International Journal of Theoretical and Applied Mathematics
Volume 2, Issue 2, December 2016, Pages: 86-92

Quarkonium Masses in the N-dimensional Space Using the Analytical Exact Iteration Method

E. M. Khokha1, M. Abu-Shady2, T. A. Abdel-Karim2

1Department of Basic Science, Modern Academy for Engineering and Technology, Cairo, Egypt

2Department of Applied Mathematics, Faculty of Science, Menoufia University, Shebin El- Kom, Egypt

Email address:

(E. M. Khokha)
(M. Abu-Shady)
(T. A. Abdel-Karim)

To cite this article:

E. M. Khokha, M. Abu-Shady, T. A. Abdel-Karim. Quarkonium Masses in the N-dimensional Space Using the Analytical Exact Iteration Method. International Journal of Theoretical and Applied Mathematics. Vol. 2, No. 2, 2016, pp. 86-92. doi: 10.11648/j.ijtam.20160202.19

Received: October 25, 2016; Accepted: November 7, 2016; Published: December 9, 2016


Abstract: The N- dimensional radial Schrödinger equation with an extended Cornell potential is solved. The analytical exact iteration method (AEIM) is applied. The energy eigenvalues are calculated in the N –dimensional space. The charmonium meson, the bottomonium meson and the  meson masses are calculated in the N-dimensional space. The special cases are obtained from the general case. The study of the effect of dimensionality number is studied. The mean value of the radius and the mean square velocity of charmonium meson, bottomonium meson, and  meson are calculated. The present results are improved in comparison with other recent studies and are in good agreement with the experimental data. Therefore, the present method with the present potential gives successfully description of heavy quarkonium properties.

Keywords: Schrödinger Equation, Cornell Potential, Analytical Exact Iteration Method


1. Introduction

Studying heavy quarkonium systems such as charmonium meson, bottomonium meson, and  meson have a vital role for comprehension the quantitative tests of (QCD) and the standard model [1]. The study of these systems is very important. A lot of heavy quarkonium systems can be studied within the Schrödinger equation [2].

The solution of Schrödinger equation with spherically symmetric potential is one of the important problems in physics. It plays a large vital role for spectroscopy, atoms, molecules, and nuclei, in particular, the properties of constituents particles and dynamics of their interactions.

Several papers have focused on the study of multi-dimensional space or N-dimensional space [3]. The study of multi-dimensional space problems is more general, and it can easily obtain the results in the lower dimensional space [4]. The effect of the N-dimensional space is studied on the energy levels of quantum mechanical systems [5]. The Hydrogen atom [6] and the harmonic oscillator [7] have been studied in the N-dimensional space. In addition, other potentials have been solved in the N-dimensional space such as Coulomb potential [8], Psedoharmonic potential [9], Kratzer potential [10], Hulthen potential [11], Poschl-Teller potential [12], Mie-type potential [13], energy dependent potential [14], forth-order inverse power potential [15], Hua potential [16], global potential [17], and Cornell potential [18].

Schrödinger equation has been solved by using numerous methods such as quasi-linearization method (QLM) [20], Hill determinant method [21], point canonical transformation (PCT) [22], and numerical methods [23-25]. The N-dimensional Schrödinger equation is solved by the Nikiforov-Uvarov (UV) method [5,11,14], (AIM) method [2], Laplace Transform method [3,8], (SUSQM) method [26], power series technique [27], and Pekeris type approximation [16,28].

In this paper, we consider the confining potential is the extended Cornell potential [19].

(1)

The aim of this paper is to calculate the solutions of the N-radial Schrodinger equation using the analytical exact iteration method (AEIM) [19,29,30]. Additionally, the present results are applied on quarkonium properties in comparison with experimental data and other recent studies.

The paper is organized as follows: In Sec. 2, the exact solution of the N-dimensional radial Schrödinger equation is derived. In Sec.3, the results are discussed. In Sec. 4, summary and conclusion are presented.

2. Exact Solution of the N-dimensional Radial Schrödinger Equation with the Extended Cornell Potential

The N-dimensional radial Schrödinger equation for two particles interacting via symmetric potential (1) takes the form [2].

(2)

where  denote the angular quantum number, the dimensional number, and  is the reduced mass of the two particles , respectively.

Inserting the transformation

(3)

Eq. (2) takes the form

(4)

Using the simplifications:

(5)

Eq. (4) reduces to

(6)

To solve the N-dimensional radial Schrödinger equation with the analytical exact iteration method (AEIM). The following ansatz for the wave function is assumed [19].

