research article exact solution of the dirac equation for the ...er a new approach for the solving...

Hindawi Publishing CorporationISRN High Energy PhysicsVolume 2013 Article ID 310392 6 pageshttpdxdoiorg1011552013310392

Research ArticleExact Solution of the Dirac Equation for the Yukawa Potentialwith Scalar and Vector Potentials and Tensor Interaction

Mona Azizi1 Nasrin Salehi1 and Ali Akbar Rajabi2

1 Department of Basic Science Shahrood Branch Islamic Azad University Shahrood 36199-43189 Iran2 Physics Department Shahrood University of Technology PO Box 3619995161-316 Shahrood Iran

Correspondence should be addressed to Nasrin Salehi salehishahroodutacir

Received 22 January 2013 Accepted 6 February 2013

Academic Editors K Cho and A Koshelev

Copyright copy 2013 Mona Azizi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We present exact solutions of the Dirac equation with Yukawa potential in the presence of a Coulomb-like tensor potential For thisgoal we expand the Yukawa form of the nuclear potential in its mesonic clouds by using Taylor extension to the power of seventhand bring out its simple form In order to obtain the energy eigenvalue and the corresponding wave functions in closed forms forthis potential (with great powers and inverse exponent) we use ansatz methodWe also regard the effects of spin-spin spin-isospinand isospin-isospin interactions on the relativistic energy spectra of nucleon By using the obtained results we have calculated thedeuteron mass The results of our model show that the deuteron spectrum is very close to the ones obtained in experiments

1 Introduction

It is well known that the exact energy eigenvalues of thebound state play an important role in quantum mechanicsIn particular the Dirac equation which describes the motionof a spin-12 particle has been used in solving many problemsof nuclear and high-energy physics Recently there has beenan increase in searching for analytic solution of the Diracequation [1ndash11] Recently tensor couplings have been usedwidely in the studies of nuclear properties [12ndash22] and theywere introduced into the Dirac equation by substitution

119875 rarr

119875minus119894119898120596120573sdot 119880(119903) [16 23] where119898 is one of the particlesmassand120596 refers to harmonic oscillator In this work we are goingto solve the relativistic Dirac equation for Yukawa potentialin fresh approaches We first discuss the Dirac equationwith a tensor potential for Yukawa potential Then we solvethe resulting equations for deuteron nuclei and obtain theparameters for it We are also able to number the mass ofDeuteron by using obtained parameters and the equation isin conformity with experimental measure

2 Yukawa Potential

The screened Coulomb potential also known as the Yukawapotential in atomic physics and theDebye-Huckel potential in

plasma physics is of interest in many areas of physics It wasoriginally used tomodel strong nucleon-nucleon interactionsdue tomeson exchange in nuclear physics by Yukawa [19 20]It is also used to represent a screened Coulomb potentialdue to the cloud of electronic charges around the nucleus inatomic physics or to account for the shielding by outer chargesof the Coulomb field experienced by an atomic electron inhydrogen plasmaHowever the Schrodinger equation for thispotential cannot be solved exactly hence various numericaland pertubative methods have been devised to obtain theenergy levels and related physical quantities [24]The genericform of this potential is given by

119880 (119909) = minus

119892

119909

119881 (119909) 119892 gt 0 (1)

that 119892 is the potential strength and

119881 (119909) = 119890

minus119896119909

(2)

In this part we expand the Yukawa potential in its mesonclouds 119909 = 119886 by using Taylor extension to the power ofseventh

2 ISRN High Energy Physics

119880 (119909) = minus

119892

119909

[119890

minus119896119886

minus 119896119890

minus119896119886

(119909 minus 119886) +

119896

2

2

119890

minus119896119886

(119909 minus 119886)

2

minus

119896

3

3

119890

minus119896119886

(119909 minus 119886)

3

+

119896

4

4

119890

minus119896119886

(119909 minus 119886)

4

minus

119896

5

5

119890

minus119896119886

(119909 minus 119886)

5

+

119896

6

6

119890

minus119896119886

(119909 minus 119886)

6

minus

119896

7

7

119890

minus119896119886

(119909 minus 119886)

7

+ sdot sdot sdot]

(3)

We can reduce this equation to

119880 (119909) = 119886119909

6

minus 119887119909

5

+ 119888119909

4

minus 119889119909

3

+ 119890119909

2

minus 119891119909 + ℎ minus

119871

119909

(4)

Now we have the general Yukawa equation that 119886 119887 119888 119889 119890119891 ℎ and 119871 are

119886 =

119896

7

7

119887 = minus (

7

7

119896

6

+

7

7

119896

6

)

119888 = (

21

7

119896

5

+

6

6

119896

5

+

1

5

119896

5

)

119889 = minus (

35

7

119896

4

+

15

6

119896

4

+

5

5

119896

4

+

1

3

119896

4

)

119890 = (

35

7

119896

3

+

20

6

119896

3

+

10

5

119896

3

+

6

4

119896

3

+

3

3

119896

3

+

1

2

119896

3

)

119891 = minus (

21

7

119896

2

+

15

6

119896

2

+

10

5

119896

2

+

6

4

119896

2

+

3

3

119896

2

+

1

2

119896

2

)

ℎ = (

7

7

119896 +

6

6

119896 +

5

5

119896 +

4

4

119896 +

3

3

119896 +

1

2

119896 + 119896)

119871 = minus (

7

7

+

6

6

+

5

5

+

4

4

+

3

3

+

1

2

+ 2)

(5)

3 Exact Analytical Solution of the DiracEquation for Yukawa Potential

We are interested in the particular solution of the Dirac equa-tionwith the Yukawa interaction potential (4) In this sectionwe anticipate the answer and then obtain the parameters forhighest nuclei deuteron A deuteron consists of a neutronand a proton Let us assume one nucleon is fixed and anothernucleon ismoving around the center ofmass At last by usingsingle particle model we investigate the effects of one pionexchange for a single nucleon The Dirac Hamiltonian is

119867 = sdot119875 + 120573 (119898 + 119878) + 119881 minus 119894120573 sdot 119880 (6)

where 119878 is the scalar potential119881 is the vector potential and119880is the tensor partWe have Δ = 119881minus119878 andΣ = 119881+119878TheDiracequation can be solved exactly for the cases Δ = 0 and Σ = 0 sdot

= (

0120590

120590 0

) and 120573 = ( 119868 00 minus119868

)which are the usual Dirac matricesWewill indicate upper and lower components of Dirac spinorwith Ψ

