Exact Solution of the Dirac Equation for the Coulomb Potential Plus NAD Potential by Using the Nikiforov-Uvarov Method

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1 Adv. Studies Theor. Phys., Vol. 6, 01, no. 15, Exact Solution of the Dirac Equation for the Coulomb Potential Plus NAD Potential by Using the Nikiforov-Uvarov Method S. Bakkeshizadeh 1 and V. Vahidi Department of Physics, Faculty of Science Urmia University, Urmia, Iran Abstract In this paper, the solutions of the Dirac equation for a diatomic molecule in a non-central potential are investigated analytically. The potential consist of the Coulomb potential plus a novel angle-dependent (NAD) potential. The Dirac equation is separated into radial and angular parts, and energy eigenvalues and eigenfunctions are derived by using Nikiforov-Uvarov (NU) method. PACS: Ge; Db Keywords: Dirac equation; Coulomb potential; NAD potential; Nikiforov- Uvarov method 1. Introduction The Dirac equation is the most perfect example of a relativistic equation which is able to describe in a simple manner relativistic effects due to the speed and those of the spin of particles. In recent years, considerable attention has been paid to exactly solvable Dirac equation [1]. In fact, the Dirac equation is exactly solvable only for very few interaction and the solutions usually comes with a strong constraint on the potentials []. For example, some authors assumed that the scalar potential is equal to the vector potential and obtained the exact solutions of the Dirac equation with some typical potential by using 1 somayehbakkeshizadeh@ymail.com

2 734 S. Bakkeshizadeh and V. Vahidi different methods. These investigations include the harmonic oscillator [3], the triaxial and axially deformed harmonic oscillators potential [4], Eckart potential [5,6], Woods-Saxon potential [7], Hulthen potential [8], pseudoharmonic oscillator [9], ring-shaped Kratzer-type potential [10], ring-shaped non-spherical oscillator [11], double ring-shaped oscillator [1], Hartmann potential [13, 14], Rosen-Morse-type potential [15], generalized symmetrical double-well potential [16], Scarf-type potential [17], and Davidson potential [18], etc. These methods include the standard method, supersymmetry quantum mechanics [5], the Nikiforov-Uvarov (NU) method [19] and others. The concept of the Coulomb gives us a very good first approximation for understanding the spectroscopy and the structure of diatomic molecules in their ground electronic states. Recently, Berkdemir [0] proposed a novel angle-dependent (NAD) potential and obtained the exact solutions of the Schrodinger equation for the Coulomb and harmonic oscillator potentials add NAD potential. An important aspect of the use of the NAD potential is to study the rotational-vibrational dynamics of a diatomic molecule in noncentral potentials. Moreover, rotational-vibrating energy states of a diatomic molecule can be exactly calculated by means of a radial potential connected by the NAD potential. The purposes of this paper is to investigate the contribution of the parameters come from NAD potential into the energy spectrum of a diatomic molecule in the Coulomb potential. To make this analysis, the NAD potential is added to the radial parts of the Coulomb as an angle dependent part. The Coulomb potential plus the NAD potential is given in the following form, V (r) = K ( r + h γ + βsin θ + ηsin 4 ) θ. (1) μr sin θcos θ where r represents spherical coordinates r, θ and ϕ, also γ,β,η and K arbitrary constant values and μ denote the mass particle. The solution of the Dirac equation for this combined potential is exactly obtained by using a systematical solution method which is introduced by Nikiforov-Uvarov (NU). The NU method is used to solve Schrodinger, Dirac, Klein-Gordon and Duffin-Kemmer- Petiau wave equations for certain kind of potentials [1-5]. This work is organized as follows: in section, the NU method is given briefly. In section 3 we consider the separation of variables for the Dirac equation. Sections 4, 5 devoted to the exact solutions of the radial and angular Dirac equation by the NU method. Finally, we present a brief discussion of the results achieved.

3 Exact solution of the Dirac equation 735. Nikiforov-Uvarov Method The second-order differential equations whose solutions are the special functions can be solved by using the NU method. This method was purposed to solve the second-order differential equation of hypergeometric-type and in this method the differential equations can be written in the following form, d Ψ(s) ds + τ(s) dψ(s) + σ(s) Ψ(s) =0, () σ(s) ds σ (s) where σ(s) and σ(s) are polynomials, at most second degree, and τ(s) is a first degree polynomial. By writing the general solution as Ψ(s) = ϕ(s)y(s), we obtain a hypergeometric type equation, d y(s) ds + τ(s) dy(s) + λ y(s) =0. (3) σ(s) ds σ(s) The function φ(s) is defined as a logarithmic derivative, φ (s) φ(s) = π(s) σ(s), (4) where y(s) is the hypergeometric type function whose polynomial solutions are given by Rodrigues relation, d n y n (s) = a n ρ(s) ds n [σn (s)ρ(s)], (5) where a n is a normalization constant, and ρ(s) is the weight function satisfying the following equation, (ρσ) = τρ. (6) The function π(s) and the parameter Λ required for this method are defined as ) π(s) = σ τ ( σ τ ± σ + kσ, (7) Λ=k + π (s). (8) In the NU method, π(s) is a polynomial with the parameter s and the determination of k is the essential point in the calculation of π(s). For finding the value of k, the expression under the square root most be square of a polynomial, so we have a new eigenvalue equation, n(n 1) d σ(s) Λ=λ n = τ, (9) ds where the derivation of the function τ(s) = τ(s) + π(s) should be negative, and by comparing Eqs. (8) and (9), we obtain the energy eigenvalues.

