An approximate solution of Dirac-Hulthén problem

JOURNAL OF MATHEMATICAL PHYSICS 48, 082302 共2007兲

An approximate solution of Dirac-Hulthén problem with pseudospin and spin symmetry for any ␬ state A. Soylu Department of Physics, Nigde University, 51350 Nigde, Turkey and Department of Physics, Erciyes University, 38039 Kayseri, Turkey

O. Bayraka兲 Department of Physics, Bozok University, 66200 Yozgat, Turkey and Department of Physics, Erciyes University, 38039 Kayseri, Turkey

I. Boztosun Department of Physics, Erciyes University, 38039 Kayseri, Turkey 共Received 12 April 2007; accepted 10 July 2007; published online 9 August 2007兲

For any spin-orbit quantum number ␬, the analytical solutions of the Dirac equation are presented for the Hulthén potential by applying an approximation to spin-orbit coupling potential for the case of spin symmetry, ⌬共r兲 = C = const, and pseudospin symmetry ⌺共r兲 = C = const. The bound state energy eigenvalues and the corresponding spinors are obtained in the closed forms. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2768436兴

I. INTRODUCTION

The pseudospin symmetry 共PSS兲 with the nuclear shell model has been introduced many years ago1,2 and it has been widely used to explain a number of phenomena in nuclear physics and related areas 共see Refs. 3 and 4 and reference therein兲. It has been shown that the quasidegenerate pseudospin doublets in nuclei arise from the near equality of the magnitude of the attractive scalar and repulsive vector potentials, i.e., VS ⬃ VV.3,4 The PSS is seen as a quasidegeneracy of the doublet single-particle states and characterized with the quantum numbers 共n , l , j = l + 1 / 2兲 and 共n − 1 , l + 2 , j = l + 3 / 2兲, where n, l, and j are the single-particle radial, orbital, and total angular momentum quantum numbers, respectively. The total angular momentum is written by a s 5 The PSS and spin pseudoangular momentum ˜ᐉ = ᐉ + 1 and a pseudospin ˜s = 1 / 2 as j = ˜ᐉ +˜. 3,5 symmetry occur for ⌺共r兲 = C = const and 䉭共r兲 = C = const in the Dirac equation, respectively, and these have been investigated in the several nuclei for a few potentials such as harmonic oscillator,6–10 Morse,11,12 and Wood-Saxon.13,14 On the other hand, the Hulthén potential16 has considerable significance to various applications in many areas of physics such as nuclear and particle physics, atomic physics, condensed matter, and chemical physics 共see Ref. 17 and the references therein兲 in the nonrelativistic and relativistic regions. Therefore, it would be interesting and important to solve Dirac equation for the Hulthén potential since it has been extensively used to describe the bound and the continuum states of the interaction systems. The solution of DiracHulthén problem has been obtained by using different methods and approximations for s wave analytically 共␬ = 0兲.18–20 However, this potential has not been solved for ␬ ⫽ 0 state. In order to find the analytical or quasianalytical solutions for such potential with any ␬, one has to resort to use some approximations.21–25 Therefore, in this paper, our aim is to solve the Dirac-Hulthén problem for any spin-orbit quantum number ␬. In order to obtain the relativistic bound state energy eigenvalues and the corresponding Dirac spinors, we use a different and more practical method, called the asymptotic

a兲

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082302-2

Soylu, Bayrak, and Boztosun

J. Math. Phys. 48, 082302 共2007兲

iteration method 共AIM兲,26 by using an approximation to the centrifugal-like term within the framework of the pseudospin and the spin symmetry concept. In the next section, we present AIM with all necessary formulas to perform our calculations. In Sec. III, a brief introduction of the usual Dirac formalism is presented. In Sec. III A, we perform the spin symmetric solution of the Dirac-Hulthén problem for any ␬ state and in Sec. III B, we also investigate the relativistic bound state eigenvalues and the corresponding spinors of Dirac particles for ␬ ⫽ 0. Finally, Sec. IV is devoted to the summary and conclusion. II. THE ASYMPTOTIC ITERATION METHOD