119860

and Ψ119861

Expressing the Dirac matrices in terms ofthe Paulimatrices we obtain the following coupled equations[25 26]

sdot119875Ψ119860

+ (119881 minus 119878 minus 119898)Ψ119861

+ 119894 sdot 119909119880 (119909)Ψ119860

= 119864Ψ119861

(7a)

sdot119875Ψ119861

+ (119881 + 119878 + 119898)Ψ119860

minus 119894 sdot 119909119880 (119909)Ψ119861

= 119864Ψ119860

(7b)

We assume that 119881 119878 and 119880 are radial potentials Using theidentity

sdot119875 =

( sdot )

119909

2

( sdot ) ( sdot119875) =

( sdot )

119909

2

(minus119894119909

120597

120597119909

+ 119894 sdot119871)

(8)

With (7a) and (7b) we find the following second-orderdifferential equations for Ψ

119860

and Ψ119861

119875

2

Ψ119860

+ (119880

2

+

119889119906

119889119909

+

2119906

119909

+

(119889Δ119889119909)

(119864 + 119898 minus Δ)

119880)Ψ119860

+ (4119880 + 2

(119889Δ119889119909)

(119864 + 119898 minus Δ)

)(

119878 sdot

119871

2

)Ψ119860

minus

(119889Δ119889119909)

(119864 + 119898 minus Δ)

120597Ψ119860

120597119909

= (119864 + 119898 minus Δ) (119864 + 119898 minus Σ)Ψ119860

(9a)

119875

2

Ψ119861

+ (119880

2

minus

119889119906

119889119909

minus

2119906

119909

minus

(119889Σ119889119909)

(119864 minus 119898 minus Σ)

119880)Ψ119861

+ (minus4119880 + 2

(119889Σ119889119909)

(119864 minus 119898 minus Σ)

)(

119878 sdot

119871

2

)Ψ119861

+

(119889Σ119889119909)

(119864 minus 119898 minus Δ)

120597Ψ119861

120597119909

= (119864 + 119898 minus Δ) (119864 minus 119898 minus Σ)Ψ119861

(9b)

Here 119878 stands for the (12) spin operator and

119871 for orbitalangular momentum operator The Hamiltonian operatorcommutes with the total angular momentum operator whichis given by

119869 =119871 +

119878 and with the parity operator Therefore

the eigenfunctions defined by Hamiltonian can be expressedas

Ψ119899119895119898119896

= (

119894

119892119899119896

(119909)

119909

120601

119897

119895119898119896

()

119891119899119896

(119909)

119909

120601

119897

119895119898(minus119896)

()

) (10)

where 120601119897119895119898119896

() denotes the spin spherical harmonics Thequantum number 119896 is related to ℓ and 119895 as follows

119896 =

minus (ℓ + 1) = minus (119895 +

1

2

) 119895 = ℓ +

1

2

ℓ = + (119895 +

1

2

) 119895 = ℓ minus

1

2

(11)

ISRN High Energy Physics 3

For spin spherical harmonics we have

sdot 120601

119897

119895119898119896

= 120601

119897

119895119898(minus119896)

(12)

Equations (9a) and (9b) reduced to a set of coupled equationsfor the radial wave functions 119892

119896

and 119891120581

(

119889

2

119889119909

2

minus

119896 (119896 + 1)

119909

2

+

2119896

119909

119880 minus

119889119880

119889119909

minus 119880

2

)119892119896

(119909)

+

119889Δ119889119909

(119864 + 119898 minus Δ)

(

119889

119889119909

+

119896

119909

minus 119880)119892119896

(119909)

= minus (119864 + 119898 minus Δ) (119864 minus 119898 minus Σ) 119892119896

(119909)

(13a)

(

119889

2

119889119909

2

minus

119896 (119896 minus 1)

119909

2

+

2119896

119909

119880 +

119889119880

119889119909

minus 119880

2

)119891119896

(119909)

+

119889Σ119889119909

(119864 minus 119898 minus Δ)

(

119889

119889119909

minus

119896

119909

+ 119880)119891119896

(119909)

= minus (119864 + 119898 minus Δ) (119864 minus 119898 minus Σ)119891119896

(119909)

(13b)

Using the following relations [13]

2119878 sdot

119871 =

119869

2

minus119871

2

minus119878

2

sdot119871120601

119897

119895119898119896

= minus (119896 + 1) 120601

119897

119895119898119896

sdot119871120601

119897

119895119898(minus119896)

= minus (119896 + 1) 120601

119897

119895119898(minus119896)

(14)

for particle of119898mass and 119864 energy we have

119878 = 119881 =

1

2

Σ

Σ (119909) = 119886119909

6

minus 119887119909

5

+ 119888119909

4

minus 119889119909

3

+ 119890119909

2

minus 119891119909 + ℎ minus

119871

119909

Δ (119909) = 0

119880 (119909) = minus

1

119909

(15)

We assume the case 119878(119909) = 119881(119909) [27ndash31] and consider 119896(119896 +1) = ℓ(ℓ+1) then the upper component in (13a) is as follows

(

119889

2

119889119909

2

minus

ℓ (ℓ + 1) + 2 + 2119896

119909

2

minus Σ (119903) (119864 + 119898))119892119896

(119909)

= (119898

2

minus 119864

2

) 119892119896

(119909)

(16)

As we know in deuteron two nucleons have in D-state so

[

119889

2

119889119909

2

minus

ℓ (ℓ + 1) + 2 minus 2 (ℓ + 1)

119909

2

]119892119896

(119909)

= ((1198861

119909

6

minus1198871

119909

5

+1198881

119909

4

minus1198891

119909

3

+1198901

119909

2

minus1198911

119909+ℎ1

minus

1198711

119909

)

minusE)119892119896

(119909)

(17)

Therefore we have a Schrodinger-like equation [31]

119892

10158401015840

1

(119909) = [(1198861

119909

6

minus1198871

119909

5

+1198881

119909

4

minus1198891

119909

3

+1198901

119909

2

minus1198911

119909+ℎ1

minus

1198711

119909

)

minusE +

ℓ (ℓ minus 1)

119909

2

]

(18)

By setting

119886 (1198641

+ 119898) = 1198861

119887 (1198641

+ 119898) = 1198871

119888 (1198641

+ 119898) = 1198881

119889 (1198641

+ 119898) = 1198891

(19)

119890 (1198641

+ 119898) = 1198901

119891 (1198641

+ 119898) = 1198911

ℎ (1198641

+ 119898) = ℎ1

119871 (1198641

+ 119898) = 1198711

(1198641

2

minus 119898

2

) = E

(20)

In order to solve (18) we suppose the following form for thewave function

119892 (119909) = 119872 (119909) 119890

119885(119909)

(21)

Now for the functions119872(119909) and119885(119909) respectively we makeuse of the following ansatz [27 32 33]

