^{1}

^{3}.

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.

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 [

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 [

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

Especially with the work of [

Neural networks (NNs) are universal functions approximators which are useful in solving differential equations.

Trial solutions of artificial neural networks involve a single independent variable regardless of the dimension of the problem.

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.

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 [

In the present paper, we solve Schrödinger equation with Coulomb-plus-linear potential via ANN system. In Section

The neural network (NN) is constructed as a model of simply designed computing unit, called

A model of single neuron.

All the input signals are summed up as

A model of multilayer neural networks.

This is an example of feed forward neural network. Feed forward neural networks are the most popular architectures due to their structural flexibility, good representational capabilities, and availability of a large number of training algorithm [

We consider the application of the ANN to a quantum mechanical calculation. We will follow the formalism which was developed in [

Let us consider a multilayer perceptron with

Once the derivatives of the error with respect to the network parameters have been defined, any minimization technique can be carried out. In this work, we used a feed forward neural network with a back propagation algorithm.

By employing this approach it is possible to obtain energy eigenvalues of the Cornell potential. Before obtaining Cornell potential eigenvalues, it will be useful to give an example.

The Yukawa potential has an important role in various branches of physics. For example, in plasma physics, it is known as the Debye-Hückel potential; in solid-state physics and atomic physics, it is called the Thomas-Fermi or the screened Coulomb potential and has a role in the nucleon-nucleon interaction arising out of the one-pion-exchange mechanism in nuclear physics [

We parametrize trial function as

The radial part of the Schrödinger equation for Coulomb-plus-linear potential is

Putting

Comparison of eigenvalues of

| | | [ | [ |
---|---|---|---|---|

0 | 0 | | | |

1 | | | | |

2 | | | | |

3 | | | | |

4 | | | | |

5 | | | |

Comparison of eigenvalues of

| | | [ | [ |
---|---|---|---|---|

0 | 0 | | | |

1 | | | | |

2 | | | | |

3 | | | | |

4 | | | | |

5 | | | |

It can be seen from Tables

The Hamiltonian for this potential is

Comparison of eigenvalues of

| | | [ | [ |
---|---|---|---|---|

0 | 0 | | | |

1 | | | | |

2 | | | | |

3 | | | | |

4 | | | | |

5 | | | |

Comparison of eigenvalues of

| | | [ | [ |
---|---|---|---|---|

0 | 0 | | | |

1 | | | | |

2 | | | | |

3 | | | | |

4 | | | | |

5 | | | |

It can be seen from Tables

The Hamiltonian for this potential is

Comparison of eigenvalues of

| | | [ | [ |
---|---|---|---|---|

0 | 0 | | | |

1 | | | | |

2 | | | | |

3 | | | | |

4 | | | | |

5 | | | |

Comparison of eigenvalues

| | | [ | [ |
---|---|---|---|---|

0 | 0 | | | |

1 | | | | |

2 | | | | |

3 | | | | |

4 | | | | |

5 | | | |

It can be seen from Tables

The Hamiltonian for this potential is

Comparison of ground state eigenvalues for

| | [ | [ | | | [ | [ |
---|---|---|---|---|---|---|---|

0.2 | 2.16731633527814569 | 2.167316 | 2.167316208772717 | 0.1 | 2.25367805603652147 | 2.253678 | 2.253678098810761 |

0.4 | 1.98850421340265894 | 1.988504 | 1.988503899750869 | 0.3 | 2.07894965124100145 | 2.078949 | 2.078949440194840 |

0.6 | 1.80107438954743241 | 1.801074 | 1.801073805646947 | 0.5 | 1.89590541258401687 | 1.895904 | 1.895904238476994 |

0.8 | 1.60440950560142422 | 1.604410 | 1.604408543236585 | 0.7 | 1.70393557501874021 | 1.703935 | 1.703934818031980 |

1.0 | 1.39787578582763672 | 1.397877 | 1.397875641659907 | 0.9 | 1.50241669742698608 | 1.502415 | 1.502415495453739 |

1.2 | 1.18083812306874152 | 1.180836 | 1.180833939744787 | 1.1 | 1.29071042316348541 | 1.290709 | 1.290708615983606 |

1.4 | 0.95264360884520136 | 0.952644 | 0.952640495218560 | 1.3 | 1.06817386102041328 | 1.068171 | 1.068171244486971 |

1.6 | 0.71266200850041624 | 0.712662 | 0.712657680461034 | 1.5 | 0.83416589280625410 | 0.834162 | 0.834162211049953 |

1.8 | 0.46026600634169799 | 0.460266 | 0.460260113873608 | 1.7 | 0.58805423050569871 | 0.588049 | 0.588049168557953 |

Quarkonium systems are an ideal area for clarifying our understanding of QCD. They probe nearly all the energy regimes of QCD from high energy region to low energy region. In the high energy region, an expansion of coupling constant is possible and perturbative QCD is applicable. In the low energy region, such an expansion is not possible in the coupling constant and therefore nonperturbative methods need to be used. Besides that nonrelativistic QCD (NRQCD) approximation is also used for spectroscopy, decay, and production of heavy quarkonium [

In this section we obtained spin-averaged mass spectra charmonium, bottomonium, and bottom-charmed system by solving nonrelativistic Schrödinger equation. It is possible to obtain full spectra for quarkonium systems including relativistic effects, spin-spin, and spin-orbit interactions. Since most of these contributions are really small compared to the given potential, even by neglecting those effects one can find results that are close to the experimental data.

The related Cornell potential is [

In Tables

Charmonium spectra. All results are in GeV.

State | This work | [ | Exp. [ |
---|---|---|---|

1S | | | |

2S | | | |

3S | | | |

1P | | | |

2P | | | |

3P | | | - |

1D | | | |

Bottomonium spectra. All results are in GeV.

State | This work | [ | Exp. [ |
---|---|---|---|

1S | | | |

2S | | | |

3S | | | |

4S | | | |

1P | | | |

2P | | | |

3P | | | |

1D | | | |

Bottom-Charmed spectra. All results are in GeV.

State | This work | [ | Exp. [ |
---|---|---|---|

1S | | | |

2S | | | |

3S | | | |

4S | | | |

1P | | | |

2P | | | |

3P | | | |

1D | | |

As can be seen in Tables

In this paper, we applied ANN method to deal with the solution of the Schrödinger equation with Coulomb-plus-linear potential. This potential belongs to the nonsolvable potentials class which has an exactly/analytically solvable part, together with a modifying term. We obtained the eigenvalues of Schrödinger equation and spin-averaged mass spectra of charmonium, bottomonium, and bottom-charmed systems. The obtained eigenvalues and heavy quarkonium spectra are in agreement with the theoretical studies and available experimental data.

The feed forward ANNs method has a good property of function approximation. A function approximation problem is to select or find a function among a well-defined functions set that closely matches a target function in a task specific way. This form employs a feed forward neural network as the basic approximation element, whose parameters (weights and biases) are adjusted to minimize an appropriate error function. In this study, the wave function is represented by the feed forward ANN and its inputs are taken as coordinate values. A trial solution is written as a feed forward neural network which contains adjustable parameters (weights and biases) and eigenvalue is refined to the known solutions by training the neural network.

This study would be useful for the exact or quasiexact spectra of a few body systems. It is also possible in principle to handle many body problems but such problems will impose much more heavier computational load and other difficulties such as convergence of sigmoid functions [

No data were used to support this study.

The author declares that they have no conflicts of interest.