

-40

-80

-100

v 0=15

-120

v 0=20

(a)

-140

v 0=25

-160 -180

2

3

4

5

6

v1(fm)

7

8

9

-30

v 0=45 -40

v 0=50

(b)

-50

v 0=55 v 0=60

v 0=30 1

-20

-60

10

0

5

10

15

20

25

30

35

40

v1(fm)

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-5 -10

E4-SH (fm-1) 0,0

-15 -20

v 0=70 -25

v 0=75

(c)

-30

v 0=80 v 0=85

0

10

20

30

40

50

60

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-35

v1(fm)

Fig. 2 The variation of the ground state binding energy for the Schrödinger equation on the parameter v1 for different values of v0 for the fixed value of α = 0.08 fm−1, for two-nucleon (a), three-nucleon (b) and four-nucleon (c) bound systems. 0

0

N=2 N=3 N=4

(a)

-60 -80

-50

-100

N=2 N=3 N=4

-100

-120

EN-SH (fm-1) 0,0

-40

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EN-SH (fm-1) 0,0

-20

5

10

15

20

v1(fm)

25

30

35

40

-150 15

20

(b)

30

40

v0

50

60

70

80

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Fig. 3 Comparison between the Schrödinger ground state binding energy for the two-nucleon, three-nucleon and four-nucleon bound systems versus different values of (a) v1 and (b) v0 for the fixed value of α=0.08 fm−1.

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In Figure 3 we investigated the ground state binding energy in reference to Eq. (15) for two, three and four-nucleon bound systems versus different values of the parameters v1 and v0 for the fixed value of α=0.08fm−1. It is clear that as v0 and v1 increase, the ground state binding energy of the system has an increasing and decreasing behavior, respectively. The agreement between the energy values for the PNU and SUSYQM method versus the different values of v0 and v1 for a fixed value of α=0.01 fm−1 are shown in Tables 1 and 2.

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Table. 1. Comparison between the state energies for the PNU and SUSYQM methods versus different values of v1 for the fixed value of α=0.01 fm−1, v0=70, h=1, m=1 fm-1 and N=2 (in the non-relativistic system).

v1

E0,0 PNU

E1,1 SUSY

PNU

E2,1 SUSY

PNU

SUSY

10.0 -97.9300 -97.9300

-63.2190 -63.2371

-46.8231 -46.8365

15.0 -67.9972 -67.9972

-47.3927 -47.4063

-36.4320 -36.4424

20.0 -52.2498 -52.2498

-38.1501 -38.1610

-30.0700 -30.0786

25.0 -42.4969 -42.4969

-32.0382 -32.0474

-25.7221 -25.7295

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Table. 2. Comparison between the state energies for the PNU and SUSYQM method versus different values of v1 for the fixed value of α=0.01 fm−1, v1=10 fm, h=1, m=1 fm-1 and N=2 (in the non-relativistic system).

80.0 85.0 90.0 95.0

E0,0 PNU -127.9200 -144.4150 -161.9100 -180.4050

E1,1

SUSY -127.9200 -144.4150 -161.9100 -180.4050

E2,1

PNU SUSY -82.5993 -82.6199 -93.2596 -93.2816 -104.5668 -104.5901 -116.5208 -116.5454

PNU -61.1919 -69.0962 -77.4806 -86.3449

SUSY -61.2072 -69.1125 -77.4978 -86.3631

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v0

4 Relativistic investigation for the N-particle system by the PNU method

The D-dimensional time-independent arbitrary l-states radial Klein-Gordon equation with scalar and vector potentials S(r) and V (r), where r = |r| describes a spin-less particle (such as an α particle), it takes the general form [28, 29]: 1 c

