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Analytical solution of the Schrodinger equation with spatially varying effective mass for generalised Hylleraas potential.

1. Introduction

The study of quantum mechanical systems within the framework of effective position dependent mass has been the subject of much activity in recent years. The Schroodinger equation with position-dependent (nonconstant) mass provides an interesting and useful model for the description of many physical problems. The most extensive use of such an equation is in the physics of semiconductor nanostructures [1,2], quantum dots [3], [sup.3]He clusters [4], quantum liquids [5], semiconductor heterostructures [6, 7], and so forth.

The solutions of nonrelativistic wave equations with constant mass have been extended to the position dependent mass in recent studies [8-11]. A general formalism for energy spectra and wave functions was found in nonrelativistic problems by using point canonical transformation [8]. Compared to the constant mass wave equation, the position-dependent mass Schrodinger equation is more complex. It is difficult to obtain its analytical solution as usual. Several authors have studied the effects of the position-dependent mass on the solutions of the Schrodinger equation. A position-dependent effective mass, m(x) = [m.sub.1] x m(x), associated with a quantum mechanical particle constitutes a useful model for the study of various potentials such as Morse potential [12-18], hard-core potential [18], Scarf potential [19-21], Poschl-Teller potential [22, 23], spherically ring-shaped potential [24], Hulthen potential [25], Kratzer potential [26], and Coulomb-like potential [27, 28]. Different techniques have been developed to obtain its exact solutions, such as factorization methods [29], Nikiforov-Uvarov (NU) methods [30], and supersymmetric quantum mechanics [31]. The position-dependent effective mass might have impact on high-energy physics [31].

The objective of this paper is to investigate the position-dependent effective mass Schroodinger equation for the generalised Hylleraas potential [32, 33] by using the Nikiforov-Uvarov (NU) method (Figure 1) [30]. Hylleraas potential is used to describe the interaction between two atoms in a diatomic molecule. We have also investigated the solutions of Hulthen potential and Woods-Saxon potential.

The plan of the present paper is as follows. In Section 2, the Nikiforov-Uvarov method is summarized. Section 3 is devoted to the solution of the position-dependent effective mass Schrodinger equation. In Sections 4 and 5, the Hulthen potential and Woods-Saxon potential are discussed, respectively. The paper is ended with a summary.

2. Nikiforov-Uvarov Method

The NU method is a useful technique to solve the second-order linear differential equations with special orthogonal functions [34]. In this method, after employing an appropriate coordinate transformation s = s(x), the nonrelativistic Schrodinger equation [d.sup.2][psi]/d[x.sup.2] + (E - V(x))[psi] = 0, ([??] = 2 m = 1) can be written in the following form:


where the prime denotes the differentiation with respect to s, [sigma](s) and [??](s) are polynomials, at most of second degree, and [??](s) is a polynomial, at most of first degree. In order to obtain a particular solution to 1), we set the following wave function as a multiple of two independent parts:

[psi](s) = [phi](s)[y.sub.n](s). (2)

Equation (2) transformed 1) to a hypergeometric-type equation:

[sigma](s) [y".sub.n](s) + [tau] (s) [y'.sub.n] (s) + [lambda][y.sub.n] (s) = 0, (3)

where first part of (2), [phi](s), has a logarithmic derivative:

[phi]'/[phi](s) = [pi](s)/[sigma](s), (4)

and second part of (2), [y.sub.n](s), is the hypergeometric-type function whose polynomial solution satisfies the Rodrigues relation:

[y.sub.n] (s) = [[C.sub.n]/[rho](s)[[d.sup.n]/d[s.sup.n]][[[sigma].sup.n](s)[rho](s)], (5)

where [C.sub.n] is normalization constant and the weight function [rho](s) satisfies the relation as

[d/ds][[sigma](s) [rho](s)] = [tau](s) [rho](s). (6)

The function [pi](s) and the eigenvalue [lambda] required in this method are defined as


k = [lambda] - [pi]' (s). (8)

Hence, the determination of k is the essential point in the calculation of [pi](s), for which the discriminant of the square root in (7) is set to zero. Also, the eigenvalue equation defined in (8) takes the following new form:

[lambda] = [[lambda].sub.n] = -n[tau]'(s) - [n(n - 1)/2] [sigma]"(s), n = 0,1,2, ..., (9)

[tau](s) = [??](s) + 2[pi]n (s), [tau]' (s) < 0. (10)

Since [rho](s) > 0 and [sigma](s) > 0, the derivative of [tau](s) should be negative [30], which helps to generate the essential condition for any choice of proper bound state solutions. In addition, the energy eigenvalues are obtained from (8) and 9).

