Advances in High Energy Physics

Volume 2015, Article ID 137038, 8 pages

http://dx.doi.org/10.1155/2015/137038

## Exact Analytical Solution of the -Dimensional Radial Schrödinger Equation with Pseudoharmonic Potential via Laplace Transform Approach

^{1}Kodalia Prasanna Banga High School (H.S), South 24 Parganas, Sonarpur 700146, India^{2}Department of Physics Education, Hacettepe University, 06800 Ankara, Turkey

Received 16 September 2015; Revised 1 December 2015; Accepted 7 December 2015

Academic Editor: Ming Liu

Copyright © 2015 Tapas Das and Altuğ Arda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The second-order -dimensional Schrödinger equation with pseudoharmonic potential is reduced to a first-order differential equation by using the Laplace transform approach and exact bound state solutions are obtained using convolution theorem. Some special cases are verified and variations of energy eigenvalues as a function of dimension are furnished. To give an extra depth of this paper, the present approach is also briefly investigated for generalized Morse potential as an example.

*The author Tapas Das wishes to dedicate this work to his wife Sonali for her love and care*

#### 1. Introduction

Schrödinger equation has long been recognized as an essential tool for the study of atoms, nuclei, and molecules and their spectral behaviors. Much effort has been spent to find the exact bound state solution of this nonrelativistic equation for various potentials describing the nature of bonding or the nature of vibration of quantum systems. A large number of research workers all around the world continue to study the ever fascinating Schrödinger equation, which has wide application over vast areas of theoretical physics. The Schrödinger equation is traditionally solved by operator algebraic method [1], power series method [2, 3], or path integral method [4].

There are various other alternative methods in the literature to solve Schrödinger equation such as Fourier transform method [5–7], Nikiforov-Uvarov method [8], asymptotic iteration method [9], and SUSYQM [10]. The Laplace transformation method is also an alternative method in the list and it has a long history. The LTA was first used by Schrödinger to derive the radial eigenfunctions of the hydrogen atom [11]. Later Englefield used LTA to solve the Coulomb, oscillator, exponential, and Yamaguchi potentials [12]. Using the same methodology, the Schrödinger equation has also been solved for various other potentials, such as pseudoharmonic [13], Dirac delta [14], and Morse-type [15, 16] and harmonic oscillator [17] specially on lower dimensions.

Recently, -dimensional Schrödinger equations have received focal attention in the literature. The hydrogen atom in five dimensions and isotropic oscillator in eight dimensions have been discussed by Davtyan and coworkers [18]. Chatterjee has reviewed several methods commonly adopted for the study of -dimensional Schrödinger equations in the large limit [19], where a relevant expansion can be used. Later Yáñez et al. have investigated the position and momentum information entropies of -dimensional system [20]. The quantization of angular momentum in -dimensions has been described by Al-Jaber [21]. Other recent studies of Schrödinger equation in higher dimension include isotropic harmonic oscillator plus inverse quadratic potential [22], -dimensional radial Schrödinger equation with the Coulomb potential [23]. Some recent works on -dimensional Schrödinger equation can be found in the references list [24–31].

These higher dimension studies facilitate a general treatment of the problem in such a manner that one can obtain the required results in lower dimensions just dialing appropriate . The pseudoharmonic potential is expressed in the form [32] where is the dissociation energy with the force constant and is the equilibrium constant. The pseudoharmonic potential is generally used to describe the rotovibrational states of diatomic molecules and nuclear rotations and vibrations. Moreover, the pseudoharmonic potential and some kinds of it for -dimensional Schrödinger equation help to test the powerfulness of different analytical methods for solving differential equations. To give an example, the dynamical algebra of the Schrödinger equation in -dimension has been studied by using pseudoharmonic potential [33]. Taşeli and Zafer have tested the accuracy of expanding of the eigenfunction in a Fourier-Bessel basis [34] and a Laguerre basis [35] with the help of different type of polynomial potentials in -dimension.