AIM 共Refs. 26–32兲 is proposed and applied to solve the second-order differential equations of the form y n⬙共x兲 = ␭0共x兲y n⬘共x兲 + s0共x兲y n共x兲,

共1兲

where ␭0共x兲 ⫽ 0 and the prime denotes the derivative with respect to x. The variables s0共x兲 and ␭0共x兲 are sufficiently differentiable. To find a general solution to this equation, we differentiate Eq. 共1兲 with respect to x and find y n⵮共x兲 = ␭1共x兲y n⬘共x兲 + s1共x兲y n共x兲,

共2兲

where ␭1共x兲 = ␭0⬘共x兲 + s0共x兲 + ␭20共x兲, s1共x兲 = s0⬘共x兲 + s0共x兲␭0共x兲.

共3兲

Similarly, the second derivative of Eq. 共1兲 yields y 共4兲 n 共x兲 = ␭2共x兲y n⬘共x兲 + s2共x兲y n共x兲,

共4兲

where ␭2共x兲 = ␭1⬘共x兲 + s1共x兲 + ␭0共x兲␭1共x兲, s2共x兲 = s1⬘共x兲 + s0共x兲␭1共x兲.

共5兲

Equation 共1兲 can be easily iterated up to 共k + 1兲th and 共k + 2兲th derivatives, k = 1 , 2 , 3 , . . .. Therefore, we have 共x兲 = ␭k−1共x兲y n⬘共x兲 + sk−1共x兲y n共x兲, y 共k+1兲 n 共x兲 = ␭k共x兲y n⬘共x兲 + sk共x兲y n共x兲, y 共k+2兲 n

共6兲

where

⬘ 共x兲 + sk−1共x兲 + ␭0共x兲␭k−1共x兲, ␭k共x兲 = ␭k−1 ⬘ 共x兲 + s0共x兲␭k−1共x兲, sk共x兲 = sk−1

共7兲

which are called as the recurrence relation. From the ratio of the 共k + 2兲th and 共k + 1兲th derivatives, we have ␭k共x兲关y n⬘共x兲 + 关sk共x兲/␭k共x兲兴y n共x兲兴 共x兲 y 共k+2兲 d n = . ln关y 共k+1兲 共x兲兴 = 共k+1兲 n dx y n 共x兲 ␭k−1共x兲关y n⬘共x兲 + 关sk−1共x兲/␭k−1共x兲兴y n共x兲兴

共8兲

For sufficiently large k, if

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082302-3

J. Math. Phys. 48, 082302 共2007兲

An approximate solution of Dirac-Hulthén problem

sk共x兲 sk−1共x兲 = = ␣共x兲, ␭k共x兲 ␭k−1共x兲

共9兲

which is the “asymptotic” aspect of the method, then Eq. 共8兲 is reduced to d ␭k共x兲 ln关y 共k+1兲 , 共x兲兴 = n dx ␭k−1共x兲

共10兲

which yields

冉冕

共x兲 = C1 exp y 共k+1兲 n

冊

冉冕

␭k共x兲 dx = C1␭k−1共x兲exp ␭k−1共x兲

冊

关␣共x兲 + ␭0共x兲兴dx ,

共11兲

where C1 is the integration constant and the right hand side of Eq. 共11兲 is obtained by using Eqs. 共9兲 and 共10兲. By inserting Eq. 共11兲 into Eq. 共6兲, the first-order differential equation is obtained as

冉冕

y n⬘共x兲 + ␣共x兲y n共x兲 = C1 exp

冊

关␣共x兲 + ␭0共x兲兴dx .

共12兲

This first-order differential equation can easily be solved and the general solution of the Eq. 共1兲 can be obtained as

冉冕

y n共x兲 = exp −

x

␣共x1兲dx1

冊冋

冕 冉冕 x

C2 + C1

x1

exp

冊 册

关␭0共x2兲 + 2␣共x2兲兴dx2 dx1 .