119872(119909) =

1 if ground stateV

prod

119894=1

(119909 minus 119886

V119894

) if V gt 0

119885 (119909) = minus

1

4

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909 + 120575 ln119909

(22)

For a particular grand angular quantum number 120574 thereare different solutions which are labeled by 120592 (120592 determinesthe number of the nodes of the wave function) By substitu-tion of119872(119909) and 119885(119909) into (21) and then taking the second-order derivative of the obtained equation we can get

119892

10158401015840

(119909) = [119885

10158401015840

+ 119885

10158402

+

119872

10158401015840

+ 2119872

1015840

119885

1015840

119872

] (23)

We consider the ground state which is called the 0th nodesolution of the above differential equation By equating (18)and (23) for 120592 = 0 it can be found that

120572 = radic1198861

120573 = minus

1198871

2

radic1198861

120578 =

1198881

minus 1205731

2120572

120591 =

minus1198891

minus 2120573120578

2120572

120575 = ℓ minus 1 E = 120578 (1 minus 2120575) minus 120591

2

minus ℎ1

(24)

The upper component is as follows

1198921

(119909) = 1198730

119909

ℓminus1 exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909) (25)

and another component of wave function is

1198911

(119909) =

( sdot119875 + 119894 sdot 119880)

(119864 + 119898)

1198921

(119909) (26)


By substituting (25) into (26) we obtained the followingequation

1198911

(119909) =

minus119894 sdot

(119864 + 119898)

[

119889

119889119909

minus 119894 sdot119871 minus 119880]119873

0

119909

ℓminus1

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909)

(27)

Therefore the total wave function is

Ψ = 1198730

(

1

minus119894 sdot

(119864 + 119898)

[120572119909

3

+ 120573119909 + 120578119909 + 120591 +

1

119909

]

)

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578 minus 120591119909)

(28)

Now we have to obtain parameters 119886 119887 119888 119889 119890 119891 ℎ and 119871 byusing of (5)

119887 =

14

119896

119886 119888 =

105

119896

2

119886 119889 =

440

119896

3

119886 119890 =

2555

119896

4

119886

119891 =

6846

119896

5

119886 ℎ =

8659

119896

6

119886

(29)

We know that 119896 = 1119909 = 0714 (fmminus1) [34] by using of (24)we can get amount of parameters 120572 120573 120578 and 120591

120572 = radic1198861

120573 = minus98radic1198861

120578 = 8308radic1198861

120591 = 8308radic1198861

(30)

The bonding energy for deuteron is reported 2224MeV [35]This is useful for calculating the numeric values of parametersin (30)

120576 = 119864

2

minus 119898

2

= 120578 (1 minus 2120575) minus 120591

2

minus ℎ1

(31)

By substituting the amounts of (31) and ℓ = 1 we obtainedthe numeric values of parameters in (30) The parameters arereported in Table 1 while the parameters of Yukawa potentialare reported in Table 2

4 The Hyperfine Interaction

As the Baryons have spin and isospin the deuteron neutronand proton can interact together We introduce the hyperfineinteraction potential The complete potential is

⟨119867int (119909)⟩ = 119881 (119909) + ⟨119867119878 (119909)⟩ + ⟨119867119868 (119909)⟩ + ⟨119867119878119868 (119909)⟩ (32)

In this work we have added the hyperfine interactionpotentials (119867

119878

(119909) 119867119868

(119909) and119867119878119868

(119909)) which yield propertiesvery close to the experimental results By regarding 119881(119909) asthe nonpertubative potential and the other terms in (32) aspertubative ones according to this explanation The spin andisospin potential contains a 120575-like term We have modified it

by a Gaussian function of the nucleons pair relative distance[36]

119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878sum(

119878119894

sdot119878119895

) (33)

where 119878119894

is the spin operator of the 119894th nucleon and 119909 is therelative nucleon pair coordinate 119860

119904

and 120590119878

are constants120590119878

= 287 fm and 119860119878

= 674 fm2 [34 37] We know that thedeuteron consists of two nucleons Where 1 indicate the firstnucleon and 2 indicate the second nucleon We have

119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878(

1

2

) [119878

2

minus 119878

2

1

minus 119878

2

2

] (34)

Furthermore we add two hyperfine interaction terms to theHamiltonian nucleon pairs similar to (33) [38]

119867119868

= 119860119878

1

(radic120587120590119868

)

3

119890

minus119909

2120590

2

119868sum(

119868119894

sdot119868119895

) (35)

where 119868119894

is the isospin operator of the 119894th nucleon and theconstants are described as 120590

119868

= 345 fm and 119860119868

= 517 fm2[37 39] The another one is a spin-isospin interaction givenby similar (33) [38 40]

119867119878119868

= 119860119878119868

1

(radic120587120590119878119868

)

3

119890

minus119909

2120590

2

119878119868sum(

119878119894

sdot119878119895

) (119868119894

sdot119868119895

) (36)

The fitted parameters are 120590119878119868

= 231 fm and119860119878119868

= minus1062 fm2[35 41] where 120595

120574

is the perturbed wave function and wewrite it as

120595120574

= 120595

1015840

120574

+ sum

120574

1015840=120574

⟨120595

0

120574

1015840 | 119867int | 1205950

120574

1015840⟩120595

10158400

120574

1015840

119864

0

120574

minus 119864

10158400

120574

(37)

The deuteronmass are given by two nucleonsmasses and theeigenenergies of the Dirac equation119864 (119864 is a function of 120578 120575120591 ℎ119898) with the first-order energy correction from potential119867int can be obtained by using the unperturbed wavefunction(33) (35) and (36)The total potential for the ground state aswell as the other states can be written as [40 42]

⟨119867int⟩ =int120595

lowast

120574

119867in1205951205741199092

119889119909 119889Ω

int120595

lowast

120574

120595120574

119909

2

119889119909 119889Ω

(38)

We first assume 120574 = 0 The potentials can be extracted from(33) (35) and (36)

119867119878

= 010059 (MeV)

119867119868

= 004782 (MeV)

119867119878119868

= minus050053 (MeV)

(39)

Now we can calculate the deuteron mass according to thefollowing formula

119872119863

= 119898119899

+ 119898119901

+ 119864120592120574

+ ⟨119867int⟩ (40)


Table 1 The best adaption with experimental value is 1198861

= 306136 120572 = 174967 120573 = minus17146815 120578 = 145362999 120591 =

minus14102380 and 119864 (MeV) = 219020

1198861

120572 120573 120578 120591 119864 (MeV)306131 174965 minus17146628 145361420 minus14102227 289658306134 174967 minus17146785 145362749 minus14102356 249699306135 174967 minus17146805 145362915 minus14102372 234861306136 174967 minus17146815 145362999 minus1410238 21902030638 174967 minus17147648 145377006 minus1410234 183278