4 736 S. Bakkeshizadeh and V. Vahidi 3. Dirac equation and separation in spherical coordinates The Dirac equation with scalar and vector potentials is [α.p + β(μ + s(r))]φ(r) =[E V ]φ(r), (10) ( ) ( ) 0 σ I 0 p = i, α =, β =. (11) σ 0 0 I where σ and I are vector Pauli spin matrix and identity matrix, respectively. p is a momentum, s and V are scalar and vector potentials (here we assume h = c = 1). In Pauli-Dirac representation ( ) ϕ(r) φ(r) =. (1) χ(r) Substituting Eqs. (11) and (1) into Eq. (10) we have σ.pχ(r) =[E V μ s(r)]ϕ(r), (13) σ.pϕ(r) =[E V + μ s(r)]χ(r). (14) With equal scalar and vector potential the above equations become σ.pχ(r) =[E μ V ]ϕ(r), (15) χ(r) = σ.p ϕ(r). E + μ (16) By eliminating χ(r) between these two, we have [ p +(E + μ)v (r) ] ϕ(r) = [ E μ ] ϕ(r). (17) In spherical coordinate the wave function is written as follows ϕ(r) = U(r) H(θ)e imϕ, m =0, ±1, ±,... (18) r By substituting Eq. (18) into Eq. (17) and using the separation of variables, for H(θ) and U(r) we have the following equations d H(θ) + cosθ [ ( dh(θ) m γ + βsin dθ sinθ dθ sin θ +a θ + ηsin 4 )] θ L H(θ) =0, sin θcos θ (19) d [ U dr μ A ak + L ] U(r) =0. (0) r μr where L is the separation constant, a, A are defined as a = E + μ, (1) A = μ E. ()

5 Exact solution of the Dirac equation Eigenvalues and eigenfunctions of the polar angel equation By introducing a new variable x = sin θ, Eq. (19) becomes d H ( 3x) dh + dx x(1 x) dx 1 4x (1 x) ( x (L+aη)+x(m aβ+l) (m +aγ))h(x) =0. (3) Comparing with Eq. () the following expressions are obtained τ = 3x, σ =x(1 x), σ = x (L+aη)+x(m aβ +L) (m +aγ). (4) Putting them in Eq. (7) the function π is π = x ±1 x (1 + 4(aη + L) 8k)+x(8k 4(m + L aβ))+4(m +aγ), (5) According to the NU method, the expression in the square root must be the square of the polynomial. So, one can find new possible functions for each k as [( 1+8a(η+β+γ) m +aγ)x+ m +aγ], π = x ± 1 ( m +a(β+γ) L) fork 1 = + 1 (m +aγ)[1+8a(η+β+γ)]. [( 1+8a(η+β+γ)+ m +aγ)x m +aγ], fork = (m +a(β+γ) L) 1 (m +aγ)[1+8a(η+β+γ)].. (6) In Eq. (6), one of the four possible forms of π is finding the negative derivation of τ given by equation τ = τ +π. Other forms are not suitable physically. Therefore, the most suitable form of π is selected as π = x 1 [( ) ] 1+8a(η + β + γ)+ m +aγ x m +aγ, (7) +a(β+γ) L) For k = (m 1 (m +aγ)[1+8a(η + β + γ)]. Hence, τ(x) is obtained as follows ( τ(x) = + m +aγ ) ( ) x a(η + β + γ)+ m +aγ. (8) The key rule of the derivative of τ appears in Eq. (9) which is a polynomial of degree Λ = λ n = nτ n(n 1) σ, where Λ denotes k + π from Eq. (8). Consequently, Λ and λ n are obtained, respectively, Λ= (m +a(β +γ) L) 1 ) ((1 + 8a(η + β + γ))(1 + m +aγ)+ m +aγ, (9)

