Analytical Solution of the Schrödinger Equation with Spatially Varying Effective Mass for Generalised Hylleraas Potential

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 Schrödinger 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], quantumdots [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 andwave functionswas found in nonrelativistic problems by using point canonical transformation [8]. Compared to the constant mass wave equation, the position-dependent mass Schrödinger 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 Schrödinger equation. A positiondependent effective mass,m(x) = m 1 ⋅ m(x), associated with a quantummechanical 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], PöschlTeller potential [22, 23], spherically ring-shaped potential [24], Hulthén 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 positiondependent effective mass might have impact on high-energy physics [31]. The objective of this paper is to investigate the positiondependent effective mass Schrödinger equation for the generalised Hylleraas potential [32, 33] by using the NikiforovUvarov (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 Hulthén 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 Schrödinger equation. In Sections 4 and 5, the Hulthén potential and Woods-Saxon potential are discussed, respectively. The paper is ended with a summary.


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 Schrödinger 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], 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][9][10][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 Schrödinger 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 Schrödinger equation. A positiondependent effective mass, ( ) = 1 ⋅ ( ), associated with a quantum mechanical particle constitutes a useful model for the study of various potentials such as Morse potential [12][13][14][15][16][17][18], hard-core potential [18], Scarf potential [19][20][21], Pöschl-Teller potential [22,23], spherically ring-shaped potential [24], Hulthén 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 positiondependent effective mass might have impact on high-energy physics [31].
The objective of this paper is to investigate the positiondependent effective mass Schrödinger 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 Hulthén 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 Schrödinger equation. In Sections 4 and 5, the Hulthén potential and Woods-Saxon potential are discussed, respectively. The paper is ended with a summary.  Schrödinger equation 2 / 2 + ( − ( )) = 0, (ℏ = 2 = 1) can be written in the following form: where the prime denotes the differentiation with respect to , ( ) and̃( ) are polynomials, at most of second degree, and ( ) 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: Equation (2) transformed (1) to a hypergeometric-type equation: where first part of (2), ( ), has a logarithmic derivative: and second part of (2), ( ), is the hypergeometric-type function whose polynomial solution satisfies the Rodrigues relation: where is normalization constant and the weight function ( ) satisfies the relation as The function ( ) and the eigenvalue required in this method are defined as = − ( ) .
Hence, the determination of is the essential point in the calculation of ( ), 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: Since ( ) > 0 and ( ) > 0, the derivative of ( ) 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).
Here, we consider the following mass distribution: The most extensive use of such kind 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 Schrödinger equation, where 1 is a constant mass (Figure 2). Obviously it has the exponential form. The above mass function is convergent ( ) → 1 (finite), when → ∞.
In order to reduce the above equation (15) Using (15) Comparing (20) with (1) Substituting these polynomials into (7), we have For physical solutions, it is necessary to choose 4

Conclusion
We have applied the NU method derived for the exponentialtype potentials to obtain the bound state solutions of the effective Schrödinger 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.