Table 2 The value of the Yukawa potential parameters

119886 (fmminus7) 119887 (fmminus6) 119888 (fmminus5) 119889 (fmminus4) 119890 (fmminus3) 119891 (fmminus2) ℎ (fmminus1)021961 430589 4524045 26553556 21597519 81092235 143625603

By substituting119898119899

= 119898119901

= 938 (MeV) = 465 (fmminus1) energyeigenvalue (119864

120592120574

) and the expectation values of119867int in (40) wehave

119872119863

= 187783808 (MeVc2) (41)

By comparing the experimental amount of deuteronmass (119872

119863

= 1875612 (MeVc2) = 929 (fmminus1)) with ourcalculatedmass for deuteronwe found that a good agreementhas been obtained by our model [35]

5 Conclusion

In this work we offer a new approach for the solving ofYukawa potential We expand this potential around of itsmesonic cloud that gets a new form with great powers andinverse exponent We calculate wave function of the Diracequation for a new form of Yukawa potential including acoulomb-like tensor potential By considering the effects ofpertubative interaction potential we see that the theoreticaland experimental masses are in complete agreement Theseimprovements in reproduction of deuteron mass is obtainedby using a suitable form for confinement potential and exactanalytical solution of the Dirac equation for our proposedpotential Finally one can use this model for another nucleiby relativistic or nonrelativistic equation and get the proper-ties of them and the amount of ldquo119892rdquo the strength of nuclearforce

References

[1] H Motavali ldquoBound state solutions of the Dirac equationfor the Scarf-type potential using Nikiforov-Uvarov methodrdquoModern Physics Letters A vol 24 no 15 pp 1227ndash1236 2009

[2] M R Setare and Z Nazari ldquoSolution of Dirac equations withfive-parameter exponent-type potentialrdquo Acta Physica PolonicaB vol 40 no 10 pp 2809ndash2824 2009

[3] L-H Zhang X-P Li and C-S Jia ldquoAnalytical approximationto the solution of the Dirac equation with the Eckart potentialincluding the spin-orbit coupling termrdquo Physics Letters A vol372 no 13 pp 2201ndash2207 2008

[4] C-L Ho ldquoQuasi-exact solvability of Dirac equation withLorentz scalar potentialrdquo Annals of Physics vol 321 no 9 pp2170ndash2182 2006

[5] H Panahi and L Jahangiry ldquoShape invariant and Rodriguessolution of the Dirac-shifted oscillator and Dirac-Morse poten-tialsrdquo International Journal of Theoretical Physics vol 48 no 11pp 3234ndash3240 2009

[6] A D Alhaidari ldquo1198712 series solution of the relativistic Dirac-Morse problem for all energiesrdquo Physics Letters A vol 326 no1-2 pp 58ndash69 2004

[7] F D Adame and M A Gonzalez ldquoSolvable linear potentials inthe Dirac equationrdquo Europhysics Letters vol 13 no 3 pp 193ndash198 1990

[8] A S de Castro and M Hott ldquoExact closed-form solutions ofthe Dirac equation with a scalar exponential potentialrdquo PhysicsLetters A vol 342 no 1-2 pp 53ndash59 2005

[9] Y Chargui A Trabelsi and L Chetouani ldquoExact solution of the(1+1)-dimensional Dirac equationwith vector and scalar linearpotentials in the presence of a minimal lengthrdquo Physics LettersA vol 374 no 4 pp 531ndash534 2010

[10] A S de Castro ldquoBound states of theDirac equation for a class ofeffective quadratic plus inversely quadratic potentialsrdquo Annalsof Physics vol 311 no 1 pp 170ndash181 2004

[11] J-Y Guo J Meng and F-X Xu ldquoSolution of the Dirac equationwith special Hulthen potentialsrdquoChinese Physics Letters vol 20no 5 pp 602ndash604 2003

[12] H Akcay and C Tezcan ldquoExact solutions of the dirac equationwith harmonic oscillator potential including a coulomb-liketensor potentialrdquo International Journal ofModern Physics C vol20 no 6 pp 931ndash940 2009

[13] H Akcay ldquoDirac equation with scalar vector quadratic poten-tials and coulomb-like tensor potentialrdquo Physics Letters A vol373 pp 616ndash620 2009

[14] O Aydogdu and R Sever ldquoExact pseudospin symmetric solu-tion of the Dirac equation for pseudoharmonic potential in thepresence of tensor potentialrdquo Few-Body Systems vol 47 no 3pp 193ndash200 2010

[15] M Eshghi ldquoDirac-Hua problem including a Coulomb-liketensor interactionrdquo Advanced Studies inTheoretical Physics vol5 no 12 pp 559ndash571 2011

[16] G Mao ldquoEffect of tensor couplings in a relativistic Hartreeapproach for finite nucleirdquo Physical Review C vol 67 Article ID044318 12 pages 2003

[17] P Alberto R Lisboa M Malheiro and A S de Castro ldquoTensorcoupling and pseudospin symmetry in nucleirdquo Physical ReviewC vol 71 Article ID 034313 2005


[18] R J Furnstahl J J Rusnak and B D Serot ldquoThe nuclear spin-orbit force in chiral effective field theoriesrdquo Nuclear Physics Avol 632 no 4 pp 607ndash623 1998

[19] H Yukawa ldquoOn the interaction of elementary particles IrdquoProceedings of the Physico-Mathematical Society of Japan vol 17pp 48ndash57 1935

[20] H Yukawa ldquoOn the interaction of elementary particles IIrdquoProceedings of the Physico-Mathematical Society of Japan vol 19pp 1084ndash1093 1937

[21] S Zarrinkamar A A Rajabi and H Hassanabadi ldquoDiracequation for the harmonic scalar and vector potentials andlinear plus Coulomb-like tensor potential the SUSY approachrdquoAnnals of Physics vol 325 no 11 pp 2522ndash2528 2010

[22] H Hassanabadi E Maghsoodi S Zarrinkamar and H Rahi-mov ldquoDirac equation for generalized Poschl-Teller scalarand vector potentials and a Coulomb tensor interaction byNikiforov-Uvarov methodrdquo Journal of Mathematical Physicsvol 53 no 2 Article ID 022104 2012

[23] M Moshinsky and A Szczepaniak ldquoThe Dirac oscillatorrdquoJournal of Physics A vol 22 no 17 pp L817ndashL819 1989

[24] H Bahlouli M S Abdelmonem and I M Nasser ldquoAnalyticaltreatment of the Yukawa potentialrdquo Physica Scripta vol 82Article ID 065005 2010