3. Position-Dependent Effective Mass Schrodinger Equation

In general, working on position-dependent effective mass Hamiltonians is inspired by the von Roos Hamiltonian [35] proposal with [??] = [2m.sub.0] = 1:


where [??] = 2[m.sub.0] = 1 and m(x) is the dimensionless form of the function m(x) = [m.sub.1] x m(x). The ambiguity parameters are constrained by the relation [alpha] + [beta] + [gamma] = -1 and we have the following time-independent Schroodinger equation from (11):

H[phi](x) [equivalent to][-[[partial derivative].sub.x](1/m(x))[[partial derivative].sub.x] + [V.sub.eff] (x) - E] [phi] (x) = 0, (12)

where the effective potential is

[V.sub.eff](x) = V(x) + [1/2]([beta] + 1) [m"(x)/[m.sup.2](x)] - [[alpha]([alpha] + [beta] + 1) + ([beta] + 1)][[m'.sup.2](x)/[m.sup.3](x)], (13)

where primes denote derivatives. Thus Schrodinger equation takes the form

(-[1/m(x)][[d.sup.2]/d[x.sup.2]] + [m'(x)/[m.sup.2](x)][d/dx] + [V.sub.eff](x) - E) [phi](x) = 0. (14)

Using the transformation [36], [phi](x) = [m.sup.v](x)[psi](x) in (14), we have


where V(x) is the Hylleraas potential [29, 30] given by

V(x) = [V.sub.1][[a + [e.sup.[lambda]x]]/[b + [e.sup.[lambda]x]]] - [V.sub.2][[d + [e.sup.[lambda]x]]/[b + [e.sup.[lambda]x]]], (16)

where a([not equal to] b), b, and d([not equal to] b) are the Hylleraas parameters, [V.sub.1], [V.sub.2] are the potential depths, and -[infinity] < x < [infinity].

Here, we consider the following mass distribution:

m(x) = [M.sub.1]/[1 + b[e.sup.[lambda]x]]. (17)

The most extensive use of such knd of mass is in the physics of semiconductor quantum well structures [11]. The motion of electrons in them can often be described by the envelope function effective-mass Schrodinger equation, where [m.sub.1] is a constant mass (Figure 2).

Obviously it has the exponential form. The above mass function is convergent m(x) [right arrow] [m.sub.1](finite), when x [right arrow] [infinity]. In order to reduce the above equation (15) into Nikiforov-Uvarov equation, we make the transformation s = 1/(1 + b[e.sup.-[lambda]x]), (0 [less than or equal to] s [less than or equal to] 1):



m'(x)/m(x) = [lambda](1 - s),

m"(x)/m(x) = [[lambda].sup.2](1 - s)(1 - 2s). (19)

Using (15)-(19), we have

[[d.sup.2][psi]/d[s.sup.2]] + [[2v - (2v + 1) s]/s(1 - s)][d[psi]/ds]

+ [1/[s.sup.2][(1 - s).sup.2]][([[zeta].sup.2] - P)[s.sup.2] + (Q, 2[[zeta].sup.2])s + ([[zeta].sup.2] - R)][psi] = 0. (20)

Comparing (20) with 1), we have


Substituting these polynomials into (7), we have


For physical solutions, it is necessary to choose

[pi](s) = [zeta](1 - s) - ([square root of R] + [epsilon]) s + [square root of R]

if k = Q - 2R -2[epsilon][square root of R]. (23)

The origin of the Nikiforov-Uvarov method is negative sign of derivative of [tau]'(s) because the condition [tau]'(s) < 0 helps to generate energy eigenvalues and corresponding eigenfunctions. Therefore t(s) becomes

[tau](s) = 1 + 2[square root of R] - 2(2 + 2[square root of R] + 2[epsilon])s. (24)