2 2

E



V  r   Mc 2  S  r   1D...1 D12  r   0 2

2

n,

,

AN US

2D  1D...1 D12  r  

PT

ED

M

(31) where Enl, M and 2D denote the K-G energy, the mass and the D-dimensional Laplacian, respectively. The Klein-Gordon equations of relativistic quantum mechanics have the correct nonrelativistic limits. Moreover, the nonrelativistic limit is obtained by taking ε−M∼E, where |E|˂˂M [32]. So we could obtain a K-G equation in hyper-radial form for an n-identical bond system, like the time-independent hyper-radial Schrödinger equation in Section 2 by using Jacobi coordinates. If the scalar and vector potentials S(rij) and V(rij) are a two-body potential of interaction, we can expand them in the hyper-spherical harmonics formalism. We define a set of Jacobi coordinates for rij, as in Section 2. In addition, x is a D-dimensional position vector in Jacobi coordinates. In the case where the scalar and vector potential have equal magnitudes, V(x) = S(x), by Jacobi relative considerations, the time-independent hyper-radial Schrödinger-like equation in Ddimensions [6] becomes:  D  2 x

2



E

2

 M 2c 4  2

c

2

 E  Mc V 2

2

2

c

2



 x   n  x   0

,

 

AC

(32) with



CE

2  D 1 d d   2  x dx  dx 

V  x    v0

ex ex  v 1  ex  1 x 2

.

(33)

By choosing a common ansatz for the wave function in the form of   x   x   

 D  3 2

D 1

2

 x ,  n 

, from Eq. (31) with a hyper-central potential we derive

10

ACCEPTED MANUSCRIPT

d2n  x  dx 2

2 2 4   E  Mc2   v ex  v ex E  M c    2 1 2 2 2 2  0 1  ex  c c x2   

     1      n  x   0 . (34)  x2   

By applying the approximations presented in the previous section for the centrifugal term and the hyper-central potential along with introducing the new variable of the form s = exp(−αx), Eq. (34) can be reduced to

 E  Mc  v 2

2

c 

2

2

2



CR IP T

2 2 4  2 E  M c  1  s   2 2 2 c   

1  s  d   d 2 1  2 2 2 ds s 1  s  ds s 1  s 

 

 s 1  s   v 1 2s 2       1 s    0.

0

 

(35) Eq. (35) can be summarized as follows:

1  s   s  1  s2   s     s  0 n,   2 1 0 n,   2  s 1  s  s2 1  s 

,

AN US

n,  s  

(36) where the parameters η2, η1 and η0 are considered as follows:

1    2   v0       1 ,   

E

0   .

 M 2c4  c 2

2 2

 E  Mc  , 2

,   2

c 2

2 2

ED

(37)

2

M

2  v1 2  v0     ,

Applying the PNU method, we obtain the energy equation (by referring to reference [9, 10]) as: .

PT

  1 1 1 1 (2n  1)  2  1  0   0  (2n  1)   2 0 (2  1  0  )  20  1   0 4 4 4 4  

CE

(38) And the hyper-radial wave function can be written in the form, with reference to the PNU method in reference [26, 27] as:  D 1    2 



e  x

0



 1 1 x  2 1 0  4  2   

1  e 

 1  2 0 ,2 2 1 0  4    n

P

1  2e  x

,

AC

 n, (x)  Nx 

(39) where Nʹ is the normalization constant and the functions Pn ,  x  are the Jacobi polynomials.

11

ACCEPTED MANUSCRIPT

5 Relativistic investigation of the N-particle system by using the SUSYQM method Applying Eq. (12), Eq. (34) as summarized as  d2   2 V eff  x    n  x   E n  n  x   dx 

,

e  x e 2 x  V 2 1  e  x  1  e  x

V eff  x  V 1

E 

En

2

 M 2c 4  2

c2



2

CR IP T

(40) where ,

,

V1   V2 

2  E  Mc 2 

v 0 ,

2 2

c 2  E  Mc 2  2 2

c

AN US

(41) where,

v 1 2      1 2 .

W  x   A   B 

e  x 1  e  x



M

(42) Based on super-algebra we make following ansatz for the super-potential: .

PT

ED

(43) This satisfies the associated Riccati equation, and substituting this expression in the SUSYQM [30, 31] method, we obtain the following identity: W 2  x  W   x  V eff  x   E n .