6 738 S. Bakkeshizadeh and V. Vahidi λ n =n +n +n m +aγ + n 1+8a(η + β + γ), (n =0, 1,...) (30) Taking σ = 4. In order to find an expression which is relating to L, the right-hand sides of Eqs. (9) and (30) must be compared with each other. In this case the result obtained will depend on the NAD potential constants as well as the usual quantum numbers; ( ) ( +a(β+γ). L = 1+8a(η + β + γ) 1+n + m +aγ + 1+n + m +aγ) (31) The separation constant L in Eq. (31) contains the contributions that come from the angle-dependent part of the NAD potential. Let us now find the corresponding eigenfunctions for the angel part. According Eqs. (4) and (6), φ and as follows φ(x) =x B/4 (1 x) (1+A+B)/4, (3) ρ(x) = 1 xb/ (1 x) (A+B)/, (33) where A = 1+8a(η + β + γ) and B = m +aγ. Substituting Eq. (33) into Eq. (5), y n (x) can be found to be ( ) y n (x) =B n n x B/ (A+B)/ dn 1 (1 x) dx n xn+b/ (1 x) n+(a+b)/. (34) The polynomial solution of y n is expressed in terms of a Jacobi polynomial which is one of the orthogonal polynomials, giving P n (B/,(A+B)/) (1 x). By using H n (x) =φ(x)y(x) the solution of Eq. (3) can be written as H n (x) =C n x B/4 (1 x) (1+A+B)/4 P (B/,(A+B)/) n (1 x). (35) where C n is a normalized constant. The useful projection of Eq. (35) can also be given in terms of the confluent hypergeometric function F (α 1,β 1,γ 1,z) with parameters α 1,β 1,γ 1. The representation of this function in terms of Jacobi polynomials is P n (B/,(A+B)/) Γ(n + B/+1) (z) = n!γ(b/+1) F (α 1,β 1,γ 1, 1 z ), (36) P n (B/,(A+B)/) (1 x) = P n (B/,(A+B)/) (1 x) = Γ(n + B/+1) n!γ(b/+1) F (α 1,β 1,γ 1,x), (z =1 x) (37) Γ(n + B/+1) n!γ(b/+1) F ( n, n+b/+(a+b)/+1,b/+1,x), (38)

7 Exact solution of the Dirac equation 739 where α 1 = n, β 1 = n + B/ +(A + B)/ + 1 and γ 1 = B/ + 1. The θ-dependent wave equation in Eq. (35) becomes H n (x) =G n x B/4 (1 x) (1+A+B)/4 F ( n, n + B/+(A + B)/+1,B/+1,x). (39) where G n is a new normalized constant. 5. Eigenvalues and eigenfunctions of the radial equation Now we return to study Eq. (0) d U dr + ( ξ r b r L ) U(r) =0, (40) r where ξ = 4aμK, b =μa. To apply the NU method, Eq. (40) is compared with Eq. () and the following expressions are obtained τ =0, σ = r, σ = ξ b r L. (41) The function π is obtained by putting the above expressing in Eq. (7); π = 1 ± 1 4b r +4(k + ξ )r +(1+4L). (4) According to the NU method, the expression in the square root must be the square of the polynomial, so k 1, = ξ ± b (1 + 4L). (43) We can find four possible functions π for each k as { 1 [1±(br+ 1+4L)], for k 1 = ξ π = + 1 [1±(br 1+4L)], for k = ξ } b (1+4L), b (1+4L).. (44) For the polynomial of τ = τ +π which [ ( has a negative derivative we select )] k = ξ b (1 + 4L) and π = 1 1 br 1+4L with this selection and Λ = k + π, τ and Λ can be written as, respectively, τ =1 ( br 1+4L ), (45) ( ) Λ= ξ b L). (46) Comparing the definition of λ n in Eq. (9) with the Eq. (46), we obtain the energy equation ( E μ ) 8k μ + ( ) (E + μ) =0. (47) n L

8 740 S. Bakkeshizadeh and V. Vahidi where L = 1+8a(η + β + γ) ( 1+n + m +aγ ) + ( 1+n + m +aγ ) + a(β + γ) Let us now find the radial wavefunctions for this potential. Using σ and π Eqs. (4) and (6), the following expressions are obtained ρ = r 1+4L e br, (48) Then from Eq. (5) one has φ = r 1 (1+ 1+4L) e br. (49) y n (r) =a n r 1+4L br dn ( e r n+ 1+4L e br). (50) dr n Where a n is a normalized constant. Thus the wavefunctions U(r) can be obtained as U(r) =a n r 1 (1+ [ 1+4L) e br r 1+4L br dn ( e r n+ 1+4L e br)]. (51) dr n And it can be written as the generalized Laguerre polynomials U(r) =N n r 1 (1+ 1+4L) e br L 1+4L n (br), (5) with normalization condition 0 U (r)dr = 1, we have found the normalization constant N n = n! (b) 1+4L Γ(n + 1+4L +1). (53) where b = μ (μ E ). 6. Conclusions In this paper, we have proposed a new exactly solvable potential which consists of the Coulomb potential plus a novel angle-dependent (NAD) potential and obtained the bound state solution of the Dirac equation by the NU method for a diatomic molecule. The angular and radial wavefunctions and energy equation are given by Eqs. (39) and (5) and (47), respectively. We know that it is possible to study the Coulomb potential plus a novel angle-dependent potential and to solve exactly the Schrodinger, Dirac and Klein-Gordon equations for this system. Possible studies along this line are in progress.

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