Advances in High Energy Physics

Volume 2019, Article ID 3105373, 9 pages

https://doi.org/10.1155/2019/3105373

## Cornell Potential: A Neural Network Approach

Physics Department, Faculty of Arts and Sciences, Ondokuz Mayis University, 55200 Samsun, Turkey

Correspondence should be addressed to Halil Mutuk; moc.liamg@kutumlilah

Received 1 January 2019; Accepted 10 February 2019; Published 28 February 2019

Academic Editor: Sally Seidel

Copyright © 2019 Halil Mutuk. 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

We solved Schrödinger equation with Cornell potential (Coulomb-plus-linear potential) by using neural network approach. Four different cases of Cornell potential for different potential parameters were used without a physical relevance. Besides that charmonium, bottomonium and bottom-charmed spin-averaged spectra were also calculated. Obtained results are in good agreement with the reference studies and available experimental data.

#### 1. Introduction

The Schrödinger equation with Coulomb-plus-linear potential (Cornell potential) has received a great deal of attention as an important nonrelativistic model for the study of quark-antiquark systems, namely, mesons [1–18]. Especially Coulomb-plus-linear potential was the first potential model to study heavy quarkonium systems and inspired other phenomenological models. The quarkonium spectrum is a substantial field to improve our understanding about nature of QCD. The Cornell potential reads aswhere and are some constants. At small quark-antiquark distances, the Cornell potential goes like which is known as Coulomb-like term and at large distances it goes as , which is known as linear term. The Coulomb-like term arises from one-gluon exchange and linear term presumably arises from higher order effects [19]. Up to now, the linear term has not been calculated from the first principles of QCD.

Aside from its physical relevance, the Schrödinger equation with Coulomb-plus-linear potential has been studied with pure mathematical techniques. Hall used the method of potential envelopes to construct general upper and lower bounds on the eigenvalues of the Hamiltonian representing a single particle in Coulomb-plus-linear potential [20]. In [21], Chaudhuri et al. used Hill determinant method to bound state eigenvalue problem with Coulomb-plus-linear potential. Plante and Antippa solved the Schrödinger equation for a quark-antiquark system interacting via a Coulomb-plus-linear potential and obtained the wave functions as power series, with their coefficients given in terms of the combinatorics functions [22]. In [23], the authors presented a numerically precise treatment of the Crank–Nicolson method with an imaginary time evolution operator in order to solve the Schrödinger equation. Although such models have been studied to solve Schrödinger equations numerically or analytically, exact solutions of Schrödinger equation with Cornell potential are still unknown.

Artificial neural networks (ANNs) can be used as an alternative method to solve both ordinary and partial differential equations. ANN is a parallel distributed processor in which numerous numbers of simply designed computing units exist. These units are called* neurons*. Its massive connections between neurons can be used for storing various types of informations, particularly the ones classified as* knowledge* or* experience* [24, 25]. This information is acquired as interneuron connection strengths, or synaptic weights.

Especially with the work of [26, 27], ANNs are being used widely for solving ordinary and partial differential equations. ANNs have many advantages compared to existing semianalytic and numerical methods. The main advantages of using neural networks to solve differential equations can be stated as follows [28]:(i)Neural networks (NNs) are universal functions approximators which are useful in solving differential equations.(ii)Trial solutions of artificial neural networks involve a single independent variable regardless of the dimension of the problem.(iii)The approximate solutions are continuous over all the domain of integration. In contrast, the numerical methods provide solutions only over discrete points and the solution between these points must be interpolated.(iv)The computational complexity does not increase considerably with the number of sampling points and with the number of dimensions in problem.

Being an eigenvalue problem, Schrödinger equation had got attention by ANNs at the beginning of 90s. In [29], the authors taught a neural network the solution of the two dimensional Schrödinger equation for some model potential energy functions. In work of Androsiuk et al. [30], they presented computer simulations of a neural network capable of learning to perform transformations generated by the Schrödinger equation required to find eigenenergies of two dimensional harmonic oscillator. Sugawara presented a new approach for solving the Schrödinger equation based on genetic algorithm and artificial neural network. The method was tested in the calculation of one-dimensional harmonic oscillator and other model potentials such as Morse potential [31].

In the present paper, we solve Schrödinger equation with Coulomb-plus-linear potential via ANN system. In Section 2, we introduce the formalism of ANN method and describe how it can be applied to the quantum mechanical calculations. In Section 3, we give and discuss numerical results for the eigenvalues of Coulomb-plus-linear potential and in Section 4 we summarize our findings.

#### 2. Formalism

The neural network (NN) is constructed as a model of simply designed computing unit, called* neuron*. Figure 1 illustrates a simple model of a single neuron with multiple inputs and one output.