共13兲

For a given potential, the radial Schrödinger equation is converted to the form of Eq. 共1兲. Then, s0共x兲 and ␭0共x兲 are determined and sk共x兲 and ␭k共x兲 parameters are calculated by the recurrence relations given by Eq. 共7兲. The termination condition of the method in Eq. 共9兲 can be arranged as ⌬k共x兲 = ␭k共x兲sk−1共x兲 − ␭k−1共x兲sk共x兲 = 0,

k = 1,2,3, . . . ,

共14兲

where k shows the iteration number. For the exactly solvable potentials, the energy eigenvalues are obtained from the roots of the Eq. 共14兲 and the radial quantum number n is equal to the iteration number k for this case. For nontrivial potentials that have no exact solutions, for a specific n principal quantum number, we choose a suitable x0 point, determined generally as the maximum value of the asymptotic wave function or the minimum value of the potential,26,27,31,32 and the approximate energy eigenvalues are obtained from the roots of the Eq. 共14兲 for sufficiently great values of k with iteration for which k is always greater than n in these numerical solutions. The general solution of Eq. 共1兲 is given by Eq. 共13兲. The first part of Eq. 共13兲 gives us the polynomial solutions that are convergent and physical, whereas the second part of Eq. 共13兲 gives us nonphysical solutions that are divergent. Although Eq. 共13兲 is the general solution of Eq. 共1兲, we take the coefficient of the second part 共C1兲 as zero, in order to find the square integrable solutions. Therefore, the corresponding eigenfunctions can be derived from the following wave function generator for exactly solvable potentials:

冉冕

y n共x兲 = C2 exp −

x

冊

sn共x1兲 dx1 , ␭n共x1兲

共15兲

where n represents the radial quantum number. III. ANALYTICAL SOLUTION OF DIRAC-HULTHÉN PROBLEM

The Dirac wave equation5,9,13–15 for a single particle with mass M in a scalar S共rជ兲 and a vector potential V共rជ兲 can be given as in unit ប = c = 1.

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082302-4

J. Math. Phys. 48, 082302 共2007兲

Soylu, Bayrak, and Boztosun

ជ . pជ + ␤共M + S共rជ兲兲 + V共rជ兲兴␺共rជ兲 = E␺共rជ兲, 关␣

共16兲

where pជ and E are momentum operator and the total relativistic energy of the system, respectively. The total angular momentum operator Jˆ and spin-orbit matrix operator Kˆ = −␤共␴ˆ . Lˆ + 1兲 commute ជ and ␤ are 4 ⫻ 4 Dirac matrices, i.e., with the Dirac Hamiltonian. ␣

ជ, pជ = − iⵜ

␣ជ =

冉 冊

ជi 0 ␴ , ជ␴i 0

␤=

冉 冊 0

I

0 −I

共17兲

,

ជ i=x,y,z are 2 ⫻ 2 Pauli matrices: where I is the 2 ⫻ 2 unit matrix and ␴ ␴x =

冉 冊 0 1 1 0

,

␴y =

冉 冊 0 −i i

0

,

␴z =

冉 冊 1

0

0 −1

共18兲

.

The Dirac spinors may be written according to the upper 共large兲 f n␬ and lower 共small兲 gn␬ components as follows:

␺n␬共rជ兲 =

冉 冊

冉

冊

Fn␬共r兲/rY ᐉjm共␪, ␾兲 f n␬ = , ˜ g n␬ iGn␬共r兲/rY ᐉjm共␪, ␾兲

共19兲

˜

where Y ᐉjm共␪ , ␾兲 and Y ᐉjm共␪ , ␾兲 are the spin and pseudospin spherical harmonics. n is the radial quantum number, and m is the projection of the angular momentum on the z axis. The orbital angular momentum quantum numbers l and ˜l refer to the spin and pseudospin quantum numbers, respectively. For a given spin-orbit quantum number ␬ = ± 1 , ± 2 , . . ., the total angular momentum, the orbital angular momentum, and pseudo-orbital angular momentum are given by j = 兩␬兩 − 1 / 2, l = 兩␬ + 1 / 2兩 − 1 / 2, and ˜l = 兩␬ − 1 / 2兩 − 1 / 2, respectively. Substituting Eq. 共19兲 to Eq. 共16兲, we can immediately obtain two coupled ordinary differential equation for radial parts of Dirac eigenfunctions as follows:

