Position Dependent Mass for the Hulthén plus Hyperbolic Cotangent Potential

Transcription

1 Bulg. J. Phys. 38 (011) Position Dependent Mass for the Hulthén plus Hyperbolic Cotangent Potential Tansuk Rai Ganapat Rai Khemka High School, 3, Rabindra Sarani, Liluah, Howrah-71104, West Bengal, India. Received 15 August 011 Abstract. Exactly solvable Schrödinger equations with a position-dependent mass are proposed. Nikiforov-Uvarov method is used to obtain the energy eigenvalues and the corresponding wavefunctions. PACS codes: w, Ge, Fd, Pb 1 Introduction Schrödinger equations with a position-dependent mass is a very useful model in many applied branches of modern physics. Special applications in condensed matter are found in the investigation of electronic properties of semiconductors [1-], quantum wells and quantum dots [3-6], 3 He clusters [7], quantum liquids [8], graded alloys and semiconductor heterostructures [9], etc. In these cases, the envelope wavefunction actually provides a microscopic view of the motion of carrier electrons with position-dependent mass. In theoretical researches, many different methods have been used in the study of systems with constant mass such as the Lie algebraic techniques[10], point canonical transformation, factorization method [11] and supersymmetric quantum mechanics together with shape invariance techniques [1-13]. Recently, some of these developments have been generalized to the systems with position-dependent effective mass and number of interesting results have been discovered [14-19]. The objective of this paper is to investigate the position-dependent effective mass Schrödinger equation by using a transformation of the wave function. We applied the Nikiforov-Uvarov method to solve the position-dependent effective mass Schrödinger equation in the presence of the Hulthén plus hyperbolic cotangent potential. The plan of the present paper is as follows. In Section, the Nikiforov-Uvarov method is summarized. Section 3 is devoted to the solution of the position c 011 Heron Press Ltd. 357

2 dependent effective mass Schrödinger equation. The paper is ended with a summary. Nikiforov-Uvarov Method In this method[0], the one dimensional Schrödinger equation is reduced to an equation by suitable coordinate transformation x = x(s), ψ (s) + τ(s) σ(s) ψ (s) + σ(s) σ ψ(s) = 0, (1) (s) where σ(s) and σ(s) are polynomials, at most of second degree, and τ(s) is a polynomial, at most of first degree. In order to obtain a particular solution to Eq. (1), we set the following wave function as a multiple of two independent parts: ψ(s) = φ(s)y(s). () Using Eq. (1) and Eq. () we have σ(s)y (s) + τ(s)y (s) + λy(s) = 0, (3) which demands the following conditions to be satisfied: φ (s) φ(s) = π(s) σ(s) (4) τ(s) = τ(s) + π(s), τ (s) < 0. (5) The condition τ (s) < 0 helps to generate physical solutions. The λ in (3) satisfies the following second-order differential equation: λ = λ n = nτ (s) n(n 1) σ (s), n = 0, 1,,... (6) It is to be noted that λ or λ n are obtained from a particular solution of the form y(s) = y n (s) d n y n (s) = C n ρ(s) ds n [σn (s)ρ(s)], (7) where C n is normalization constant and the weight function ρ(s) satisfies the relation as d [σ(s)ρ(s)] = τ(s)ρ(s). (8) ds On the other hand, in order to find the eigenfunctions, φ n (s) and y n (s) in Eqs. (4) and (7) and eigenvalues λ n in Eq. (6), we need to calculate the functions 358 ( σ τ π(s) = ) ± (σ ) τ σ + kσ (9)

3 Position Dependent Mass for the Hulthén plus Hyperbolic Cotangent Potential k = λ π (s). (10) In principle, since π(s) has to be a polynomial of degree at most one, the expression under the square root sign in Eq.(9) can be put into order to be the square of a polynomial of first degree, which is possible only if its discriminant is zero. Thus, the equation for k obtained from the solution of Eq.(9) can be further substituted in Eq.(10). In addition, the energy eigenvalues are obtained from Eqs.(6) and (10). 3 Position-Dependent Effective Mass Schrödinger Equation The general Hermitian position-dependent effective mass Hamiltonian, is given by [1] [ 1 ( M α (x) d dx Mβ (x) d dx Mγ (x)+ M γ (x) d dx Mβ (x) d dx Mα (x) ) ] + V (x) ϕ(x) = Eϕ(x), (11) where = m 0 = 1 and M(x) is the dimensionless form of the function m(x) = m 0 M(x). The ambiguity parameters are constrained by the relation α + β + γ = 1 and we have the following time-independent Schrödinger equation from Eq. (11) Hϕ(x) [ d dx where the effective potential is ( 1 d M(x) dx ) ] + V eff (x) E ϕ(x) = 0, (1) V eff (x) = V (x) + 1 (β + 1) M (x) M (x) [α(α+β+1)+(β+1)]m (x) M 3 (x). (13) Thus Schrödinger equation takes the form ( 1 d M(x) dx + M ) (x) d M (x) dx + V eff(x) E ϕ(x) = 0. (14) Using the transformation[], ϕ(x) = M η (x)ψ(x) in Eq. (14), we have { d dx (η (x) d M 1)M (η(η ) + α(α + β + 1) + β + 1) M(x) dx M ( ) 1 M } (x) + (β + 1) η + M(x)(V (x) E) ψ(x) = 0, (15) M(x) 359