[25] M R Shojaei and A A Rajabi ldquoHypercentral constituent quarkmodel and the hyperfine dependence potentialrdquo Iranian Journalof Physics Research vol 7 no 2 2007

[26] A A Rajabi ldquoA three-body force model for the harmonic andanharmonic oscillatorrdquo Iranian Journal of Physics Research vol5 no 2 2005

[27] N Salehi and A A Rajabi ldquoProton static properties by usinghypercentral constituent quark model and isospinrdquo ModernPhysics Letters A vol 24 no 32 pp 2631ndash2637 2009

[28] A D Alhaidari H Bahlouli and A Al-Hasan ldquoDirac andKlein-Gordon equations with equal scalar and vector poten-tialsrdquo Physics Letters A vol 349 no 1ndash4 pp 87ndash97 2006

[29] A A Rajabi ldquoHypercentral constituent quark model andisospin for the baryon static propertiesrdquo Journal of SciencesIslamic Republic of Iran vol 16 pp 73ndash79 2005

[30] N Salehi and A A Rajabi ldquoA new method for obtaining thespectrum of baryon resonances in the Killingbeck potentialrdquoPhysica Scripta vol 85 Article ID 055101 2012

[31] C-Y Chen ldquoExact solutions of the Dirac equation with scalarand vector Hartmann potentialsrdquo Physics Letters A vol 339 no3-5 pp 283ndash287 2005

[32] A A Rajabi ldquoA method to solve the Schrodinger equationfor any power hypercentral potentialsrdquo Communications inTheoretical Physics vol 48 no 1 pp 151ndash158 2007

[33] N Salehi A A Rajabi and Z Ghalenovi ldquoSpectrum of strangeand nonstrange baryons by using generalized gursey radicatimass formula and hypercentral potentialrdquoActa Physica PolonicaB vol 42 no 6 pp 1247ndash1259 2011

[34] S S M Wong ldquoIntroductory Nuclear Physicsrdquo John Wiley ampSons 1998

[35] N Salehi A A Rajabi and Z Ghalenovi ldquoCalculation ofnonstrange baryon spectra based on symmetry group theoryand the hypercentral potentialrdquo Chinese Journal of Physics vol50 no 1 p 28 2012

[36] L Y Glozman W Plessas K Varga and R F WagenbrunnldquoUnified description of light- and strange-baryon spectrardquoPhysical Review D vol 58 no 9 Article ID 094030 1998

[37] M De Sanctis M M Giannini E Santopinto and A VassalloldquoElectromagnetic form factors in the hypercentral CQMrdquo Euro-pean Physical Journal A vol 19 no 1 pp 81ndash85 2004

[38] L Y Glozman and D O Riska ldquoThe spectrum of the nucleonsand the strange hyperons and chiral dynamicsrdquo Physics Reportvol 268 no 4 pp 263ndash303 1996

[39] M F de la Ripelle ldquoA confining potential for quarksrdquo NuclearPhysics A vol 497 pp 595ndash602 1989

[40] MM Giannini E Santopinto and A Vassallo ldquoAn overview ofthe hypercentral Constituent QuarkModelrdquo Progress in Particleand Nuclear Physics vol 50 no 2 pp 263ndash272 2003

[41] H Hassanabadi ldquoA simple physical approach to study spectrumof baryonsrdquo Communications in Theoretical Physics vol 55 no2 p 303 2011

[42] HHassanabadi A A Rajabi and S Zarrinkamar ldquoSpectrum ofbaryons and spin-isospin dependencerdquo Modern Physics LettersA vol 23 no 7 pp 527ndash537 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


119880 (119909) = minus

119892

119909

[119890

minus119896119886

minus 119896119890

minus119896119886

(119909 minus 119886) +

119896

2

2

119890

minus119896119886

(119909 minus 119886)

2

minus

119896

3

3

119890

minus119896119886

(119909 minus 119886)

3

+

119896

4

4

119890

minus119896119886

(119909 minus 119886)

4

minus

119896

5

5

119890

minus119896119886

(119909 minus 119886)

5

+

119896

6

6

119890

minus119896119886

(119909 minus 119886)

6

minus

119896

7

7

119890

minus119896119886

(119909 minus 119886)

7

+ sdot sdot sdot]

(3)

We can reduce this equation to

119880 (119909) = 119886119909

6

minus 119887119909

5

+ 119888119909

4

minus 119889119909

3

+ 119890119909

2

minus 119891119909 + ℎ minus

119871

119909

(4)

Now we have the general Yukawa equation that 119886 119887 119888 119889 119890119891 ℎ and 119871 are

119886 =

119896

7

7

119887 = minus (

7

7

119896

6

+

7

7

119896

6

)

119888 = (

21

7

119896

5

+

6

6

119896

5

+

1

5

119896

5

)

119889 = minus (

35

7

119896

4

+

15

6

119896

4

+

5

5

119896

4

+

1

3

119896

4

)

119890 = (

35

7

119896

3

+

20

6

119896

3

+

10

5

119896

3

+

6

4

119896

3

+

3

3

119896

3

+

1

2

119896

3

)

119891 = minus (

21

7

119896

2

+

15

6

119896

2

+

10

5

119896

2

+

6

4

119896

2

+

3

3

119896

2

+

1

2

119896

2

)

ℎ = (

7

7

119896 +

6

6

119896 +

5

5

119896 +

4

4

119896 +

3

3

119896 +

1

2

119896 + 119896)

119871 = minus (

7

7

+

6

6

+

5

5

+

4

4

+

3

3

+

1

2

+ 2)

(5)

3 Exact Analytical Solution of the DiracEquation for Yukawa Potential

We are interested in the particular solution of the Dirac equa-tionwith the Yukawa interaction potential (4) In this sectionwe anticipate the answer and then obtain the parameters forhighest nuclei deuteron A deuteron consists of a neutronand a proton Let us assume one nucleon is fixed and anothernucleon ismoving around the center ofmass At last by usingsingle particle model we investigate the effects of one pionexchange for a single nucleon The Dirac Hamiltonian is

119867 = sdot119875 + 120573 (119898 + 119878) + 119881 minus 119894120573 sdot 119880 (6)

where 119878 is the scalar potential119881 is the vector potential and119880is the tensor partWe have Δ = 119881minus119878 andΣ = 119881+119878TheDiracequation can be solved exactly for the cases Δ = 0 and Σ = 0 sdot

= (

0120590

120590 0

) and 120573 = ( 119868 00 minus119868

)which are the usual Dirac matricesWewill indicate upper and lower components of Dirac spinorwith Ψ