冉

d ␬ + Fn␬共r兲 = 关M + En␬ − V共r兲 + S共r兲兴Gn␬共r兲, dr r

冊

共20兲

冉

d ␬ − Gn␬共r兲 = 关M − En␬ + V共r兲 + S共r兲兴Fn␬共r兲. dr r

冊

共21兲

and

Eliminating Gn␬共r兲 in Eq. 共20兲 and Fn␬共r兲 in Eq. 共21兲, we obtain the second-order differential equation for the lower and upper components of the Dirac wave function as follows:

冋

册

d⌺/dr共d/dr − ␬/r兲 d2 ␬共␬ − 1兲 − − 共M + En␬ − ⌬共r兲兲共M − En␬ + ⌺共r兲兲 − Gn␬共r兲 = 0 dr2 r2 M − En␬ + ⌺共r兲 共22兲

and

冋

册

d⌬/dr共d/dr + ␬/r兲 d2 ␬共␬ + 1兲 − 共M + En␬ − ⌬共r兲兲共M − En␬ + ⌺共r兲兲 + Fn␬共r兲 = 0, 2 − 2 dr r M + En␬ − ⌬共r兲 共23兲

where ⌺共r兲 = V共r兲 + S共r兲 and ⌬共r兲 = V共r兲 − S共r兲.

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082302-5

J. Math. Phys. 48, 082302 共2007兲

An approximate solution of Dirac-Hulthén problem

A. Spin symmetric solution of the Hulthén potential for any ␬ state

In the case of exact spin symmetry 共d⌬共r兲 / dr = 0, i.e., ⌬共r兲 = C = const兲, Eq. 共23兲 becomes

冋

册

d2 ␬共␬ + 1兲 − − 共M + En␬ − C兲共M − En␬ + ⌺共r兲兲 Fn␬共r兲 = 0, dr2 r2

共24兲

where ␬ = ᐉ for ␬ ⬍ 0 and ␬ = −共ᐉ + 1兲 for ␬ ⬎ 0. The energy eigenvalues depend on n and ᐉ, i.e., En␬ = E共n , ᐉ共ᐉ + 1兲兲, which is well known as the exact spin symmetry.4,13 We assume that ⌺共r兲 is the Hulthén potential,16 which is defined as V共r兲 = − Ze2␦

e −␦r , 1 − e −␦r

共25兲

where Z and ␦, respectively, are the atomic number and the screening parameter, determining the range for the Hulthén potential. The Hulthén potential behaves like the Coulomb potential near the origin 共r → 0兲, but in the asymptotic region 共r Ⰷ 1兲, the Hulthén potential decreases exponentially, therefore its capacity for bound states is smaller than the Coulomb potential. Equation 共24兲 cannot be solved analytically for the Hulthén potential with any ␬ state. So we have to use an approximation for the ␬共␬ + 1兲 / r2 term similar to approximation used for the centrifugal term by other authors.21–25 In this approximation, 1 / r2 ⬇ ␦2e−␦r / 共1 − e−␦r兲2 is used for the centrifugal-like term. This term is valid in small ␦ and ␬ values. We have used the relation as follows:

␬共␬ + 1兲 e −␦r 2 ⬇ ␦ ␬ 共 ␬ + 1兲 . r2 共1 − e−␦r兲2

共26兲

Inserting the Hulthén potential and this new term into Eq. 共24兲 and using the following Ansätze, in order to make the differential equation more compact

␦r = x,

␧2 =

M 2 − En2␬ − C共M − En␬兲

␤2 =

Ze2共M + En␬ − C兲

.

