J-matrix calculation of Tsallis entropy for Hellmann potential

Abstract The Tsallis entropy for a system that interrelates with the Hellmann potential is calculated and discussed. The calculation is done in the position and momentum spaces using the J-matrix method. Most of our outcomes are reported for the first time. The results are compared with the available literature results.


Introduction.
About thirty years ago, nonadditive entropy known as "Tsallis entropy q S " was first proposed by Tsallis himself [1]. Accordingly, the concept that related with nonadditive statistical mechanics has found applications in a variety of disciplines including physics, fluids, informatics, linguistics, among others [2,3]. We refer to Ref [4] for recent, more historical details, and applications. The Tsallis entropy is useful in cases where there are strong correlations between different microstates in a system. It is defined as the q-generalized entropy, in the form: The Tsallis index q plays crucial rules in nonadditive entropy since it is considered a way of characterizing a system's correlationsparticularly how strong they are. It is either more or less than one in such systems. When the correlations in a system are weak or non-existent, q approaches 1 and q S becomes additive and reduces to the usual Shannon entropy S [5]: The index q is usually used as a fitting parameter in case of systems that are not well enough understood.
It is known that q S , as S , is a basic quantity measuring the expected amount of information contained in the probability distribution in respected space. In other words, it gives a complete description of the In addition to the application of nonadditive entropy in different fields [2][3][4], we are going to apply it in atomic systems. In doing so, let us consider two probabilistically independent and isolated systems A and B System A has the Coulombic potential (CP), / Z r , and system B has the Yukawa potential (YP) [7,8], / r Ze r P . In the case when the two systems are merged with each other, they form a new combined system C , which interacts with the Hellmann potential (HP) [9][10][11]. The Hellmann potential (HP) consists of a summation of CP and YP as: where Z is the nuclear charge and P is the screening parameter.
It is our aim to numerically calculate q S , in position ( r -) and momentum ( p -) spaces, and consequently calculate the q -values that satisfies the nonadditive relation, Eq. (2). The numerical calculation will be done using the J-matrix method [12][13][14] for different values of P , where c P P .

Background theory
It is understood that ( ) U r is important in calculating the information entropies. Thus, in incorporating the J-matrix method to calculate ( ) U r , we start with ( ) U r of the stationary and nonrelativistic quantum systems as: that fulfill the normalization condition, ( ) 1 d U ³ r r (6) The wave functions ^( ) < r in (5) are the bounded solutions of the Schrödinger equation.
Symbolically, the nonrelativistic Schrödinger wave equation (in atomic units) is written in the form: where Ĥ is the full Hamiltonian,^È are the associated eigenvalues, and ˆ( , , ) V r A ) is the used potential with parameters , A , which will be suppressed in the following discussion for simplicity. The V r A ) is the used potential with parameters , A , which will be suppressed in the following discussion for simplicity. The suppressed parameters have critical values above which no bound states exist.
In spherical coordinates ( , , ) where ( ) n R r ( ) ( is the eigenfunction of the nonrelativistic radial Schrödinger equation: Here, N is the dimension of the matrix, and O is a positive length scale parameter. In Laguerre basis set, we define:  is the normalization constant, with the 3-term recursion relation for the coefficient of transfer the reference Hamiltonian ˆo H into tridiagonal matrix form. (iii) The potential matrix elements are calculated numerically using the Gauss quadrature method.
Thus, ( ) U r of the n m n m -state can be written as: In p-space, to formulate the density ( ) With the information of ( ) q S r and ( ) q S p , we can check the BBM-expression [15], which is given in the three dimensional form as: Eq. (16) represents the relation of the total entropy to the summation of entropies 1 ( ) S r and 1 ( ) S p . Table 1 shows, for YP and HP, the variation of the energy eigenvalues and the Tsallis entropy 1 S as a function of P , for the 1s-state of the hydrogen atom (Z = 1) in r -and p -spaces. For CP [16], where 0 P , the analytical value 1

Results and discussion
and we are going to calculate the value of ' q which satisfy the above equation. In Table 1 and for 1 q , the value of ' q , as a function of P , are calculated for the constrain that satisfy by the Eq. (18). The values of ' q are given in the last column in each space. It is noticed that ' q decrease as P increases. The value of ' 1 q ! indicates the strong correlations of 1 S in the given system with respect to q and P .
The values given in Table 1  S ,which means the delocalization of ( ) r U increases with increasing P , as shown clearly in table 1. 4. With increasing P , the ' q value decreases from 1.36 to 1.34 in the case of the r -space, and decreases from 1.06 to 0.99 in the case of the p -space. This implies the correlation is large in the position space but not in momentum space. 5. In the range 0 P to 0.5, the values of 1 S are additive in the p -space, but not in the r -space.

Conclusion
We have conveyed the outcomes of our numerical studies of Tsallis entropy for 1s-state of the hydrogen atom 1 Z in the cases CP, YP and HP. The J-matrix method is implemented for the mentioned potentials in spherical coordinates. Many distinctive features have been realized in the case of this study.
For example, it was found that the summation of the entropy in r -and p -spaces of the Tsallis entropy fulfills the BBM expression. The scaling laws for the calculated Tsallis entropy are given as a function of the parameters q and P . The scaling laws clearly show the correlation between the calculated quantities S are additive in the p -space, but not in the r -space. In view of our discussions, it is hoped that this opening study of the nonadditive entropy will motivate stimulating studies, theoretically and experimentally, with other combined potentials in the atomic field.