Supersymmetric Approach for Eckart Potential Using the NU Method

Transcription

1 Adv. Studies Theor. Phys., Vol. 5, 011, no. 10, Supersymmetric Approach for Eckart Potential Using the NU Method H. Goudarzi 1 and V. Vahidi Department of Physics, Faculty of Science Urmia University, Urmia, P.O. Box: 165, Iran Abstract Using supersymmetric approach, the Schrödinger equation for various potentials can be exactly solved. In this work, the solutions of Schrödinger equation for Eckart potential is exactly obtained via the Nikiforov-Uvarov method. The corresponding energy eigenstates and energy eigenvalue of Hamiltonian analytically are found. PACS: 1.0.-m; m; Et Keywords: Supersymmetric quantum mechanics; Eckart potential; Nikiforov- Uvarov method 1. Introduction Supersymmetric quantum mechanics (SUSYQM) is a framework used to determine energy eigenvalues and eigenfunctions of quantum mechanical Hamiltonians. However, many authors have developed various methods to solve the Schrödinger equation with taking into account different form of potentials in the Hamiltonian[1]. Supersymmetric approach was developed later by Infeld and Hull to classify the analytically solvable problems []. Recently many researchers have worked on solving these equation with physical potentials including Morse potential [3-5], Pöschl-Teller potential [6,7], Hulthén potential [8,9], Non-central potential [10], Woods-Saxon potential [11,1], Rosen-Morse (trigonometric) potential and Scarf potential[13]. In the present work, we introduce an alternative, elegant and simple method for an algebraic solution of the Schrödinger equation with Eckart potential. This method is called the Nikiforov-Uvarov (NU) method [14] and is based on solving the second-order linear differential equations by reducing to a generalized equation of hypergeometric type [10]. In particular, there have been 1 h.goudarzi@urmia.ac.ir, goudarzia@phys.msu.ru

2 470 H. Goudarzi and V. Vahidi several applications of the NU method in exactly solving of Schrödinger equation with some well-known potentials [15-17], Dirac, Klein-Gordon and Duffin- Kemmer-Petiau equations for Coulomb type potential [18-0]. This paper is organized as follows. In section, we give a summary of the supersymmetric quantum mechanics. In section 3, we review the Nikiforov- Uvarov (NU) method briefly. In section 4, by calculating superpotential and some required functions according NU method, the explicit relations for energy spectrum and also wave functions for Schrödinger equation with Eckart potential are obtained. In section 5, we discuss the results.. Summary of SUSYQM Supersymmetric algebra allows us to write Hamiltonians as [1] H ± = h m dx + V ±(x), (1) where the Supersymmetric partner potentials V ± in terms of the superpotential W (x) are given by d V ± (x) =W (x) ± The superpotential has a definition W (x) = h d m dx h dw (x) m dx. () [ ln ψ (0) 0 (x) ], (3) where ψ (0) 0 (x) denotes the ground state wave function that satisfies the relation [ ] ψ (0) m x 0 (x) =N 0 exp W (x )dx, (4) h where N 0 -is a normalization constant. The Hamiltonian H ± can also be written in terms of the bosonic operators A and A + as, H ± = A A ±, (5) where A ± = ± h d + W (x). (6) m dx It is remarkable that the energy eigenvalues of H and H + are identical except for the ground state. In the case of unbroken supersymmetry, the ground state energy of the Hamiltonian H is zero (E (0) 0 = 0) [1]. In the

3 Supersymmetric approach for Eckart potential 471 factorization of the Hamiltonian, Eqs. (1), (5) and (6) are used, respectively. Hence, we obtain for physical Hamiltonian H 1 (x) = h d m dx + V 1(x) =(A + 1 A 1 )+E(0) 1. (7) Comparing each side of Eq. (7) term by term, we get the Riccati equation for the superpotential W 1 (x) W 1 W 1 = m h ( V1 (x) E (0) ) 1. (8) Let us now construct the supersymmetric partner Hamiltonian H as H (x) = h d m dx + V (x) =(A A+ )+E(0), (9) and Riccati equation takes W W = m h ( V (x) E (0) ). (10) Similarly, one can write in general the Riccati equation and Hamiltonians by iteration as Wn ± W n = m ( ) h Vn (x) E n (0) =(A ± n A n )+E(0) n, (11) and where H n (x) = h d m dx + V n(x) =A + n A n + E(0) n ; n =1,, 3,... (1) A ± n = ± h d m dx + d dx ( ln ψ (0) n (x) ). (13) Because of the SUSY unbroken case, the partner Hamiltonian satisfy the following expressions [1,] E (0) n+1 = E (1) n, with E (0) 0 =0; n =0, 1,,... (14) and also the wave function with the same eigenvalue can be written as [1] with n = A ψ (0) n+1, (15) ψ (1) ψ (0) E (0) n n+1 = A+ ψ n (1). (16) E (0) n This procedure is known as the hierarchy of Hamiltonians.