119860

and Ψ119861

Expressing the Dirac matrices in terms ofthe Paulimatrices we obtain the following coupled equations[25 26]

sdot119875Ψ119860

+ (119881 minus 119878 minus 119898)Ψ119861

+ 119894 sdot 119909119880 (119909)Ψ119860

= 119864Ψ119861

(7a)

sdot119875Ψ119861

+ (119881 + 119878 + 119898)Ψ119860

minus 119894 sdot 119909119880 (119909)Ψ119861

= 119864Ψ119860

(7b)

We assume that 119881 119878 and 119880 are radial potentials Using theidentity

sdot119875 =

( sdot )

119909

2

( sdot ) ( sdot119875) =

( sdot )

119909

2

(minus119894119909

120597

120597119909

+ 119894 sdot119871)

(8)

With (7a) and (7b) we find the following second-orderdifferential equations for Ψ

119860

and Ψ119861

119875

2

Ψ119860

+ (119880

2

+

119889119906

119889119909

+

2119906

119909

+

(119889Δ119889119909)

(119864 + 119898 minus Δ)

119880)Ψ119860

+ (4119880 + 2

(119889Δ119889119909)

(119864 + 119898 minus Δ)

)(

119878 sdot

119871

2

)Ψ119860

minus

(119889Δ119889119909)

(119864 + 119898 minus Δ)

120597Ψ119860

120597119909

= (119864 + 119898 minus Δ) (119864 + 119898 minus Σ)Ψ119860

(9a)

119875

2

Ψ119861

+ (119880

2

minus

119889119906

119889119909

minus

2119906

119909

minus

(119889Σ119889119909)

(119864 minus 119898 minus Σ)

119880)Ψ119861

+ (minus4119880 + 2

(119889Σ119889119909)

(119864 minus 119898 minus Σ)

)(

119878 sdot

119871

2

)Ψ119861

+

(119889Σ119889119909)

(119864 minus 119898 minus Δ)

120597Ψ119861

120597119909

= (119864 + 119898 minus Δ) (119864 minus 119898 minus Σ)Ψ119861

(9b)

Here 119878 stands for the (12) spin operator and

119871 for orbitalangular momentum operator The Hamiltonian operatorcommutes with the total angular momentum operator whichis given by

119869 =119871 +

119878 and with the parity operator Therefore

the eigenfunctions defined by Hamiltonian can be expressedas

Ψ119899119895119898119896

= (

119894

119892119899119896

(119909)

119909

120601

119897

119895119898119896

()

119891119899119896

(119909)

119909

120601

119897

119895119898(minus119896)

()

) (10)

where 120601119897119895119898119896

() denotes the spin spherical harmonics Thequantum number 119896 is related to ℓ and 119895 as follows

119896 =

minus (ℓ + 1) = minus (119895 +

1

2

) 119895 = ℓ +

1

2

ℓ = + (119895 +

1

2

) 119895 = ℓ minus

1

2

(11)



sdot 120601

119897

119895119898119896

= 120601

119897

119895119898(minus119896)

(12)


119896

and 119891120581

(

119889

2

119889119909

2

minus

119896 (119896 + 1)

119909

2

+

2119896

119909

119880 minus

119889119880

119889119909

minus 119880

2

)119892119896

(119909)

+

119889Δ119889119909

(119864 + 119898 minus Δ)

(

119889

119889119909

+

119896

119909

minus 119880)119892119896

(119909)


(119909)

(13a)

(

119889

2

119889119909

2

minus

119896 (119896 minus 1)

119909

2

+

2119896

119909

119880 +

119889119880

119889119909

minus 119880

2

)119891119896

(119909)

+

119889Σ119889119909

(119864 minus 119898 minus Δ)

(

119889

119889119909

minus

119896

119909

+ 119880)119891119896

(119909)


(119909)

(13b)


2119878 sdot

119871 =

119869

2

minus119871

2

minus119878

2

sdot119871120601

119897

119895119898119896

= minus (119896 + 1) 120601

119897

119895119898119896

sdot119871120601

119897

119895119898(minus119896)

= minus (119896 + 1) 120601

119897

119895119898(minus119896)

(14)


119878 = 119881 =

1

2

Σ

Σ (119909) = 119886119909

6

minus 119887119909

5

+ 119888119909

4

minus 119889119909

3

+ 119890119909

2

minus 119891119909 + ℎ minus

119871

119909

Δ (119909) = 0

119880 (119909) = minus

1

119909

(15)


(

119889

2

119889119909

2

minus

ℓ (ℓ + 1) + 2 + 2119896

119909

2

minus Σ (119903) (119864 + 119898))119892119896

(119909)

= (119898

2

minus 119864

2

) 119892119896

(119909)

(16)


[

119889

2

119889119909

2

minus

ℓ (ℓ + 1) + 2 minus 2 (ℓ + 1)

119909

2

]119892119896

(119909)

= ((1198861

119909

6

minus1198871

119909

5

+1198881

119909

4

minus1198891

119909

3

+1198901

119909

2

minus1198911

119909+ℎ1

minus

1198711

119909

)

minusE)119892119896

(119909)

(17)


119892

10158401015840

1

(119909) = [(1198861

119909

6

minus1198871

119909

5

+1198881

119909

4

minus1198891

119909

3

+1198901

119909

2

minus1198911

119909+ℎ1

minus

1198711

119909

)

minusE +

ℓ (ℓ minus 1)

119909

2

]

(18)

By setting

119886 (1198641

+ 119898) = 1198861

119887 (1198641

+ 119898) = 1198871

119888 (1198641

+ 119898) = 1198881

119889 (1198641

+ 119898) = 1198891

(19)

119890 (1198641

+ 119898) = 1198901

119891 (1198641

+ 119898) = 1198911

ℎ (1198641

+ 119898) = ℎ1

119871 (1198641

+ 119898) = 1198711

(1198641

2

minus 119898

2

) = E

(20)


119892 (119909) = 119872 (119909) 119890

119885(119909)

(21)


119872(119909) =

1 if ground stateV

prod

119894=1

(119909 minus 119886

V119894

) if V gt 0

119885 (119909) = minus

1

4

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909 + 120575 ln119909

(22)


119892

10158401015840

(119909) = [119885

10158401015840

+ 119885

10158402

+

119872

10158401015840

+ 2119872

1015840

119885

1015840

119872

] (23)


120572 = radic1198861

120573 = minus

1198871

2

radic1198861

120578 =

1198881

minus 1205731

2120572

120591 =

minus1198891

minus 2120573120578

2120572


2

minus ℎ1

(24)


1198921

(119909) = 1198730

119909


120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909) (25)


1198911

(119909) =

( sdot119875 + 119894 sdot 119880)