Motivated by these types of works, in this present paper we discuss the exact solutions of the -dimensional radial Schrödinger equation with pseudoharmonic potential using the Laplace transform approach. To make this paper self-contained we briefly outline Laplace transform method and convolution theorem in the next section. Section 3 is for the bound state spectrum of the potential system. Section 4 is devoted to the results and discussion where we derive some well known results for special cases of the potential. The application of the present method is shown in Section 5 where we briefly show how generalized Morse potential could be solved. Finally the conclusion of the present work is placed in Section 6.

#### 2. Laplace Transform Method and Convolution Theorem

The Laplace transform or of a function is defined by [36, 37] If there is some constant such that for sufficiently large , the integral in (2) will exist for . The Laplace transform may fail to exist because of a sufficiently strong singularity in the function as . In particular The Laplace transform has the derivative properties where the superscript denotes the th derivative with respect to for and with respect to for .

The inverse transform is defined as . One of the most important properties of the Laplace transform is that given by the convolution theorem [38]. This theorem is a powerful tool to find the inverse Laplace transform. According to this theorem if we have two transformed functions and , then the product of these two is the Laplace transform of the convolution , where So the convolution theorem yields Hence If we substitute , then we find the important consequence .

#### 3. Bound State Spectrum

The -dimensional time-independent Schrödinger equation for a particle of mass with orbital angular momentum quantum number is given by [39] where and denote the energy eigenvalues and potential. within the argument of the th state eigenfunctions denotes angular variables . The Laplacian operator in hyperspherical coordinates is written as where is known as the hyperangular momentum operator.

We chose the bound state eigenfunctions that are vanishing for and . Applying the separation variable method by means of the solution , (8) provides two separated equations: where is known as the hyperspherical harmonics, and the hyperradial or in short the “radial” equation where is the separation constant [40, 41] with .

In spite of taking (1) we take the more general form of pseudoharmonic potential [42] where , and are three parameters that can take any real value. If we set , , and , (13) converts into the special case which we have given in (1). Taking this into (12) and using the abbreviations we obtain We are looking for the bound state solutions for with the following properties: and . Let us assume a solution of type with . Here the term ensures the fact that and is expected to behave like .

Now changing the variable as and taking from (15) we obtain In order to get an exact solution of the above differential equation we remove the singular term by imposing the condition So we have where is taken as the positive solution of , which can be easily found by little algebra from (17). It is worth mentioning here that the condition given by (17) is necessary to get an analytical solution because otherwise only approximate or numerical solution is possible. Introducing the Laplace transform with the boundary condition and using (4), (18) can read where . This parameter can have integer or noninteger values and there will be integer or noninteger term(s) in energy eigenvalue according to values of which can be seen below.

The solution of the last equation can be written easily as where is a constant. The term is a positive real number as we restrict ourselves to the choice . Now, since is positive and , then the second factor of (20) could become negative if and thus its power must be a positive integer to get singled valued eigenfunctions. This will also exclude the possibility of getting singularity in the transformation. So we have Using (21), we have from (20) where and . In order to find , we find [43] Therefore using (7) and (23), we have The integration can be found in [44], which gives where is the confluent hypergeometric functions. Now using the beta function , can be written: So we have the radial eigenfunctions where is the normalization constant. It should be noted that because of the “quantization condition” given by (21), it is possible to write the radial eigenfunctions in a polynomial form of degree , as converges for all finite with and is not a negative integer or zero.

Now the normalization constant can be evaluated from the condition [45] To evaluate the integration the formula is useful here, where are the Laguerre polynomials. It should be remembered that the formula is applicable only if is a positive integer. Hence identifying , , and we have So using the following formula for the Laguerre polynomials, we write the normalization constant Finally, the energy eigenvalues are obtained from (21) along with (17) and (14): and we write the corresponding normalized eigenfunctions as or Finally, the complete orthonormalized eigenfunctions of the -dimensional Schrödinger equation with pseudoharmonic potential can be given by

We give some numerical results about the variation of the energy on the dimensionality obtained from (32) in Figure 1. We summarize the plots for as a function of for a set of physical parameters by taking , especially.