1. Introduction

The investigations of the analytical solutions of the Schrödinger equation with position-dependent mass is one of the important topics in quantum mechanical sciences. The concept of the local mass has interesting features in the large scale characteristics of the universe and in gravitational field theories.^[1–4] The position-dependent mass distribution also has received a great deal of attention in more terrestrial sciences, such as the material science, condensed matter physics, and semiconductors nanosubstructures,^[5–7] quantum wells and quantum dots,^[8–10] quantum liquids,^[11] impurities in crystals,^[12] and ³He clusters.^[13,14]

The Hermitian Hamiltonian of von Roos with local mass distribution m = m(r) is given by,^[5]

where P denotes the momentum operator and V(r) is an arbitrary potential. Also, α, β, and γ are the ambiguity parameters satisfying the dimensional constrain α +β +γ = −1. Several sets of the values of ambiguity parameters are suggested in the literature.^[15–19] Here, we apply the ambiguity parameter set suggested by BenDaniel and Duke (α = γ,β = −1).^[15] This model ensures the continuities required at the heterojunction boundaries between two crystals.

Amongst the distinct methods that have been applied to solve the Schrödinger equation, one may mention the point-canonical transformation method,^[20–23] the factorization scheme,^[24,25] the supersymmetry approach,^[26–28] the algebraic way,^[29] the power series expansion,^[30,31] the transfer matrix method,^[32,33] the asymptotic iteration method,^[34–36] and the Nikiforov–Uvarov approach.^[37–40]

One of the most effective methods for solving the Schrödinger equation with different sorts of spherically symmetric potentials is the Laplace transformation method.^[41] The advantage of this method is that a second-order differential equation reduces to a first-order differential equation. It was Schrödinger who used this technique for the first time in quantum physics in order to solve the radial eigenfunction of a hydrogen atom.^[42] The method has become commonly employed ever since to solve various kinds of spherically symmetric potentials.^[43–50]

One of the significant potentials is the Morse potential which is applied for describing the vibrational and rotational movements of the diatomic molecules.^[51] Solutions of the Schrödinger equation with the Morse potential are investigated by applying different methods in Refs. [52]–[61].

Here we introduce an analytical bound state solution of the Schrödinger equation with position-dependent mass. We apply the Laplace transformation method to obtain the eigenvalues and the eigenfunctions in a closed form. The point-canonical transformation method also has been applied^[56,62] in order to get rid of the first derivative of the wave function. In this way, we can suitably separate the terms contained in the mass function from the terms of the field derivatives. Then, by applying some simple constraints, on the terms included in the mass function, we construct the mass function in such a way that the corresponding equation can be solved via the Laplace transformation method used in Refs. [3] and [46]. The obtained mass distribution is a physical one, that is positive-valued, convergent, and well localized around the equilibrium position of a typical molecule. The mass function depends on the ordinary parameters of the Morse potential. The obtained mass function also depends on the energy spectrum of the Morse oscillator.

The organization of this paper is as follows. In Section 2, we firstly introduce the initial form of the mass function with undetermined parameters. Then by considering the desired form of the Schrödinger equation which is most solvable with the Laplace transforms, we setup some constraints on our mass parameters to find the final form of the mass distribution. The parameters of the Morse potential become untouched. Then, in Section 3 we solve the equation and introduce a possible bound state solution of the model. Finally, some results and discussions are presented in Section 4.

2. A possible mass distribuion

For the BenDaniel–Duke Hamiltonian, the one-dimensional Schrödinger equation, in position representation, is given bycolumnbreak

The first derivative can be eliminated from the right-hand side of Eq. (2) by using the transformation:

This technique is based on the point-canonical transformation method.^[56,62] Substituting Eq. (3) into Eq. (2), one obtains

In the case of the Morse potential,^[51] equation (4) turns into the following equation

Here

and r₀ is the equilibrium position of the molecules and the prime sign denotes differentiation with respect to x. The parameter D describes the depth of the potential and the dimensionless parameter α characterizes the potential acting range. Applying

where m₀ is a mass dimensional parameter, equation (5) turns into

where

Here, we use the following effective mass distribution

where τ, γ, and η are constant parameters. Inserting Eq. (10) into Eq. (8) yields

with this providing that the constants D₁, D₂, and D₃ must be

Also, the mass parameters in Eq. (10) must be of the following form

Substituting the above equations into Eq. (10) leads to

The mass function (14) is positive and convergent for y < y_c. However, y has an upper limit y_max = k e^α which can be found for x_min = −1 in Eq. (7). Therefore, this condition is well satisfied for y_max < y_c.

3. Bound state solutions

In the presence of the mass function (14) our Schrödinger equation turns into Eq. (11). In order to have finite solutions at large values of y, we should apply the following ansatz

then equation (11) becomes

Here we use the Laplace transformation method to solve Eq. (16),^[3,46] By applying Laplace transform^[41]

the following equation can be obtained

where and from Eq. (12) D₃ < 0. Equation (17) is a first-order differential equation and its solutions are in the form

where N′ is a constant. Here the term

is a multi-valued function. In order to have a single valued wave function we impose the condition

Now inserting Eqs. (7), (9), and (12) into Eq. (19), the eigenvalues of the bound states can be obtained as

In order to find the eigenfunctions, we should apply inverse Laplace transforms to Eq. (18). However, in order to do so, it needs to be expanded in power series. This leads to

After applying the inverse Laplace transform, we find

The series expansion of the confluent hypergeometric functions is given as

Comparing Eq. (22) with Eq. (23) yields

Inserting Eq. (24) into Eq. (15) and then Eq. (3) and using Eq. (14), leads to

where

and equations (9) and (20) have also been used. Here N_n is the normalization factor.

4. Discussions

This study has introduced a possible mass distribution for the Morse potential, which has an analytical bound state solution via the Laplace transformation method. We have found a closed form for the eigenvalue and eigenfunction of the corresponding one dimensional Schrödinger equation. The prominent point of this study is the considered mass distribution which depends on the quantum number n. In fact, using Eqs. (12) and (26), the mass distribution Eq. (14) yields

This mass function is convergent and positive for

However, the upper bound of y is given by

Therefore, by considering Eq. (26), our mass function is meaningful under the condition

or

This relation establishes a condition for the starting point of the quantum number n. The case n ≥ 0 works only for α < 0.693. In the case of H₂ and HCl molecules with α = 1.440 and α = 2.380 respectively, conditions (28a) and (28b) require n ≥ 1. However for I₂, with α = 4.954, we must have n ≥ 2. In order to have a schematic representation of the considered mass function, the distribution m/m₀ is plotted in Fig. 1 for some typical molecules.

According to Fig. 1, the mass function is well concentrated around r = r₀.

Figure (2a) shows the energy spectra for three molecules H₂ (D = 38292 cm,⁻¹), HCl (D = 37244 cm⁻¹), and I₂ (D = 12550 cm⁻¹). Figure(2b) presents the solutions of the eigenfunctions for H₂ molecules (k = 34.9) and for different quantum numbers n.