(7)

Where

(8a)

(8b)

From Eq. (7), we obtain

(9)

Comparing Eqs. (6) and (9) yields

(10)

At n=0, By substituting Eqs. (8) into Eq. (10) gives

(11)

By comparing the corresponding powers of r on both sides of Eq. (11), one obtains

(12a)

(12b)

(12c)

(12d)

(12e)

To get well- behaved solutions at the origin and the infinity,  is taken in the following from Eq. (12d) as

(13)

From Eqs. (5), (12a), (12e) and (13). The ground state energy is:

(14)

Where the coefficient c satisfy this restriction

(15)

Substitution of, ,  and  from Eqs. (12a), (12b) and (13) together with Eqs. (3), (7) and (8) gives the following ground state wave function:

(16)

Secondly, for the first node (n=1), we use  and g(r) from Eq. (8b) to solve Eq. (10) gives:

(17)

Then, the relations between the potential parameters and the coefficients  and  are given by:

(18a)

(18b)

(18c)

(18d)

(18e)

(18f)

(18g)

Where  are obtained from the constraint relations,

(19a)

(19b)

So, one obtain

(19c)

Then, the energy eigenvalue is:

(20)

The corresponding wave function is:

(21)

The second node (n=2), we use and g(r) from Eq. (8b) to solve Eq. (10) gives:

(22)

Therefore, the relations between the potential parameters and the coefficients, ,  are given by:

(23a)

(23b)

(23c)

(23d)

(23e)

(23f)

(23g)

The coefficients  are given by the constraint relations,

(24a)

(24b)

Then, the energy eigenvalue is:

(25)

The corresponding wave function is:

(26)

Similarly, the iteration method is repeated many times. Thus, the exact energy formula for the confining potential (1) with any arbitrary n state is obtained:

(27)

The corresponding wave functions for any n state are:

(28)

where the relations between the parameters of potential and the coefficients

(29a)

(29b)

,

(29c)

(29d)

(29e)

And so on

Finally, from Eqs. (27) and (12d) together we can write the energy eigenvalue of Eq. (4) in N-dimensional as

(30)

3. Discussion of Results

3.1. Mass Spectra of Heavy Quarkonia in the N-dimensional Space

In this section, the properties of charmonuim meson, bottomonium meson, and  meson are calculated, in which the quarkonium meson have quark and antiquark masses. The following relation [1] is used for determining quarkonium masses in the N-dimensional space

(31)

By substituting Eq. (30) into Eq. (31), the quarkonium mass in the N-dimensional space takes the following form

(32)

Thus, the charmonium mass is given by:

(33)

Similarly, the bottomonium mass is calculated from Eq. (33) by replacing

Table 1. Mass spectra of charmonuim for parameters (= 1.209 GeV, a = 0.0673, b = 0.0895, d = 0.001).

State P. Work Exp. [27] [2] [4] N=4 N=5 N=6
1S 3.095481 3.096 3.078 3.078 3.096 3.331703 3.567735 3.803718
1P 3.567735 - 3.415 3.415 3.433 3.803718 4.039683 4.275638
1D 4.039683 - 3.581 4.187 3.686 4.275638 4.511588 4.747534
2S 3.567354 3.686 3.749 3.752 3.767 3.803575 4.039607 4.27559
2P 4.039607 3.773 3.917 4.413 3.910 4.27559 4.511555 4.74751
3S 4.039226 4.040 4.085 5.297 3.984 4.275448 4.511479 4.747463
4S 4.511098 4.263 4.589 6.407 4.150 4.74732 4.983351 5.219335

Table 2. Mass spectra of bottomonium for parameters (=4.823 GeV, a= 0.09347, b= 0.34457, d= 0.001 ).

State P. Work Exp. [27] [2] [4] N=4 N=5 N=6
1S 9.74473 9.46 9.510 9.510 9.460 9.884619 10.02406 10.16338
1P 10.02406 - 9.862 9.862 9.840 10.16338 10.30266 10.44192
1D 10.30266 - 10.038 10.627 10.023 10.44192 10.58116 10.7204
2S 10.02315 10.023 10.214 10.214 10.140 10.16304 10.30248 10.4418
2P 10.30248 - 10.390 10.944 10.160 10.4418 10.58108 10.72034
3S 10.30158 10.355 10.566 11.726 10.280 10.44147 10.5809 10.72023
4S 10.580 10.580 11.094 12.834 10.420 10.71989 10.85933 10.99865

Table 3. Mass spectra of  meson for parameters (=1.209 GeV, =4.823 GeV, a= 0.2793, b= 0.999, d= 0.001 ).

