Variational and exact solutions of the wavefunction

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Variational and exact solutions of the wavefunction at origin (WFO) for heavy quarkonium by using a global potential

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Phys. Scr. 80 (2009) 065003 (5pp)

doi:10.1088/0031-8949/80/06/065003

Variational and exact solutions of the wavefunction at origin (WFO) for heavy quarkonium by using a global potential G R Boroun and H Abdolmalki Department of Physics, Razi University, Kermanshah 67149, Iran E-mail: [email protected]

Received 24 August 2009 Accepted for publication 9 October 2009 Published 18 November 2009 Online at stacks.iop.org/PhysScr/80/065003 Abstract ¯ c¯c and b¯c massive quarks In this paper, we discuss the wavefunction at origin (WFO) for bb, systems. Using the variational method and an exact computation of the Schrödinger equation, we computed the WFO and also expectation values √ of h1/r i, hr i for massive mesons, while we had global potential in the form of V (r ) = k( r + αr ). We show that the final results of this potential include and cover all other results by other potential models (Cornell, Martin, logarithmic, Lichtenberg, etc). Also, the energy values obtained from this model have the highest level compared with other models. PACS numbers: 03.65.Ge, 12.39.Jh, 12.39.Pn

1. Introduction Maybe it is possible to solve any question in any scale using computers, but the important problem is to analyze the physical bases of exact solutions through approximation methods. Indeed, perturbation theory is an approximate way of problem solution that is used only when we know the solution of a question whose Hamiltonian is significantly similar to our solution. But there are a few physical systems with definite Hamiltonians that we cannot solve exactly, because there is no close relation between problems. Thus, these problems cannot be solved by the perturbation method, because this method is not very exact at the initial step. In addition, the WKB method is semiclassical, as it is not much applicable for quantum problems as well. Therefore, the variational method is a good tool for estimation of the base state energy and then the wavefunction at the origin [5]. This method does not require a particular Hamiltonian to be solved exactly. Besides, through this method, we can calculate the eigenfunction of energy systems that have a definite Hamiltonian and an indefinite eigen value. Our goal in this work is to use the variational method and to obtain the solution for massive quark systems. In order to do this, we need the WFO. We test its validity by comparing it with other results.

Nowadays massive quark systems have a great role in particle physics. It was believed that the top quark was very important, but, because of their short lifetime, they do not participate in hadron structures. Therefore, only c and b flavors are left to take part in the hadrons production interplay. In fact, because of the improvement and progress in laboratory data, massive hadron physics is becoming an important subject in particle physics [1]. New chances have been brought up to apply non-relativity methods that are successful for bounded states in high-energy physics, which have resulted in the discovery of massive flavor mesons of ψ (3 GeV c−2 ) and ϒ¨ (10 GeV c−2 ). As the ground state of heavy q and q¯ changes in non-relativity form, systems including massive quarks would construct bounded states. So, for studying massive hadrons, evaluation of the Schrödinger equation is very important [2]. This implies that the wavefunction at origin (WFO) has an immense role in the evaluation of quarkonium production and amplitude scattering and also in the fragmentation phenomenon of massive hadrons [3, 4]. Numerical methods are used just for a small number of potentials; hence the approximation method has a great role in evaluating functions at the ground state. 0031-8949/09/065003+05$30.00

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© 2009 The Royal Swedish Academy of Sciences

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Phys. Scr. 80 (2009) 065003

G R Boroun and H Abdolmalki

Therefore, the next step is to select a proper potential which is used for strong interaction. This potential has the following general form:

2. The variational method The Schrödinger equation for the massive quarks systems can be written as 1 (1) H |ψi = − 1 + V (r ) |ψi = E|ψi, 2µ

V (r ) = −ar −α + br β + C.

Indeed this potential changes to other potentials: Martin, logarithmic, Cornell, Lichtenberg, etc, if we define quantities α, β according to these values, that is, the

where µ is the reduced mass of the q¯q system. In the variational method computation, we do not try to solve problems in a quantitative way [5]. One of the uses of this method is to find the energy quantity eigenvalue and the approximate function eigenvalue. It is defined as follows: E(ψ) =

hψ|H|ψi = hH i. hψ|ψi

α = β = 1 the Cornell potential model [8]: α = β = 0.75 the Lichtenberg potential model [9]; α = β = 0.5 the Song–Liu potential model [10]; α = β → 0 the logarithmic potential model [11]; α = 0, β = 0.1 the Martin potential model [12].