4 where V (x) is the Hulthén[3-4] plus deformed type cotangent potential given by e λx V (x) = V 1 1 qe λx + V coth q (λx). (16) In order to reduce the above Eq.(15) into Nikiforov-Uvarov equation, we make the transformation 1 s = M(x) = 1 qe λx (17) and setting A = η(η ) + α(α + β + 1) + β + 1 B = ( 1 (β + 1) η) P = A + B V1 4qλ + V 4λ Q = A + 3B V1 4qλ V 4λ + E 4λ (18) R = A + B δ = V E ( 4λ) ( ) µ = β α + β+1 also M (x) = 4λ (1 s) M(x) M. (19) (x) = 4λ (1 s)(1 s) M(x) Using Eqs. (15) (19), we have d ψ η (η + 1)s dψ + ds s(1 s) ds 1 [ Ps s (1 s) + Qs R ] ψ = 0. (0) Comparing Eq.(0) with Eq.(1), we have σ(s) = s(1 s) σ(s) = Ps + Qs R τ(s) = η (η + 1)s (1) Substituting these polynomials into Eq.(9), we have ( ) 1 (δ µ)s + µ; k = Q R + δµ π(s) = η (1 s) ±. () (δ + µ)s µ; k = Q R δµ For physical solutions, it is necessary to choose ( ) 1 π(s) = η (1 s) (δ + µ)s + µ 360 if k = k = Q R δµ. (3)

5 Position Dependent Mass for the Hulthén plus Hyperbolic Cotangent Potential The following track in this selection is to achieve the condition τ (s) < 0. Therefore, τ(s) becomes Therefore, from Eqs. (6) and (10), we have τ(s) = 1 s ((δ + µ)s µ). (4) λ = λ n = n( (δ + µ)) + n(n 1) (5) and λ = P (δ + µ) (δ + µ) + η 1 4. (6) Comparing Eqs. (5) and (6), we have ( δ + µ = n + 1 ) ± P + η. (7) Hence two energy levels are E n = 4λ [ ( n + 1 (β ) qv V 1 ± 4qλ α(α + β + 1) ) ( α + β + 1 ) + V. (8) For acceptable solution it is required that 1 η + µ δ when 1 η + µ < 0, δ > 0 and 1 η + µ δ when 1 η + µ > 0, δ < 0. From Eqs. (5), (8), and (4), we obtain the weight function and from Eqs. (4), (1), and (3), we have ρ(s) = s µ (1 s) δ (9) φ(s) = s (1 η)+µ (1 s) δ. (30) Now using the properties of Jacobi Polynomial [5] P n (µ,δ) (x) = ( 1)n (1 x) µ (1 + x) δ d n [ (1 x) n+µ n n! dx n (1 + x) n+δ] P n (µ,δ) (1 s) = ( )n (s) µ (1 s) δ d n [ s n+µ n n! dx n (1 s) n+δ] (31) The wave functions are obtained from Eqs. (), (7), (5-30) ψ n (s) = N n s (1 η)+µ (1 s) δ P (µ,δ) n (1 s), (3) 361

6 where N n is normalization constant to be determined from the normalization condition ψ n (s) ds = 1. (33) Two different forms of Jacobi Polynomials [5] are n ( )( ) n + µ n + δ P n (µ,δ) (x)= n ( 1) n p (1 x) n p (1 + x) p, (34) p n p p=0 P n (µ,δ) (x)= Γ(n+µ+1) n ( ) ( ) r n Γ(n+µ+δ+r+1) x 1, (35) n!γ(n+µ+δ+1) r Γ(r+µ+1) where r=0 ( ) n n! = r r!(n r)! = Γ(n + 1) Γ(r + 1)Γ(n r + 1) P (µ,δ) n (1 s) = ( 1) n Γ(n + µ + 1)Γ(n + δ + 1) n ( 1) p s n p (1 s) p p!(n p)! Γ(p + δ + 1)Γ(n + µ p + 1) p=0 (36) P n (µ,δ) (1 s) = ( 1) n Γ(n + µ + 1) Γ(n + µ + δ + 1) n r=0 ( 1) r r!(n r)! Γ(n + µ + δ + r + 1) s r (37) Γ(µ + r + 1) 1 = Nn Γ(n + µ + ( 1)n 1) Γ(n + δ + 1) Γ(µ + δ + 1) n ( 1) p+r Γ(n + µ + δ + r + 1)I n (p, r) p!r!(n p)!(n r)!γ(p+δ+1)γ(r+δ+1)γ(n+µ p+1), (38) where p,r=0 I n (p, r)= 4 Conclusion s n+µ+r p (1 s) p+δ+1 ds= (n+µ+r p)!(p+δ)!. (39) (n+µ+δ+r p)! In this paper, the solutions of the position-dependent effective mass Schrödinger equation for the Hulthén plus hyperbolic cotangent potential have been investigated by Nikiforov-Uvarov method. It has been shown that, both the wavefunctions and the corresponding energy spectra of the system have an exact. An appropriate mass function has been introduced for solving the position-dependent effective mass Schrödinger equation. 36

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