共27兲

d2Fn␬共x兲 e−x e−x 2 2 − + − ␧ + ␤ ␬ 共 ␬ + 1兲 Fn␬共x兲 = 0. dx2 共1 − e−x兲 共1 − e−x兲2

共28兲

␦2

,

The radial Dirac equation takes the following form:

冋

␦

册

If we rewrite Eq. 共28兲 by using a new variable of the form s = e−x, we obtain

冋

册

␬共␬ + 1兲 d2Fn␬共s兲 1 dFn␬共s兲 ␤2 ␧2 + − + − + Fn␬共s兲 = 0. 2 2 ds s ds s s共1 − s兲 s共1 − s兲2

共29兲

The wave function has to respect the boundary conditions, i.e., Fn␬共⬁兲 = 0 at s = 0 for x → ⬁ and Fn␬共0兲 = 0 at s = 1 for x → 0; therefore the reasonable physical wave function we propose is as follows: Fn␬共s兲 = s␧共1 − s兲␬+1 f n␬共s兲.

共30兲

If we insert this wave function into Eq. 共29兲, we have the second-order homogeneous linear differential equations as in the following form:

冋

册

冋

册

共2␧ + ␬ + 2兲␬ + 2␧ − ␤2 + 1 共2␧ + 2␬ + 3兲s − 共2␧ + 1兲 df n␬共s兲 d2 f n␬共s兲 + = f n␬共s兲, ds2 s共1 − s兲 ds s共1 − s兲 共31兲 which is now amenable to an AIM solution. By comparing this equation with Eq. 共1兲, we can write the ␭0共s兲 and s0共s兲 values and by means of Eq. 共7兲, we may calculate ␭k共s兲 and sk共s兲. This gives

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082302-6

J. Math. Phys. 48, 082302 共2007兲

Soylu, Bayrak, and Boztosun

␭0共s兲 =

s0共s兲 =

␭1共s兲 =

冉

冉

冊

共2␧ + 2␬ + 3兲s − 共2␧ + 1兲 , s共1 − s兲

冊

共2␧ + ␬ + 2兲␬ + 2␧ − ␤2 + 1 , s共1 − s兲

2 + 6␧ − 7s − 2␬s − ␤2s + 12s2␬ − 18␧s − 6␧s␬ s2共− 1 + s兲2 +

12␧s2 + 11s2 + 4␧2 + ␬2s + ␤2s2 + 4␧2s2 − 8␧2s + 6␧s2␬ + 3␬2s2 s2共− 1 + s兲2

s1共s兲 =

共32兲

共2␬ + 2␧ − ␤2 + 2␧␬ + ␬2 + 1兲共− 2 + 5s + 2␧s + 2␬s − 2␧兲 s2共− 1 + s兲2 ... .

Combining these results with the quantization condition given by Eq. 共14兲 yields s 0␭ 1 − s 1␭ 0 = 0 ⇒ ␧ 0 =

␤2 − 1 − 2␬ − ␬2 , 2共␬ + 1兲

s 1␭ 2 − s 2␭ 1 = 0 ⇒ ␧ 1 =

␤2 − 4 − 4␬ − ␬2 , 2共␬ + 2兲 共33兲

␤2 − 9 − 6␬ − ␬2 , s 2␭ 3 − s 3␭ 2 = 0 ⇒ ␧ 2 = 2共␬ + 3兲 ... . When the above expressions are generalized, the eigenvalues turn out as ␧ n␬ =

冉

冊

␤2 − 共n + ␬ + 1兲2 , 2共n + ␬ + 1兲

Using Eqs. 共27兲 and 共34兲, we obtain energy eigenvalues En␬ as follows:

冋

共M − En␬兲共M + En␬ − C兲 = ␦2 Ze2

共34兲

n = 0,1,2,3, . . . .

共M + En␬ − C兲 − 共n + ␬ + 1兲 2␦共n + ␬ + 1兲

册

2

.