Therefore, from (8) and 9), we have

[lambda] = Q - 2R - 2[epsilon][square root of R] - [zeta] -([square root of R] + [epsilon]),

[lambda] = [[lambda].sub.n] = 2n([square root of R] + [epsilon]) + n(n + 1). (25)

Comparing (25), we have

[square root of R] + [epsilon] = -(n + [1/2]) + [square root of (P - v + 1/4)],

[square root of R] - [epsilon] = [Q-P]/-(n + 1/2) + [square root of (P - v + [1/4])]. (26)

Using (26) and (18), we have

[[epsilon].sub.2] = [1/4][2n + 1 + 2 [square root of R] - [square root of (P - v + [1/4])]]. (27)

Hence the energy becomes


The last term of the square bracket must be positive for Ben-Daniel and Duke's model [37] ([alpha] = v = 0, [beta] = -1) (Figure 3). From (6), (21), and (24) we obtain the weight function

[rho](s) = [s.sup.2[square root of R]] [(1 - s).sup.2[epsilon]], (29)

and from (4), (21), and (23) we have

[phi](s) = [s.sup.[square root of R] + [zeta]] [(1 - s).sup.[epsilon]. (30)

Now we use the properties of Jacobi polynomial [35]:


where [P.sup.(a,b).sub.n] (a > -1, b > -1) is the Jacobi polynomial. The wave functions (Figure 4) are obtained from (2), (5), and ((29)-(31)):

[[psi].sub.n](s) = [N.sub.n][s.sup.[square root of R]+[zeta]][(1 - s).sup.[epsilon]][P.sup.(2[square root of R]2[epsilon]).sub.n] (1 - 2s), (32)

where [N.sub.n] is normalization constant to be determined from the normalization condition:

[[integral].sup.[infinity].sub.-[infinity]][[absolute value of [[psi].sub.n](x)].sup.2]dx = 1 = [[integral].sup.1.sub.0][[absolute value of [[psi].sub.n](s)].sup.2]ds. (33)

For acceptable solution it is required that [absolute value of [square root of R] + [zeta]] [greater than or equal to] [epsilon] when [square root of R] + [zeta] < 0, [epsilon] > 0 and [square root of R] + [zeta] [less than or equal to] [absolute value of [epsilon]] when [square root of R] + [zeta] > 0, [epsilon] < 0.

4. Hulthen Potential

We set the conditions [V.sub.2] = [V.sub.1], a = 1 + d, and b = -q; the potential in (16) reduces to Hulthen potential (Figure 5) [38-41]:

V(x) = -[V.sub.1][[e.sup.-[lambda]x]/[1 - q[e.sup.-[lambda]x]]]. (34)

Then the energy becomes


with 0 [less than or equal to] n < [infinity]. It is exactly the same result in the literature [38] for [m.sup.1] = 1.

5. Woods-Saxon Potential

For the conditions [V.sub.1] = -[V.sub.0], a = 0, and [V.sub.2] = 0, the potential given in 16) becomes Woods-Saxon potential (Figure 6) [40-45],

V(x) = -[V.sub.0]1/[1 + b[e.sup.-[lambda]x]]. (36)

Then the energy becomes


where 0 [less than or equal to] n < [infinity].

6. Conclusion

We have applied the NU method derived for the exponential-type potentials to obtain the bound state solutions of the effective Schrodinger equation with position-dependent mass for the Hylleraas potential. Furthermore, a suitable choice of a position mass function of the exponential-like form has also been devised. Also we have shown that our results are consistent with ones obtained before.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors wish to deliver their sincere gratitude to the referee for constructive suggestions and technical comments on the paper.


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Sanjib Meyur, Smarajit Maji, and S. Debnath

Department of Mathematics, Jadavpur University, Kolkata 700032, India

Correspondence should be addressed to Sanjib Meyur;

Received 30 May 2014; Revised 19 July 2014; Accepted 21 July 2014; Published 11 August 2014

Academic Editor: Shi-Hai Dong
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Author:Meyur, Sanjib; Maji, Smarajit; Debnath, S.
Publication:Advances in High Energy Physics
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Date:Jan 1, 2014
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