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.
All the input signals are summed up as and the output signal is determined by the nonlinear activation function . In this work we use a sigmoid functionas an activation function since it is possible to derive all the derivatives of in terms of itself. This differentiability is an important aspect for the Schrödinger equation. Here we use a multilayer perceptron neural network (MLPN). The architecture of MLPN is shown in Figure 2.
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 [24]. This network consists of neurons arranged in layers in which every neuron is connected to all neurons of the next layer and the information process can be in only one way, from input to the output. The input-output properties of the neurons (perceptrons in computerized systems) can be written as where is for input, is for hidden, and is for output layers. Input to the perceptrons is given as Here, and represent the numbers of the units belonging to input and hidden layers, is the synaptic weight parameter which connects the neurons and , and represents threshold parameter for the neuron [30]. The overall response of the network is given asThe derivatives of with respect to network parameters (weights and thresholds) can be done by differentiating (9) as Neural network systems are self-training systems. Therefore parameters of training need updating in the transfer process and these equations play important roles on the learning processes of the neural networks.
2.1. Implementation of ANN to Quantum System
We consider the application of the ANN to a quantum mechanical calculation. We will follow the formalism which was developed in [26]. Consider the following differential equation:where is a linear operator, is a known function, and at the boundaries. In order to solve this differential equation a trial function can be written as which employs a feed forward neural network with parameter vectors and to be adjusted. The parameter vector corresponds to the weights and biases of the neural architecture. The functions and should be specified in a convenient way so that satisfies the boundary conditions regardless of the and values. To obtain a solution for (14), the collocation method can be used and the differential equation can be transformed into a minimization problemFor Schrödinger equation, (14) takes the form with the boundary condition . In this case, the trial solution can be written as where at boundary conditions for a range of values. By discretizing the domain of the problem, it is transformed into a minimization problem with respect to the parameters and where is the error function and can be computed as
Let us consider a multilayer perceptron with input units, one hidden layer with units and one output. This multilayer perceptron looks like Figure 2 but with just one output. For a given input vector the output of the network is where Here, denotes the weight from the input unit to the hidden unit , is the weight from the hidden unit to the output, is the bias of hidden unit , and is the sigmoid function, given in (2). The derivatives of network output can be defined as where and is the th order derivative of the sigmoid.
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.
2.1.1. A Warm up Example: Yukawa Potential
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 [34]. The Yukawa potential reads as where the parameters and are given by different expressions, depending on the problem under consideration. Yukawa potential does not admit an exact solution and therefore various approximate analytic and numerical methods. The ground state energy for , , and in reported by [34] is , by [35] is , and by [36] is .
We parametrize trial function as where denotes a feed forward artificial neural network with one hidden layer and sigmoid hidden units The minimization problem is The energy eigenvalue obtained with this scheme is which is in good agreement with the numerical results in first five digits.
3. Numerical Results and Discussion
The radial part of the Schrödinger equation for Coulomb-plus-linear potential iswhere and . We trained the network with 200 equidistance points with in the interval and solved the Schrödinger equation in four cases.
3.1. Case 1:
Putting into (29) we get The Hamiltonian of this equation is By applying the procedure which was mentioned in the previous section, we get eigenvalues of this potential. In Tables 1 and 2, we present eigenvalues of the potential for and states, respectively.
It can be seen from Tables 1 and 2 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies.
3.2. Case 2:
The Hamiltonian for this potential is In Tables 3 and 4, we present eigenvalues of the potential for and states, respectively.
It can be seen from Tables 3 and 4 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies.
3.3. Case 3:
The Hamiltonian for this potential is In Tables 5 and 6, we present eigenvalues of the potential for and states, respectively.
It can be seen from Tables 5 and 6 that eigenvalues of the radially and orbitally excited states by ANN method are in good agreement with the reference studies. In this case, numerical results of orbitally excited sates can differ in first two or three digits. The reason for this difference can be the dominance of the linear part of the potential compared to the Coulomb part.
3.4. Case 4:
The Hamiltonian for this potential is In Table 7, we present ground state eigenvalues of the potential for varying values. It can be seen from Table 7 that obtained results are in good agreement with the given studies.
3.5. Heavy Quarkonium Spectra
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 [37, 38]. For an overview of NRQCD, see [39].
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 [40] with
In Tables 8, 9, and 10, charmonium, bottomonium, and bottom-charmed spectra are presented, respectively.
As can be seen in Tables 8, 9, and 10 charmonium, bottomonium, and bottom-charmed spectra of ANN agree well with given reference and available experimental data.
4. Summary
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 [41] and availability of hardware.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that they have no conflicts of interest.