4 47 H. Goudarzi and V. Vahidi 3. A Review of 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, (17) σ(s) ds σ (s) where σ(s) and σ(s) are polynomials, at most second degree, and τ(s) isa 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. (18) σ(s) ds σ(s) The function φ(s) is defined as a logarithmic derivative, φ (s) φ(s) = π(s) σ(s), (19) 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)], (0) where a n is a normalization constant, and ρ(s) is the weight function satisfying the following equation, [ρ(s)σ(s)] = τ(s)ρ(s). (1) The function π(s) and the parameter λ required for this method are defined as ) (s) τ(s) ( π(s) = σ σ (s) τ(s) ± σ(s)+kσ(s), () λ = k + π (s). (3) 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, λ = λ n = τ n(n 1) d σ(s), (4) ds where the derivation of the function τ(s) = τ(s) + π(s) should be negative, and by comparing Eqs. (3) and (4), we obtain the energy eigenvalues.

5 Supersymmetric approach for Eckart potential Supersymmetric Solution of Schrodinger Equation with Eckart Potential Now, we consider the Schrodinger type second order differential equation with the Eckart potential e x a x a V Ec = β (1 e x a ) α1+e, (0 < x <, β<α), (5) 1 e x a a where the parameters α and β describe the depth of the potential well, which both α and β are greater than zero; a is a parameter to control the width of the potential well [3]. One can easily obtain its superpotential as W (x) = A cot αx + B csc αx, (6) where 0 αx π, A- is a constant and A>B. It is possible to neglect the second term in W (x) because of B<A. From Eq. (), we calculate its superpartners V ± (x, A) =A cot αx ± h αacsc αx. (7) m Substituting above superpartners in Hamiltonian Eq. (1) and taking alone the its minus components, the Schrödinger equation can be written explicitly as d ψ(x) dx + m h [Ẽ Zcot αx ] ψ(x) =0, (8) where Ẽ = E + hαa m, Z = A(A hα m ). By introducing a transformation Eq. (8) takes the following form s = cotαx, (9) d ψ(s) + s dψ(s) 1 + ds (1 + s ) ds (1 + s ) [β γ s ]ψ(s) =0, (30) where β = mẽ α h and γ = mz α h. Comparing Eq. (30) with Eq. (17), one can obtain the exact expressions for determined functions in NU method: σ(s) = (1 + s ), τ =s, σ(s) =(β γ s ). (31) Substituting these polynomials in Eq. (), we get for π(s) function as follows: π(s) =± (γ + k)s + k β, (3) where the constant k is determined as k 1 = γ, k = β. (33)

6 474 H. Goudarzi and V. Vahidi Thus we obtain for π(s) π(s) =± γ β, for k = γ, (34) π(s) =± β + γ s, for k = β. (35) A proper value for π(s) is chosen, so that function τ(s) =s s β + γ, (36) has a negative derivative [14]. Using Eq. (4), we can find λ = β β + γ, (37) and from Eq. (3) also we obtain λ = n n +n β + γ. (38) Hence, using Eqs. (37) and (38) the energy eigenvalues are found as Ẽ n1, = Dα h [ 1 ] m F n(1 ± F ) n, (39) where D = hαa m h α and F = 1 m 1+4γ. Now, we are going to determine the eigenfunctions for this system. By using Eqs. (19) and (1), we obtain φ(s) = (1 + s ) N, ρ(s) = (1 + s ) N, (40) where N = β + γ. Hence, hypergeometric function for solution of differential equation (18) using the polynomial Eq. (0) is obtained as y n (s) =a n (1 + s ) N dn [ (1 + s ) n (1 + s ) ] N. (41) ds n Finally, the wave functions ψ n (s) for such system can be written in the following form ψ n (s) =a n (1 + s ) N d n [ (1 + s ) ] n N, (4) ds n with s = cotαx. The normalization constant a n of wave function may be analytically calculated. 5. Conclusions To summarize, we have considered the supersymmetric exact solution of Schrödinger equation with Eckart potential by applying the NU-method. The eigenfunction and corresponding eigenvalues of this potential are analytically calculated. It seems that this supersymmetric solution via the NU-method may be more effective for obtaining exact solutions of some potentials.

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