共35兲

It can be seen in Eq. 共35兲 that the spin limit leads to quadratic eigenvalues. Therefore, the solution of this equation consists of positive and negative energy eigenvalues for each value of n and ␬. Ginocchio5 has shown that there are only positive energy eigenvalues and no bound negative energy eigenvalues exist in the spin limit. Therefore, only positive energy eigenvalues are chosen for the spin limit. Now, as indicated in Sec. II, we may immediately determine the corresponding wave functions by using Eq. 共15兲. f n␬共s兲 = 共− 1兲n

⌫共2␧n␬ + n + 1兲 F 共− n,2␧n␬ + 2␬ + 2 + n;2␧n␬ + 1;s兲. ⌫共2␧n␬ + 1兲 2 1

共36兲

Therefore, we can write the total radial wave function as below:

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082302-7

J. Math. Phys. 48, 082302 共2007兲

An approximate solution of Dirac-Hulthén problem

Fn␬共s兲 = Ns␧n␬共1 − s兲␬+1 2F1共− n,2共␧n␬ + ␬ + 1兲 + n;2␧n␬ + 1;s兲,

共37兲

where N is the normalization constant. B. Pseudospin symmetric solution of the Hulthén potential for any ␬ state

In the case of exact PSS 共d⌺共r兲 / dr = 0, i.e., ⌺共r兲 = C = const兲, Eq. 共22兲 becomes

冋

册

d2 ␬共␬ − 1兲 − − 共M + En␬ − ⌬共r兲兲共M − En␬ + C兲 Gn␬共r兲 = 0, dr2 r2

共38兲

˜ for ␬ ⬍ 0 and ␬ = ˜ᐉ + 1 for ␬ ⬎ 0 and ⌬共r兲 is the Hultèn potential. The energy eigenwhere ␬ = −ᐉ ˜ + 1兲兲. The eigenstates with j = ˜ᐉ ± 1 / 2 are degenerate values depend on n and ˜ᐉ, i.e., En␬ = E共n , ˜ᐉ共ᐉ for ˜ᐉ ⫽ 0 which is well known as the exact PSS.4,13 Equation 共38兲 cannot be solved exactly for the Hulthén potential for ␬ ⫽ 0 by using the standard methods. Therefore, an approximation has to be made for ␬共␬ − 1兲 / r2 term as in the previous section. According to this approximation, we use ␦2e−␦r / 共1 − e−␦r兲2 instead of 1 / r2 in Eq. 共38兲. Therefore, inserting Eq. 共25兲 into Eq. 共38兲 and by using the following Ansätze:

␦r = x,

␧2 =

M 2 − En2␬ + C共M + En␬兲

␦2

,

␤2 = −

Ze2共M − En␬ + C兲

␦

,

共39兲

we can obtain

冋

册

−x d2 e−x 2 2 e − ␬ 共 ␬ − 1兲 − ␧ + ␤ Gn␬共x兲 = 0. dx2 共1 − e−x兲2 1 − e−x

共40兲

If we write Eq. 共40兲 by using a new variable of the form s = e−x, we obtain

冋

册

d2 1 d ␬共␬ − 1兲 ␧2 ␤2 − + − + Gn␬共s兲 = 0. ds2 s ds s共1 − s兲2 s2 s共1 − s兲

共41兲

The wave function has to be the boundary condition, i.e., Gn␬共⬁兲 = 0 at s = 0 for x → ⬁ and Gn␬共0兲 = 0 at s = 1 for x → 0; therefore the reasonable physical wave function we propose is as follows: Gn␬共s兲 = s␧共1 − s兲␬ f n␬共s兲.