(119864 + 119898)

1198921

(119909) (26)



1198911

(119909) =

minus119894 sdot

(119864 + 119898)

[

119889

119889119909

minus 119894 sdot119871 minus 119880]119873

0

119909

ℓminus1

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909)

(27)


Ψ = 1198730

(

1

minus119894 sdot

(119864 + 119898)

[120572119909

3

+ 120573119909 + 120578119909 + 120591 +

1

119909

]

)

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578 minus 120591119909)

(28)


119887 =

14

119896

119886 119888 =

105

119896

2

119886 119889 =

440

119896

3

119886 119890 =

2555

119896

4

119886

119891 =

6846

119896

5

119886 ℎ =

8659

119896

6

119886

(29)


120572 = radic1198861

120573 = minus98radic1198861

120578 = 8308radic1198861

120591 = 8308radic1198861

(30)


120576 = 119864

2

minus 119898

2

= 120578 (1 minus 2120575) minus 120591

2

minus ℎ1

(31)




⟨119867int (119909)⟩ = 119881 (119909) + ⟨119867119878 (119909)⟩ + ⟨119867119868 (119909)⟩ + ⟨119867119878119868 (119909)⟩ (32)


119878

(119909) 119867119868

(119909) and119867119878119868



119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878sum(

119878119894

sdot119878119895

) (33)

where 119878119894


119904

and 120590119878


= 287 fm and 119860119878


119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878(

1

2

) [119878

2

minus 119878

2

1

minus 119878

2

2

] (34)


119867119868

= 119860119878

1

(radic120587120590119868

)

3

119890

minus119909

2120590

2

119868sum(

119868119894

sdot119868119895

) (35)

where 119868119894


119868

= 345 fm and 119860119868


119867119878119868

= 119860119878119868

1

(radic120587120590119878119868

)

3

119890

minus119909

2120590

2

119878119868sum(

119878119894

sdot119878119895

) (119868119894

sdot119868119895

) (36)


= 231 fm and119860119878119868

= minus1062 fm2[35 41] where 120595

120574


120595120574

= 120595

1015840

120574

+ sum

120574

1015840=120574

⟨120595

0

120574

1015840 | 119867int | 1205950

120574

1015840⟩120595

10158400

120574

1015840

119864

0

120574

minus 119864

10158400

120574

(37)


⟨119867int⟩ =int120595

lowast

120574

119867in1205951205741199092

119889119909 119889Ω

int120595

lowast

120574

120595120574

119909

2

119889119909 119889Ω

(38)


119867119878

= 010059 (MeV)

119867119868

= 004782 (MeV)

119867119878119868

= minus050053 (MeV)

(39)


119872119863

= 119898119899

+ 119898119901

+ 119864120592120574

+ ⟨119867int⟩ (40)



= 306136 120572 = 174967 120573 = minus17146815 120578 = 145362999 120591 =

minus14102380 and 119864 (MeV) = 219020

1198861





= 119898119901


120592120574


119872119863

= 187783808 (MeVc2) (41)


119863


5 Conclusion


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





sdot 120601

119897

119895119898119896

= 120601

119897

119895119898(minus119896)

(12)


119896

and 119891120581

(

119889

2

119889119909

2

minus

119896 (119896 + 1)

119909

2

+

2119896

119909

119880 minus

119889119880

119889119909

minus 119880

2

)119892119896

(119909)

+

119889Δ119889119909

(119864 + 119898 minus Δ)

(

119889

119889119909

+

119896

119909

minus 119880)119892119896

(119909)


(119909)

(13a)

(

119889

2

119889119909

2

minus

119896 (119896 minus 1)

119909

2

+

2119896

119909

119880 +

119889119880

119889119909

minus 119880

2

)119891119896

(119909)

+

119889Σ119889119909

(119864 minus 119898 minus Δ)

(

119889

119889119909

minus

119896

119909

+ 119880)119891119896

(119909)


(119909)

(13b)


2119878 sdot

119871 =

119869

2

minus119871

2

minus119878

2

sdot119871120601

119897

119895119898119896

= minus (119896 + 1) 120601

119897

119895119898119896

sdot119871120601

119897

119895119898(minus119896)

= minus (119896 + 1) 120601

119897

119895119898(minus119896)

(14)


119878 = 119881 =

1

2

Σ

Σ (119909) = 119886119909

6

minus 119887119909

5

+ 119888119909

4

minus 119889119909

3

+ 119890119909

2

minus 119891119909 + ℎ minus

119871

119909

Δ (119909) = 0

119880 (119909) = minus

1

119909

(15)


(

119889

2

119889119909

2

minus

ℓ (ℓ + 1) + 2 + 2119896

119909

2

minus Σ (119903) (119864 + 119898))119892119896

(119909)

= (119898

2

minus 119864

2

) 119892119896

(119909)

(16)


[

119889

2

119889119909

2

minus

ℓ (ℓ + 1) + 2 minus 2 (ℓ + 1)

119909

2

]119892119896

(119909)

= ((1198861

119909

6

minus1198871

119909

5

+1198881

119909

4

minus1198891

119909

3

+1198901

119909

2

minus1198911

119909+ℎ1

minus

1198711

119909

)

minusE)119892119896

(119909)

(17)


119892

10158401015840

1

(119909) = [(1198861

119909

6

minus1198871

119909

5

+1198881

119909

4

minus1198891

119909

3

+1198901

119909

2

minus1198911

119909+ℎ1

minus

1198711

119909

)

minusE +

ℓ (ℓ minus 1)

119909

2

]

(18)

By setting

119886 (1198641

+ 119898) = 1198861

119887 (1198641

+ 119898) = 1198871

119888 (1198641

+ 119898) = 1198881

119889 (1198641

+ 119898) = 1198891

(19)

119890 (1198641

+ 119898) = 1198901

119891 (1198641

+ 119898) = 1198911

ℎ (1198641

+ 119898) = ℎ1

119871 (1198641

+ 119898) = 1198711

(1198641

2

minus 119898

2

) = E

(20)


119892 (119909) = 119872 (119909) 119890

119885(119909)

(21)


119872(119909) =

1 if ground stateV

prod

119894=1

(119909 minus 119886

V119894

) if V gt 0

119885 (119909) = minus

1

4

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909 + 120575 ln119909

(22)


119892

10158401015840

(119909) = [119885

10158401015840

+ 119885

10158402

+

119872

10158401015840

+ 2119872

1015840

119885

1015840

119872

] (23)


120572 = radic1198861

120573 = minus

1198871

2

radic1198861

120578 =

1198881

minus 1205731

2120572

120591 =

minus1198891

minus 2120573120578

2120572


2

minus ℎ1

(24)