(2)

With respect to the experimental data [13], we find the constant values as √ α V (r ) = k r + + c, k = 0.705 85, r α = 0.461 22, c = 8.817 15. (8)

In the above equation, if |ψi is related to the a parameter, then E(ψ) will also depend on this parameter. Note that for any ‘arbitrary’ trial function ψtrial , the energy E given by equation (2) is always larger than the exact energy E, E(ψ) =

hψ|H|ψi > E0. hψ|ψi

Here V(r) and r −1 are in GeV. In this potential model, m b and m c are 4.803 03 and 1.3959 GeV C−2 , respectively [13]. So the expected values of potential and kinetics based on quantities ‘a’ and ‘b’ are calculated as follows: Z Z ∞ r 2 ψ ∗ V (r )ψ dr hV i = ψ ∗ V (r )ψ d3r = 4π

(3)

Substituting equation (1) in equation (3) and carrying out integration, we obtain an expression for E that has to be minimized according to the variational parameter, and we can calculate this function according to the equation ∂E = 0. ∂ai

0

Z = 4π

(4)

∞

r 2 e−2ar

0

= 5.5979a 9/4 k

Finally, after putting ai taken from the variational method into the initial wavefunction, we can find the wavefunction at the origin and then h1/r i, hr i and also expectation values of energy.

b

h √ α i k r+ + c dr r

0.177 132 0.234 996α − a 21/8 a 3/2

+ c (9)

and T=

3. Calculations The main task is to determine the WFO. It is always possible to write a Hamiltonian like this for any physics system as

P2 −1 2 d d2 = + 2 2µ 2µ r dr dr Z 4πn 2 ∞ 2 −2ar b 2 d d2 b ⇒ hT i = − r e + 2 e−2ar dr 2µ 0 r dr dr = 0.052a 2/b (1 + b)e1.39/b

hH i = hT i + hV i.

(7)

(5)

0(1/b) 0(3/b)

(10)

where h¯ = c = 1 and 0(1 + α) = α0(α). An expression for the expectation energy value can be obtained by putting hT i, hV i from equations (9) and (10) into equation (5) and then minimizing this expectation value based on the variational variable. Therefore, using the normalization condition, we can determine the shape of the wavefunction. Finally, we obtain

So the first step is to calculate hV i and hT i. To do this, we should define an optional wavefunction and a potential proportionate to the question’s condition. A wavefunction proportionate to the question’s condition is one that would become zero at r → ∞ and at the origin is nonzero. So it has a Gaussian or exponential shape. Our selected wavefunction is [6] b ψtrial = N e−ar , (6)

N=

where N is a normalizing constant, ‘a’ is the variational parameter and ‘b’ is a constant related to our problem, which takes the following known forms in the limit state:

b(2a)3/b 4π 0(3/b)

1/2

,

(11)

which defines the normalization constant for the trial wavefunction based on the ‘a’ and ‘b’ variables. So we find the WFO as |ψ(0)|2 = N 2 = WFO. (12)

b = 1 → a hydrogen-like wavefunction, b = 2 → a harmonic oscillator wavefunction, b = 4/3 → the Bing Ding wavefunction [6], b = 3/2 → the Gupta wavefunction [7].

In figures 1–6, we compare our predictions with other model data [8–17] for WFO as a function of the ‘b’ value for systems 2

Phys. Scr. 80 (2009) 065003

G R Boroun and H Abdolmalki 2.0

25

Our results Cornell Martin Log

1.5

15

3

WFO (Gev )

3

WFO (GeV )

20


10

1.0

0.5

5

0

0.0 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.4

0.6

0.8

1.0

b

1.2

1.4

1.6

1.8

2.0

2.2

b

Figure 1. The WFO values against ‘b’ for the global potential compared with another potential for the bb¯ system.

Figure 3. The same as figure 1 but for the c¯c system .

1.8

Our results Martin Cornell Log

1.7 1.6


3.0

1.4 –1

〈r 〉 (GeV )

–1

〈 r 〉 (GeV )

1.5

1.3 1.2 1.1

2.5

2.0

1.0 0.9 0.8 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

1.5

b

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

b

Figure 2. The expectation value of hr i based on the global potential compared with other results for the bb¯ system.

Figure 4. The same as figure 2 but for the c¯c system.

¯ c¯c and b¯c). It can be observed including massive quarks (bb, from these figures that one can obtain good agreement with other results. In figures 7–9, we show the energy expectation ¯ c¯c and b¯c systems. We value as a function of ‘b’ for the bb, observe that our results with our global potential are higher than other results. This implies that the 2p state is between 1s and 2s states. Also, the b¯c system energy is between the systems bb¯ and c¯c in all cases.

3.0


2.5

3

WFO (GeV )

2.0

4. Exact solution of the Schrödinger equation

1.5

1.0

0.5

We solved the Schrödinger equation exactly using the Rang–Gutta method for systems containing massive quarks and then we obtained numerical values of the WFO and expectation values of E, hr i and h1/r i for 1s, 2s and 2p states. We used Mathematica software to do this [14], and our results are shown in tables 1–6 for all states and are compared with other results.

0.0 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

b

Figure 5. The same as figure 1 but for the b¯c system.

5. Conclusion looking for in physics. We have used the global potential for the investigation of the variational method. We have obtained the WFO and expectation values of hr i and h1/r i

As is shown in all the figures, there is more regular behavior in this model and this regular behavior is the thing we are 3

Phys. Scr. 80 (2009) 065003


3.0

20


15

E (GeV)

–1

〈r 〉 (GeV )

2.5


2.0

10

5 1.5

0 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.4

b

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

b

Figure 6. The same as figure 2 but for the b¯c system. Figure 9. The same as figure 7 but for the b¯c system. Table 1. The numerical values of the radial wavefunction at the origin, |R(0)|2 = |9(0)|2 /4π, for the 1s, 2s and 2p states of the bb¯ system with respect to the global potential and compared with other potential models.