共42兲

If we insert this wave function into Eq. 共41兲, we have the second-order homogeneous linear differential equation as in the following form:

冋

册

冋

册

共2␬ + 2␧s + s − 1 − 2␧兲 df n␬共s兲 共␬2 + 2␧␬ − ␤2兲 d2 f n␬共s兲 − = − f n␬共s兲, ds2 s共s − 1兲 ds s共s − 1兲

共43兲

which is now amenable to an AIM solution. By comparing this equation with Eq. 共1兲, we can write the ␭0共s兲 and s0共s兲 values and by means of Eq. 共7兲, we may calculate ␭k共s兲 and sk共s兲. This gives ␭0共s兲 = −

共2␬ + 2␧s + s − 1 − 2␧兲 , s共s − 1兲

s0共s兲 = −

共␬2 + 2␧␬ − ␤2兲 , s共s − 1兲

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082302-8

J. Math. Phys. 48, 082302 共2007兲

Soylu, Bayrak, and Boztosun

␭1共s兲 =

2 + ␬2s + ␤2s2 − ␤2s − 4␬s − 12␧s + 2␬2s2 − 6␧␬s + 6␧ s2共s − 1兲2 +

− 4s + 2s26␧␬s2 + 6␧s2 + 6␬s2 + 4␧2 − 8␧2s + 4␧2s2 , s2共s − 1兲2

s1共s兲 =

共44兲

共␬2 + 2␧ − ␤2兲共− 2 + 3s + 2␬s + 2␧s − 2␧兲 , s2共s − 1兲2 ... .

Combining these results with the quantization condition given by Eq. 共14兲 yields s 0␭ 1 − s 1␭ 0 = 0 ⇒ ␧ 0 =

s 1␭ 2 − s 2␭ 1 = 0 ⇒ ␧ 1 =

␤2 − ␬2 , 2␬

␤2 − ␬2 − 1 − 2␬ , 2共␬ + 1兲 共45兲

␤2 − ␬2 − 4␬ − 4 , s 2␭ 3 − s 3␭ 2 = 0 ⇒ ␧ 2 = 2共␬ + 2兲 ... . When the above expressions are generalized, the eigenvalues turn out as ␧ n␬ =

␤2 − 共n + ␬兲2 . 2共n + ␬兲

共46兲

Using Eqs. 共39兲 and 共46兲, we can immediately obtain the energy eigenvalues of pseudo spin symmetric case for any ␬ by using AIM as follows:

冋

共M + En␬兲共M − En␬ + C兲 = ␦2 − Ze2

共M − En␬ + C兲 − 共n + ␬兲 2␦共n + ␬兲

册

2

.

共47兲

It can be seen in Eq. 共47兲 that similar to spin limit, the pseudospin limit leads to quadratic eigenvalues. Therefore, the solution of this equation consists of positive and negative energy eigenvalues. However, in the pseudospin limit, there are only negative energy eigenvalues and no bound positive energy states exist as argued by Ginocchio5. Now, as indicated in Sec. II, we can determine the corresponding wave functions by using Eq. 共15兲. f n␬共s兲 = 共− 1兲n

⌫共2␧n␬ + n + 1兲 2F1共− n,2␧n␬ + 2␬ + n;2␧n␬ + 1;s兲. ⌫共2␧n␬ + 1兲

共48兲

Thus, we can write the total radial wave function as below: Gn␬共s兲 = Ns␧n␬共1 − s兲␬ 2F1共− n,2共␧n␬ + ␬兲 + n;2␧n␬ + 1;s兲,

共49兲

where N is the normalization constant. IV. CONCLUSION

In this study, we have presented the bound state solutions of the Dirac equation for the Hulthén potential within the framework of AIM by applying an approximation to centrifugal-like

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082302-9

An approximate solution of Dirac-Hulthén problem

J. Math. Phys. 48, 082302 共2007兲

term. For arbitrary ␬ state, we have obtained the energy eigenvalues and Dirac spinors in the closed form for the case of the spin symmetry and exact PSS. The analytical formulas, given by Eqs. 共35兲 and 共47兲 may have interesting applications in nuclear physics and related areas. Regarding the method presented in this study, we should point out that it is a systematic one and it is very efficient and practical in solving the spin symmetric and pseudospin symmetric systems for any ␬ state.