1198921

(119909) = 1198730

119909


120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909) (25)


1198911

(119909) =

( sdot119875 + 119894 sdot 119880)

(119864 + 119898)

1198921

(119909) (26)



1198911

(119909) =

minus119894 sdot

(119864 + 119898)

[

119889

119889119909

minus 119894 sdot119871 minus 119880]119873

0

119909

ℓminus1

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909)

(27)


Ψ = 1198730

(

1

minus119894 sdot

(119864 + 119898)

[120572119909

3

+ 120573119909 + 120578119909 + 120591 +

1

119909

]

)

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578 minus 120591119909)

(28)


119887 =

14

119896

119886 119888 =

105

119896

2

119886 119889 =

440

119896

3

119886 119890 =

2555

119896

4

119886

119891 =

6846

119896

5

119886 ℎ =

8659

119896

6

119886

(29)


120572 = radic1198861

120573 = minus98radic1198861

120578 = 8308radic1198861

120591 = 8308radic1198861

(30)


120576 = 119864

2

minus 119898

2

= 120578 (1 minus 2120575) minus 120591

2

minus ℎ1

(31)




⟨119867int (119909)⟩ = 119881 (119909) + ⟨119867119878 (119909)⟩ + ⟨119867119868 (119909)⟩ + ⟨119867119878119868 (119909)⟩ (32)


119878

(119909) 119867119868

(119909) and119867119878119868



119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878sum(

119878119894

sdot119878119895

) (33)

where 119878119894


119904

and 120590119878


= 287 fm and 119860119878


119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878(

1

2

) [119878

2

minus 119878

2

1

minus 119878

2

2

] (34)


119867119868

= 119860119878

1

(radic120587120590119868

)

3

119890

minus119909

2120590

2

119868sum(

119868119894

sdot119868119895

) (35)

where 119868119894


119868

= 345 fm and 119860119868


119867119878119868

= 119860119878119868

1

(radic120587120590119878119868

)

3

119890

minus119909

2120590

2

119878119868sum(

119878119894

sdot119878119895

) (119868119894

sdot119868119895

) (36)


= 231 fm and119860119878119868

= minus1062 fm2[35 41] where 120595

120574


120595120574

= 120595

1015840

120574

+ sum

120574

1015840=120574

⟨120595

0

120574

1015840 | 119867int | 1205950

120574

1015840⟩120595

10158400

120574

1015840

119864

0

120574

minus 119864

10158400

120574

(37)


⟨119867int⟩ =int120595

lowast

120574

119867in1205951205741199092

119889119909 119889Ω

int120595

lowast

120574

120595120574

119909

2

119889119909 119889Ω

(38)


119867119878

= 010059 (MeV)

119867119868

= 004782 (MeV)

119867119878119868

= minus050053 (MeV)

(39)


119872119863

= 119898119899

+ 119898119901

+ 119864120592120574

+ ⟨119867int⟩ (40)



= 306136 120572 = 174967 120573 = minus17146815 120578 = 145362999 120591 =

minus14102380 and 119864 (MeV) = 219020

1198861





= 119898119901


120592120574


119872119863

= 187783808 (MeVc2) (41)


119863


5 Conclusion


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1198911

(119909) =

minus119894 sdot

(119864 + 119898)

[

119889

119889119909

minus 119894 sdot119871 minus 119880]119873

0

119909

ℓminus1

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578119909

2

minus 120591119909)

(27)


Ψ = 1198730

(

1

minus119894 sdot

(119864 + 119898)

[120572119909

3

+ 120573119909 + 120578119909 + 120591 +

1

119909

]

)

times exp(minus14

120572119909

4

minus

1

3

120573119909

3

minus

1

2

120578 minus 120591119909)

(28)


119887 =

14

119896

119886 119888 =

105

119896

2

119886 119889 =

440

119896

3

119886 119890 =

2555

119896

4

119886

119891 =

6846

119896

5

119886 ℎ =

8659

119896

6

119886

(29)


120572 = radic1198861

120573 = minus98radic1198861

120578 = 8308radic1198861

120591 = 8308radic1198861

(30)


120576 = 119864

2

minus 119898

2

= 120578 (1 minus 2120575) minus 120591

2

minus ℎ1

(31)




⟨119867int (119909)⟩ = 119881 (119909) + ⟨119867119878 (119909)⟩ + ⟨119867119868 (119909)⟩ + ⟨119867119878119868 (119909)⟩ (32)


119878

(119909) 119867119868

(119909) and119867119878119868



119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878sum(

119878119894

sdot119878119895

) (33)

where 119878119894


119904

and 120590119878


= 287 fm and 119860119878


119867119878

= 119860119878

1

(radic120587120590119878

)

3

119890

minus119909

2120590

2

119878(

1

2

) [119878

2

minus 119878

2

1

minus 119878

2

2

] (34)


119867119868

= 119860119878

1

(radic120587120590119868

)

3

119890

minus119909

2120590

2

119868sum(

119868119894

sdot119868119895

) (35)

where 119868119894


119868

= 345 fm and 119860119868


119867119878119868

= 119860119878119868

1

(radic120587120590119878119868

)

3

119890

minus119909

2120590

2

119878119868sum(

119878119894

sdot119878119895

) (119868119894

sdot119868119895

) (36)


= 231 fm and119860119878119868

= minus1062 fm2[35 41] where 120595

120574


120595120574

= 120595

1015840

120574

+ sum

120574

1015840=120574

⟨120595

0

120574

1015840 | 119867int | 1205950

120574

1015840⟩120595

10158400

120574

1015840

119864

0

120574

minus 119864

10158400

120574

(37)


⟨119867int⟩ =int120595

lowast

120574

119867in1205951205741199092

119889119909 119889Ω

int120595

lowast

120574

120595120574

119909

2

119889119909 119889Ω

(38)


119867119878

= 010059 (MeV)

119867119868

= 004782 (MeV)

119867119878119868

= minus050053 (MeV)

(39)


119872119863

= 119898119899

+ 119898119901

+ 119864120592120574

+ ⟨119867int⟩ (40)



= 306136 120572 = 174967 120573 = minus17146815 120578 = 145362999 120591 =

minus14102380 and 119864 (MeV) = 219020

1198861





= 119898119901


120592120574


119872119863

= 187783808 (MeVc2) (41)


119863


5 Conclusion


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





= 306136 120572 = 174967 120573 = minus17146815 120578 = 145362999 120591 =

minus14102380 and 119864 (MeV) = 219020

1198861





= 119898119901


120592120574


119872119863

= 187783808 (MeVc2) (41)


119863


5 Conclusion


References

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics


































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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