exact solutions of the klein-gordon equation for the ...

Bull. Cal. Math. Soc, 104, (5) 481-490 (2012)

EXACT SOLUTIONS OF THE KLEIN-GORDON EQUATION FOR THE MASS-DEPENDENT GENERALIZED WOODS-SAXON POTENTIAL B. BISWAS* AND S. DEBNATH f

(Received 11 January 2012) Abstract. The one-dimensional Klein-Gordon equation for the mass-dependent generalized Woods-Saxon potential with equal scalar and vector potentials are studied in this paper. The exactly normalized bound state wave function and energy expressions are obtained by using the N-U method (Nikiforov and Uvarov, 1988). We also studied the PT-symmetric and nonPT-symmetric cases for the potential and give a brief discussion on the corresponding Hamiltonian.

1. Introduction. It is well-known that Klein-Gordon equation describes the spin-zero particles and Dirac equations are applied to describe ^-spin particles in relativistic quantum mechanics. The problem of finding exact solutions of the Klein-Gordon equation for a number. of special potentials has been a line of great interest in recent years, some authors, by using different methods, studied the bound states of the Klein-Gordon equation under the condition that each of the scalar potentials is equal to its vector potential. The potentials considered include the Hulthen potential (Hu and Su, 1991 and Chen, 2004), scarf-type potential (Motavali and Abkarieh, 2010), 3D harmonic oscillator potential (SU and Ma, 1986), pseudo-harmonic oscillator potential (Goudarzi and Vahidi, 2011), ring-shaped Kratzer-type potential (Batiha, 2009), double ring-shaped oscillator potential (Ying, 2008 and Bayrak, Boztosun and Ciffcci, 2007), Hartmann potential (Musafa and Sever, 1991), generalized Poschl-Teller potential (Yasuk and Durmus, 2008), generalized symmetrical double-well potential (Olgar, 2008), Rosen-Morse type potential (Soylu, Bayrak and Boztosun, 2008 and Debnath and Biswas, 2012), Eckart potential (Taskin, Boztosun and Bayrak, 2008) etc. T.he methods considered are supersymetric approach (SUSY) (Taskin, Boztosun and Bayrak, 2008; Coopoer, Khare, Sukhatme, 1995; Aktas and Sever, 2004; Dirac, 1945 and Goudarzi and Vahidi, 2011), supersymmetric WKB approach (Chen, 2004), Nikiforov-Uvarov method (Egrifes, Demirhan and Buyukkihe, 1999; Cotfas, 2002; Yesiltas, Simsek, Sever, Tezcan, 2003; Berkdemir, Berkdemic and Sever, 2004 and Berkdeinir, 2006), the Variational method (Aygun, Bayrak and Boztosun, 2007), the Functional Analysis method (Barakat, 2006), the Shifted j^ Expansion (Soylu and Boztosun, 2008), Asymptotic Iteration method (Inci, Boztosun and Bonatsos, 2008; Bayrak, Boztosunn, Ciftci, 2007; Hu and Su, 1991; Motavalli and Akbarieh, 2010 and Debnath and Biswas, 2012). Presented at the International Conference on Dynamical Systems: Theory and Applications (ICDS 2012) during 11-14, January 2012 organised by Department of Mathematics, Centre for Mathematical Biology and Ecology, Jadavpur University and Biomathematical Society of India.

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B. BISWAS AND S. DEBNATH

In this article, we have studied the bound'state energy eigenvalues and the corresponding eigenfunctions of mass-dependent generalized Woods-Saxon potential with equal scalar and vector potentials for the s-wave in the one-dimensional radial Klein-Gordon equation with the help of the Nikiforov-Uvarov method. Further, we discuss the PT-symmetric and nonPTVsymmetric cases for the potentials and a brief property check up for the corresponding Hamiltonian. This paper is organized as follows: To make it self-contained we give a brief review of N-U method in section 2. In section 3, we consider the generalized Woods-Saxon potential and obtain its relevant K-G equation. In section 3.1, we obtain the exact solution and energy eigenvalues by N-U method considering the as rest-mass. In section 3.2, we obtain the exact solution and energy eigenvalues by N-U method considering the as variable-mass. In section 4, we discuss the PT-symmetric generalized Woods-Saxon method with corresponding Hamiltonian. In section 5, we discuss the non-PTsymmetric case with corresponding Hamiltonian. And at last, section 6 is kept for conclusion. 2. Overview of Nikiforov-Uvarov Method. The N-U method is based on solving a second order linear differential equation by reducing it to a generalized hypergeometric type. In both relativistic and nonrelativistic quantum mechanics, the wave equation with a given potential can be solved by this method by reducing the one dimensional K-G equation to an equation of the form:

Where a(s) andCT(S)are polynomials of degree atmost 2 and f(s) is a polynomial of degree atmost 1. In "order to find a particular solution to equation (1), we use the following transformation: *(s) = $(s)y(s)

(2)

Thus equation (1) reduces to a hyper-geometric type equation of the form: a{s)y"(s) + T(s)y'(s) + Xy(s) = 0 Where T(S) = (s) + 2TT(S) satisfies the condition T'(S) < 0 and TT(S) is defined as

• . (3)

in which if is a parameter. Determining K is the essential point in calculation of n[s). It is obtained by setting the discriminant of the square root equal to zero. Therefore, one gets a general quadratic equation for K. By using A = K + TT'(S) = -n-r'(s) - n(n~^a"(s)

(4)

The values of K can used for the calculation of energy eigenvalues. Polynomial solutions yn{s) are given by the Rodrigues relation

EXACT SOLUTIONS OF THE KLEIN-GORDON EQUATION FOR THE MASS-DEPENDENT. . .

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in which Bn is a normalization constant and p(s) is the weight function satisfying (6)

on the other hand, second part of the wave function