1. Introduction

A quark is an elementary particle and a fundamental constituent of matter [1-4].

Quarks combine to form composite particles called hadrons [2, 3]. Quarks are never directly observed or found in isolation; they can be found only within hadrons, such as baryons and mesons [2, 3]. Quarks have various intrinsic properties, including electric charge, mass, color charge and spin [2, 3]. There are six types of quarks, known as flavors: up, down, strange, charm, top, and bottom. Up and down quarks, have the lowest masses of all quarks [1-3].

The heavier quarks rapidly change into up and down quarks through a process of particle decay: the transformation from a higher mass state to a lower mass state. Because of this, up and down quarks are generally stable and the most common in the universe, whereas strange, charm, bottom, and top quarks can only be produced in high energy collisions (such as those involving cosmic rays and in particle accelerators) [1-3]. For every quark flavor there is a corresponding type of antiparticle, known as an antiquark, that differs from the quark only in that some of its properties have equal magnitude but opposite sign [1-3, 5].

They are simple structure less, particles with fractional electric charge; point likeand spin-1/2 and they have positive parity, see table 1. Anti quarks have negative parity. The strong interaction between the quarks originates from the gluon field. Its field quanta are the gluons [1-3, 5]. Quarks do not interact with each other directly; they do so through intermediate agents called gluons which are massless, spin-1 particles.

Table 1. Characteristics of the quarks

2. Asymptotic Freedom

According to QCD (Quantum chromo dynamics) [1, 3, 5], the force between quarks is actually rather weak as long as the quarks are close together, closer than 1fermi. One Fermi is approximately the size of a proton) [6]. This weakness of the force between nearby quarks is called "asymptotic freedom". Asymptotic freedom is a property of the QCD vacuum, which describes how it responds to an "extra quark" [6]. Because of the weakness of the force at short distances and the fact that the b and c quarks are heavy anon relativistic approach is commonly used. Many potentials are adopted for such systems.

3. Theory and Calculations

In the present work, we will use non-relativistic Schrödinger equation because the quarks b and c are heavy. The Hamiltonian equation for a bound quark and anti quark in general is

(1)

the potentials used are

where the harmonic and an harmonic terms are responsible for the confinement of the bound particles and the 1/r potential for the interaction of the charges. The Hamiltonian equation becomes

(2)

and the Schrodinger equation is

(3)

(4)

the solution of this equation is presented in many bookse.g. see Ref. [7]. The solution is a product of the center of mass and relative motions. Here we are interested in the solution of the radial part R (r) of the relative motion [8].

The redused mass for the two particles of masses

The radial Schrödinger equation is [7]

(5)

let , hence the equation becomes

(6)

(7)

we get

(8)

Where the dot means a derivative with respect to r

(9)

and rearranging equation (8). We get

(10)

we will use and follow the asymptotic iteration method for eigenvalue problems [8].

Using the same notations, we define

The choice of r can be critical to the speed of the convergence to the eigenvalues, as well as for the stability of the process. Although, we do not have at the moment a specific method to determine the best initial value of r, so we may suggest the following approach:

We note that the ground state energy of this potential can be obtained by setting

we may therefore start our iteration with obtained from

(11)

where is a function of (y, r, z, l and x) , and

where is the angular momentum quantum number , and N is dimensionless, .

for system [2,3]: we will consider the different S-state levels using the available experimental total masses [9], we know that the binding energy is equal to the experimental energy minus M according to this equation . Hence, in order to get the binding energy of the levels one must find the coefficients (A, B, r, and w) in equation 11. To do that we will use the experimental data given for the lowest three states of the bound system [9].

First, we try to find (r)

case (1)

substituting in equation (11), we get

(12)

case (2)

substituting in equation (11), we get

(13)

subtracted equation (12) from (13), we get

(14)

Substituting in equation (13), we get

(15)

similarly, for the cases of the levels N=4 and 5, we get

(16)

and

(17)

we will subtract equation (15) from (16), to get

(18)

we will subtract equation (16) from (17)

(19)

we will subtract equation (18) from (19).

Finally, the coefficients of the potential using the first experimental lower levels of the bound states [10] by solving equations 15, 16 and 17 are obtained

(20)

(21)

(22)

Having obtained the parameters of the suggested potential, we will proceed to calculate the energy levels of the bound system.

Now, we will find the value , to start our iteration and find the eigenvalues of the bound states and compare them with the experimental data. Substituting the coefficients in equation (11)

in case (1)

we get

the solution of this equation is

in case (2)

we get

the solution of this equation is

substituting the value of in the equation (11) and obtain the value of the energy

(23)

in case (3)

we get

the solution of this equation is

substituting the value of in equation (11) and obtain the value of the energy

(24)

in case (4)

we get

the solution of this equation is

substituting the value of in equation (11) and obtain the value of the energy

(25)

in case (5)

we get

the solution of this equation is

substituting the value of in equation (11) and obtain the value of the energy

(26)

in case (6)

we get

the solution of this equation is

substituting the value of in equation (11) and obtain the value of the energy

(27)

In table (2) a comparison between the experimental data and theoretical results are presented.

Similarly in the system [2, 3]

Table 2

the coefficients of the potentials are as follows

(28)

(29)

(30)

In table (3) a comparison between the experimental data and theoretical results are shown.

Table 3

4. Conclusions

In the present work we used the result of the asymptotic iteration method for eigenvalue problems [8] to make fitting of the parameters of the potential with the experimental data of the bound and system. We chose the harmonic, an harmonic plus, Coulomb, potentials to solve the non-relativistic Schrodinger equation. The results give good agreement with the experimental data [9] with maximum error 2.8% and 6.4% .