State P. Work Exp. [32] [33] [34] N=4 N=5 N=6
1S 6.277473 6.277 6.349 6.264 6.270 6.658294 7.038623 7.418830
1P 7.038623 - 6.715 6.700 6.699 7.418830 7.798987 8.179120
2S 7.037641 - 6.821 6.856 6.835 7.418462 7.798791 8.178997
2P 7.798791 - 7.102 7.108 7.091 8.178997 8.559155 8.939287
3S 7.797808 - 7.175 7.244 7.193 8.178630 8.558959 8.939165

We choose the positive sign in Eq. (33) in the present computations as in Refs. [4,31]. The free parameters of the present computations a, b and d are fitted with experimental data and Eq. (33). Ikhdair and Hamzavi [19] solved Schrodinger equation in three dimensions for the same potential of the present work and calculated the energy eigenvalues. It can easily obtain their results if we put (N=3) in Eq. (30). In addition, we apply our results to quarkonium properties. In Refs. [2,27], the potential is a special case of the present potential when we put (d = 0). Also, In Ref. [4], the potential is a special case of the present potential when setting (b=0 and d = 0).

In Table 1, the charmonium mass is computed from Eq. (33) in comparison with other theoretical studies [2,4,27], and the experimental data. In Ref. [27], the authors computed the charmonium and bottomonium masses using the power series technique for the quark-antiquark interaction potential. In Ref. [2], the authors computed the charmonium and bottomonium masses using the asymptotic iteration method (AIM) for the quark-antiquark interaction potential. In Ref. [4], the authors computed the charmonium and bottomonium masses using the Nikiforov-Uvarov (UV) method. The present results are in good agreement with experimental data and are improved in comparison with results in Refs. [2,4,27]. In Table 2, the bottomonium mass is computed. Most states of bottomonium are improved in comparison with other theoretical studies [2,4,27], and are in good agreement with experimental data. In Table 3, the meson mass is computed. The 1S state is closed with experimental data and the other states are in good agreement in comparison with Refs. [32,33,34]. The charmonium mass, the bottomonium mass, and  mass increase with increasing dimensional number due to an increase in the binding energy.

Fig. 1. The present potential is plotted as a function of distance (r) for different states, where (a) represents charmonium states, (b) represents bottomonium states, and (c) represents meson states.

In Fig. 1. The present potential is plotted as a function of distance (r) for different states for the range of distances 0.1 fm ≤ r ≤ 1 fm as in Ref. [35]. The three curves increase with increasing distance (r). Additionally, all curves shift to upper values by increasing angular quantum number. This behavior is in agreement with Ref. [35].

3.2. Radius of Bound State

The mean radius  and the mean square velocity  of the bound states of charmonium meson, bottomonium mseon and  meson are calculated by using the Virial theorem [35]. The relation of the mean-kinetic energy and the potential energy is given by:

(34)

The present potential is, reduces

(35)

The mass relation

(36)

The mean square velocity

(37)

Table 4.  and  of charmonium.

State Parameter c <r>(GeV-1) <r2>
1 S 0.31384 1.92740 0.270037
1 P 0.62740 2.61297 0.390517
1 D 0.94121 3.16224 0.476976
2 S 1.36707 2.76119 0.495803
2 P 2.04345 3.32003 0.593240
3 S 0.56582 3.10281 0.434766

Table 5.  and  of bottomonium.

State Parameter c <r>(GeV-1) <r2>
1 S 0.51071 0.70781 0.109020
1 P 1.02557 1.13682 0.153330
1 D 1.53909 1.48158 0.190298
2 S 1.19244 1.18887 0.167441
2 P 1.82675 1.54697 0.209862
3 S 0.95280 1.32665 0.146140

Table 6.  and  of  meson.

State Parameter c <r>(GeV-1) <r2>
1 S 1.35685 0.67088 0.469706
1 P 2.71809 1.07872 0.603684
1 D 3.15324 1.12857 0.658530
2 S 4.82411 1.46948 0.763969
2 P 2.53006 1.25885 0.532638

In Tables (4, 5, and 6). The quantities.  and  are computed for different states of charmonium meson, bottomonium meson and -meson from Eqs. (36) and (37). The radius of  is larger than the radius of  and smaller than the radius of . In addition, the values of radii of , are located in the interval 0.1 to 1 fm. This in agreement with Ref. [36].

4. Summary and Conclusion

In the present study, we employ the analytical exact iteration method (AEIM) for determining the energy eigenvalues and the wave functions of the multi-dimensional radial Schrödinger equation with the extended Cornell potential. The charmonium mass, bottomonium mass and  meson mass are analytically obtained in the N-dimensional space and the special cases are obtained in comparison with other studies [2,4,27]. The effect of dimensional number is studied on the mass spectra of charmonium, bottomonium and  meson in Tables (1, 2, and 3). Increasing dimensional number increases charmonium mass, bottomonium mass, and  meson mass. The obtained results are in good agreement in comparison with Refs [2,4,27] and are in agreement with the experimental data. In addition, the mean radius  and the mean square velocity  of the bound states of charmonium meson, bottomonium meson and  meson are computed in Tables (4, 5, and 6). The radius of  is larger than the radius of  and smaller than the radius of. This observation refers to one of the characteristics of quarkonium that the heavy quarkonium have smaller radii.


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