CE

(44) Now substituting Eq. (41) and Eq. (43) into Eq. (44) yields

AC

 2 x e  x  A 2   B 2   B   e      B  2 A B    x 2  1  e  x  1  e    (45) From the comparison of each side of Eq. (45), we obtain E 0,  A 2 ,

  V eff  x   E 0,  

.

 B   2A B   V 2 ,

B 

2

  B   V 1 .

(46) 12

ACCEPTED MANUSCRIPT

Solving Eqs. (46) yields

A  B

V

2

V 1

2B 

  B, 2

    4V 2 . 2 2

 e 2 x V    A 2   B 2   B    1  e  x   e 2 x V    A 2   B 2   B    1  e  x 

 

2

2

CR IP T

(47) Consequently, these super-symmetric partner potentials are obtained as  e  x  , 1  e  x    e  x     B   2A B   , 1  e  x     B   2A B  

AN US

(48) and

S  ai  V   B   i  , x  V   B    i  1 , x  ,









2 2   V V V 2 V 1 B    i  1    B   i     2 1      , S  ai        2 2   2  B    i  1      2  B   i    

ED

M

(49) where the remainder U(ai) is independent of x. By referring to the SUSYQM [3,31] method we can obtain the energy levels as follows: , E n ,  E n,  E 0,





2





 V 2 V 1  B   n     V 2 V 1  B     S  ai     2 2  2  B   n    2B  i 1  n

CE

E

 n,

PT

(50) where,

2

.

(51) Finally, the energy relation is written as

AC E n2,





 V 2 V 1  B   n    M 2c 4  2c 2   2  2  B   n  

2

.

(52) In super-symmetric quantum mechanics, the ground state eigen-function U0(x) can be written as



 0,  x   N 0 exp   W  y  dy x



.

(53) Using Eqs. (43) and (53), the eigen-function for the ground state in terms of x can be obtained as 13

ACCEPTED MANUSCRIPT

 0,  x   N 0e A x 1  e  x





B



,

0

CR IP T

(54) where N0 is a normalization constant. We carried out some calculations for the ground state in reference to Eq. (38) for two, three and four-body bound systems. We have investigated the dependence of the two-, three- and fournucleon binding energies on the parameters v0 and v1, in Figs. 4 and 5, respectively. For a particular v1, the few-nucleon binding energies are found to increase with increasing v0, as it should be. Also, it can be seen that for a particular v0, the few-nucleon binding energies decrease with increasing v1. 0

v 1=32

v 1=32

(a)

-15

-20 40

v 1=34 v 1=36 v 1=38

-10

AN US

v 1=38

-10

(b)

-5

v 1=36

E3-K-G (fm-1) 0,0

-5

E2-K-G (fm-1) 0,0

v 1=34

-15

45

50

55

60

65

v0

-20 40

70

45

50

55

60

65

70

v0

0

v 1=32

(c)

v 1=34 v 1=36 v 1=38

-10

ED

-15

M

E4-K-G (fm-1) 0,0

-5

-20 40

45

50

55

60

65

70

v0

AC

CE

PT

Fig. 4 The variation of the ground state binding energy for the K-G equation on the parameter v0 for different values of v1 for the fixed value of α = 0.08 fm−1, for (a) two-nucleon, (b) three-nucleon and (c) four-nucleon bound systems.

14

ACCEPTED MANUSCRIPT

0

0

v 0=42 v 0=44 v 0=48

-10

-15

(b)

v 0=44

-5

v 0=46

E3-K-G (fm-1) 0,0

E2-K-G (fm-1) 0,0

-5

v 0=42

(a)

v 0=46 v 0=48

-10

-15

-20 10

15

20

25

30

35

-20 10

40

15

20

25

v1(fm)

35

40

CR IP T

0

E4-K-G (fm-1) 0,0

30

v1(fm) -2

v 0=42

-4

v 0=44

-6

v 0=46

-8

v 0=48

-10

(c)

-12 -14 -16 -18 -20 10

15

20

25

30

35

40

AN US

v1(fm)

Fig. 5 The variation of the ground state binding energy for the K-G equation on the parameter v1 for different values of v0 for the fixed value of α = 0.08 fm−1, for (a) two-nucleon, (b) three-nucleon and (c) four-nucleon bound systems.