20


E (GeV)

15

10

|R(0)|2 for bb¯

1s (GeV3 )

2s (GeV3 )

2p (GeV3 )

Our present work QCD [15] Martin [16] Log [8] Cornell [17] Buchmuller [18] Lichtenberg [18]

5.97601 6.477 4.591 4.916 14.05 6.256 6.662

5.98 3.234 2.571 2.532 5.668 3.086 3.370

2.7042 1.417 1.572 1.535 2.067 – –

5

Table 2. The same as table 1 but for the c¯c system. 0 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

b

Figure 7. The expectation value of hEi based on the global potential compared with other results for the bb¯ system.

1s (GeV3 )

2s (GeV3 )

2p (GeV3 )

Our present work QCD [15] Martin [16] Log [8] Cornell [17] Buchmuller [18] Lichtenberg [18]

0.50414 0.810 0.999 0.815 1.454 0.794 1.121

0.50366 0.529 0.559 0.418 0.927 0.517 0.693

0.02670 0.075 0.125 0.078 0.131 – –

Table 3. The same as table 1 but for the b¯c system.

20


15

E (GeV)

|R(0)|2 for c¯c

10

|R(0)|2 for b¯c

1s (GeV3 )

2s (GeV3 )

2p (GeV3 )

Our work QCD [15] Martin [16] Log [8] Cornell [17] Buchmuller [18] Lichtenberg [18]

1.21126 1.642 1.710 1.508 3.184 1.603 2.128

1.21078 0.983 0.950 0.770 1.764 0.953 1.231

0.267802 0.201 0.327 0.239 0.342 – –

5

Table 4. The expectation values of hEi, hr i and h1/r i obtained from the global potential for the bb¯ system. 0 0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

b

Figure 8. The same as figure 7 but for the c¯c system.

4

bb¯ state

r (GeV−1 )

1/r (GeV)

E (GeV)

1s 2s 2p

1.823 34 3.100 41 2.446 48

0.685 79 0.486 239 0.467 284

10.093 10.4006 10.2527

Phys. Scr. 80 (2009) 065003


References

Table 5. The same as table 4 but for the c¯c system. c¯c state

r (GeV−1 )

1/r (GeV)

E (GeV)

1s 2s 2p

2.618 6 4.761 12 3.751 84

0.491 518 0.325 336 0.307 043

10.2886 10.7333 10.5493

[1] Abachi S et al (DO Collaboration) 1994 Phys. Rev. Lett. 72 2138 Abachi S et al (DO Collaboration) 1995 Phys. Rev. Lett. 74 2032 Smith M C and Willenbrock S S 1997 Phys. Rev. Lett. 79 3825 [2] Quigg C and Rosner J L 1979 Phys. Rep. 56 167 [3] Braaten E and Cheung K 1995 Phys. Rev. D 51 4819 [4] Eichten E and Quigg C 1994 Phys. Rev. D 49 5845 [5] Sukurai J J 1994 Modern Quantum Mechanics (Reading, MA: Addison-Wesley) Zettili N 2001 Modern Quantum Mechanics (New York: Wiley) [6] Ding Y B, Li X Q and Shen P N 1999 Phys. Rev. D 60 074010 [7] Gupta S N, Radford S F and Repko W W 1982 Phys. Rev. D 49 1551 [8] Eichten E et al 1975 Phys. Rev. Lett. 34 369 [9] Lichtenberg D B et al 1989 Z. Phys. C 4 1615 [10] Song X T and Liu H 1987 Z. Phys. C 34 223 [11] Quigg C and Rosner J L 1977 Phys. Lett. B 71 153 [12] Martin A 1981 Phys. Lett. B 100 511 [13] Motyka L and Zalewski K 1996 Z. Phys. C 69 343 Zalewski K 1998 Acta Phys. Polon. B 29 1 [14] Lucha W and Schorberg F F 1999 Int. J. Mod. Phys. C 10 607 [15] Buchmüller W and Tye S-H H 1981 Phys. Rev. D 24 132 [16] Martin A 1980 Phys. Lett. B 93 338 [17] Eichten E et al 1978 Phys. Rev. D 17 3090 [18] Collins S J, Imbo T D, Alex King B and Martell E C 1997 Phys. Lett. B 393 155

Table 6. The same as table 4 but for the b¯c system. b¯c state

r (GeV−1 )

1/r (GeV)

E (GeV)

1s 2s 2p

2.282 29 4.069 55 3.208 02

0.559 254 0.377 957 0.358 344

10.2083 10.6014 10.4313

from the variational and exact solutions of the Schrödinger equation and then compared these variational results with other potential models. Hence, this desired potential is a suitable potential for strong interactions. It is clear from the figures that our results for the global potential are comparable with the Cornell–logarithmic–Martin potential models. As presented in all tables, our results for WFO by the exact solution of the Schrödinger equation are within the span of results from other potentials. Also, the energy expectation values for this potential have the highest level compared with other models.

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