ACKNOWLEDGMENTS

This paper is an output of the project supported by the Scientific and Technical Research Council of Turkey 共TÜBİTAK兲, under the Project No. TBAG-106T024. One of the authors 共A.S.兲 acknowledges the financial support of TÜBİTAK-BİDEP. This work is also partially supported by the Turkish State Planning Organization 共DPT兲 under the Grant No. DPT-2006K-120470. A. Arima, M. Harvey, and K. Shimizu, Phys. Lett. 30B, 517 共1969兲. K. T. Hecht and A. Adler, Nucl. Phys. A 137, 129 共1969兲. J. N. Ginocchio, Phys. Rev. Lett. 78, 436 共1997兲. 4 J. N. Ginocchio and D. G. Madland, Phys. Rev. C 57, 1167 共1998兲. 5 J. N. Ginocchio, Phys. Rep. 414, 165 共2005兲. 6 R. Lisboa, M. Malheiro, A. S. de Castro, P. Alberto, and M. Fiolhais, Phys. Rev. C 69, 024319 共2004兲. 7 J. N. Ginocchio, Phys. Rev. C 69, 034318 共2004兲. 8 J. N. Ginocchio, Phys. Rev. Lett. 95, 252501 共2005兲. 9 J.-Y. Guo, X.-Z. Fang, and F.-X. Xu, Nucl. Phys. A 757, 411 共2005兲. 10 A. S. de Castro, P. Alberto, R. Lisboa, and M. Malheiro, Phys. Rev. C 73, 054309 共2006兲. 11 C. Berkdemir, Nucl. Phys. A 770, 32 共2006兲. 12 W.-C. Qiang, R.-S. Zhou, and Y. Gao, J. Phys. A 40, 1677 共2007兲. 13 J.-Y. Guo and Z.-Q. Sheng, Phys. Lett. A 338, 90 共2005兲. 14 Q. Xu and S.-J. Zhu, Nucl. Phys. A 768, 161 共2006兲. 15 W. Grenier, Relativistic Quantum Mechanics, 3rd ed. 共Springer, Berlin, 2000兲. 16 L. Hulthén, Ark. Mat., Astron. Fys. 28A, 5 共1942兲. 17 Y. P. Varshni, Phys. Rev. A 41, 4682 共1990兲. 18 B. Roy and R. Roychoudhury, J. Phys. A 23, 5095 共1990兲. 19 J.-Y. Guo, J. Meng, and F.-X. Xu, Chin. Phys. Lett. 20, 602 共2003兲. 20 A. D. Alhaidari, J. Phys. A 37, 5805 共2004兲. 21 R. L. Greene and C. Aldrich, Phys. Rev. A 14, 2363 共1976兲. 22 U. Myhrman, J. Phys. A 16, 263 共1983兲. 23 A. Bechlert and W. Bühring, J. Phys. B 21, 817 共1988兲. 24 B. Gonul, O. Ozer, Y. Cancelik, and M. Kocak, Phys. Lett. A 275, 238 共2000兲. 25 S.-W. Qian, B.-W. Huang, and Z.-Y. Gu, New J. Phys. 4, 13 共2002兲. 26 H. Ciftci, R. L. Hall, and N. Saad, J. Phys. A 36, 11807 共2003兲. 27 F. M. Fernández, J. Phys. A 37, 6173 共2004兲. 28 H. Ciftci, R. L. Hall, and N. Saad, J. Phys. A 38, 1147 共2005兲. 29 O. Bayrak and I. Boztosun, J. Phys. A 39, 6955 共2006兲. 30 O. Bayrak, G. Kocak, and I. Boztosun, J. Phys. A 39, 11521 共2006兲. 31 I. Boztosun, M. Karakoc, F. Yasuk, and A. Durmus, J. Math. Phys. 47, 062301 共2006兲. 32 T. Barakat, J. Phys. A 39, 823 共2006兲. 1 2 3

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