M

In Figure 6 we investigated the ground state binding energy referring to Eq. (38) for two, three and four-nucleon bound systems versus the different values of v1 and v0 for the fixed value of α=0.08 fm−1. It is clear that as v0 and v1 increases, the ground state binding energy of the system has an increasing and decreasing behavior, respectively.

EN-K-G (fm-1) 0,0

-17

-18

CE

-19

56

58

60

62

N=2 N=3 N=4

-15

PT

EN-K-G (fm-1) 0,0

N=2 N=3 N=4

(a)

-16

-20 54

-14

ED

-15

(b)

-16 -17 -18 -19

64

66

68

70

-20 10

15

20

25

30

v1(fm)

v0

AC

Fig. 6 Comparison between the K-G ground state binding energies for two-nucleon, three-nucleon and four-nucleon bound systems versus different values of (a) v0 and (b) v1 for the fixed value of α=0.08 fm−1.

The agreement between the energy values for the PNU and SUSYQM method versus the different value of v0 and v1 for the fixed value of α=0.01 fm−1, in Tables 3 and 4 are shown. Table. 3. Comparison between the state energies for the PNU and SUSYQM methods versus the different values of v1 for the fixed value of α=0.01 fm−1, v0=60, h=c=1, m=15fm-1 and N=2 (in Relativistic system). E0,0 v1

PNU

E1,1 SUSY

PNU

E2,1 SUSY

PNU

SUSY

15

ACCEPTED MANUSCRIPT

-14.923461

-14.923461

-14.370631

-14.372070

-13.630103

-13.631392

45.0

-14.866175

-14.866175

-14.005648

-14.007371

-13.077830

-12.985689

50.0

-14.718753

-14.718753

-13.363874

-13.365767

-12.192202

-12.076068

55.0 60.0

-14.251417 -13.059409

-14.251417 -13.059409

-12.356896 -11.059594

-12.358691 -11.061101

-10.934080 -9.6640071

-10.935321 -9.6650794

CR IP T

40.0

Table. 4. Comparison between the state energies for the PNU and SUSYQM methods versus different values of v1 for the fixed value of α=0.01 fm−1, v1=40 fm, h=c=1, m=15 fm-1 and N=2 (in the relativistic system).

45.0 50.0 55.0 60.0

E0,0 PNU SUSY -9.148941 -9.148941 -13.48983 -13.48983 -14.77491 -14.77491 -14.92346 -14.92346

E1,1 PNU SUSY -7.368681 -7.369785 -11.222982 -11.224712 -13.539901 -13.541808 -14.370631 -14.372070

E2,1 PNU SUSY -6.155030 -6.039553 -9.653293 -9.654503 -12.375674 -12.271220 -13.630103 -13.631392

AN US

v0

6 Conclusions

CE

PT

ED

M

We investigated the relativistic and non-relativistic few-body bound system problem by presenting the analytical solution of the D-dimensional Klein–Gordon equation with an equal magnitude scalar and vector potential and the D-dimensional Schrödinger equation, respectively. The parametric Nikiforov-Uvarov and supersymmetric quantum mechanics methods are used to obtain the energy eigen-values and wave functions for few-body bound systems. The dependence of the binding energies for the systems with two, three and four relativistic and non- relativistic nucleons on the potential parameters are shown. Also, we showed an agreement between the energy values for PNU and SUSYQM methods. Our proposed approach could be useful in investigating the relativistic and non-relativistic corrections relevant to the observables characterizing the properties of few body nuclear systems, within a simple treatment.

Reference

AC

[1] R. P. Feynman, Phys. Rev. 76, 749 (1949). [2] F. J. Dyson, Phys. Rev. 75, 486 (1949). [3] M. M. Giannini, E. Santopinto, A.Vassallo, Prog. Part. Nucl. Phys. 50, 263 (2003). [4] E. Ahmadi Pouya and A. A. Rajabi, Eur. Phys. J. Plus 131: 240 (2016). [5] S. Capstick, B. D. Keister, Phys Rev D, 51, 3598 (1995). [6] M. Aslanzadeh and A. A. Rajabi, Few-Body Syst 57 145 (2016). [7] C. F. Hou, Z. X. Zhou, Y. LI, Chin. Phys. B 8, 8 (1999). [8] Lalit K. Sharma, Pearson V. Luhanga and Samuel Chimidza, Chiang Mai J. Sci. 38, 514 (2011). [9] M. Mousavi, M. R. Shojaei, Chin. J. Phys. 54, 750–755 (2016). [10] M. Mousavi and M. R. Shojaei, Pramana - J Phys 88: 21 (2017). [11] C. S. Jia, P. Gao and X. L. Peng, J. Phys. A: Math. Gen. 39, 7737 (2006). [12] H. Feizi, A. A. Rajabi, and M. R. Shojaei, Acta Phys. Polonica B, 42, 2143 (2011).

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CR IP T

[13] A. A. Rajabi, Few-Body Systems, 37 197—213 (2005). [14] U. A. Deta, Suparmi and Cari, Adv. Studies Theor. Phys. 7 647 (2013). [15] M. M. Giannini, E. Santopinto, A.Vassallo, Nucl. Phys. A 699, 308 (2002). [16] F. Zernike, H. C. Brinkman, Proc. Kon. Ned. Acad. Wet. 33, 3 (1935) [17] d. l. Fabre M. Ripelle, Ann. Phys. N.Y. 147, 281 (1983) [18] J. Avery, Hyperspherical Harmonics: Applications in Quantum Theory (Dordrecht:Kluwer, 1989). [19] M. R. Shojaei and M. Mousavi, Adv. High Energy Phys., vol. 2016, Article ID 8314784, 12 pages, (2016). [20] M. Farrokh, M. R. Shojaeia and A. A. Rajabi, Eur. Phys. J. Plus 128 14(2013). [21] M. Hamzavi, M. Movahed, K.-E. Thylwe, A. A. Rajabi, Chin. Phys. Lett. Vol. 29, No. 8 080302 (2012). [22] B. I. Ita and A. I. Ikeuba, Journal of Atomic and Molecular Physics, vol. 2013, Article ID 582610, 4 pages, (2013). doi:10.1155/2013/582610 [23] R. L. Greene, C. Aldrich, Phys. Rev. A 14 2363 (1976). [24] M. Mousavi and M. R. Shojaei, Commun. Theor. Phys. 66 483 (2016). [25] F. J. S. Ferreira, F. V. Prudente, Physics Letters A, 377, 3027-3032 (2013). [26] M. Mousavi and M. R. Shojaei, Adv. High Energy Phys. vol. 2017, Article ID 5841701, 5 pages, (2017). doi:10.1155/2017/5841701 [27] H. Yanar and A. Havare, Adv. High Energy Phys. vol. 2015, Article ID 915796, 17 pages, (2015). doi:10.1155/2015/915796 [28] O. Bayrak, I. Boztosun, J. Phys. A: Math. Gen. 39, 6955 (2006). [29] A. A. Rajabi, M. R. Shojaei, Int. J. Phy. Sci. 6, 33 (2011). [30] A. N. Ikot, H. Hassanabadi, H. P. Obong, Y. E. Chad Umoren, C. N. Isonguyo, and B. H. Yazarloo, Chin. Phys. B Vol. 23, No. 12 120303 (2014). [31] H. Hassanabadi, S. Zarrinkamar and H. Rahimov, Commun. Theor. Phys. 56 423–428 (2011). [32] A.D. Alhaidari, H. Bahlouli, A. Al-Hasan, Physics Letters A 349 87–97(2006).

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