In this paper, an analytical model of 2p orbital helium atom has been considered to quantify the values of the Rényi and Tsallis entropies along with their theoretical aspects. The normalized radial wave function used here (in atomic units) is obtained by solving the Schrödinger equation. The complete form of the coordinate space wave function is obtained by the use of the spherical harmonics. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities constituted with the respective coordinate and momentum space wave functions have been used to compute the numerical values of the Rényi and Tsallis entropies in the coordinate and momentum space for different values of the order β varying from 2 to 10. The computed values are presented in a tabular form. Further, it is mathematically demonstrated that in the limit of order β → 1, both the Rényi and Tsallis entropies lead to the Shannon entropy. Finally, an outlook of the present work has been summarized with some concluding remarks.
In this paper, an analytical model of 2p orbital helium atom has been considered to quantify the values of the Rényi and Tsallis entropies along with their theoretical aspects. The normalized radial wave function used here (in atomic units) is obtained by solving the Schrödinger equation. The complete form of the coordinate space wave function is obtained by the use of the spherical harmonics. The momentum space wave function is obtained by taking the Fourier transform of the coordinate space wave function. The probability densities constituted with the respective coordinate and momentum space wave functions have been used to compute the numerical values of the Rényi and Tsallis entropies in the coordinate and momentum space for different values of the order β varying from 2 to 10. The computed values are presented in a tabular form. Further, it is mathematically demonstrated that in the limit of order β → 1, both the Rényi and Tsallis entropies lead to the Shannon entropy. Finally, an outlook of the present work has been summarized with some concluding remarks.
In this paper, we have presented an analytical model of two electron systems consisting of a many particle correlated wave function with some variational parameters α, λ and µ and used it to quantify the electron-electron correlation described by the wave function containing explicitly r12 (inter atomic distance between two electrons) dependent term. The single particle wave functions and the charge densities have been extracted from the said correlated wave function both for the uncorrelated and correlated systems in coordinate space and its momentum analogs have been obtained by taking the Fourier transform of the coordinate analogs. We have computed and presented the results of the numerical values of the theoretic information entropies of the Shannon entropy, Fisher information entropy, Shannon power and the FisherShannon product. The numerical values are consistently found to satisfy the Beckner, Bialynicki-Birula and Mycielski (BBM ) inequality relation; Stam-Cramer-Rao inequalities or Fisher based uncertainty relation and Fisher-Shannon product relation for the uncorrelated and correlated systems in both the coordinate and momentum spaces.
Abstract A comparative study of 1D quantum structures which allows analytic expressions for the position and momentum Rényi R ( α ) and Tsallis T ( α ) entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms the special status of the Gaussian distribution: at any parameter α it converts into the equality both Rényi and Tsallis uncertainty relations removing for the latter an additional requirement 1/2 ≤ α ≤ 1 that is a necessary condition for all other geometries. It is shown that the lowest limit of the semi-infinite range of the dimensionless parameter α where momentum components exist strongly depends on the position potential and/or boundary condition for the position wave function. Asymptotic limits reveal that in either space the entropies R ( α ) and T ( α ) approach their Shannon counterpart, α = 1, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are also indicated.
Abstract Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (rc) for the latter. In a CHA, at small rc, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate rc, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.
A comparative study of the atomic Rényi entropies for the exponential-cosine screened Coulomb potential (ECSCP), has been discussed. The study has been done by solving the Schrödinger equation in the Hylleraas - space, and within the framework of Ritz’s variational principle. Different wave functions for the 1s 2 -state of the helium isoelectronic series, in the Hylleraas-space with few variational parameters, have been used to calculate the Shannon and Rényi entropies. They are used to create a comparative study and discuss the scaling behavior of different physical quantities close to the ionization limit. The 3-dimension visualization of the Shannon and Rényi charge densities, as a function of the nuclear charge Z and the screening parameter μ , will be explained and discussed. Comparison with literature results is made, whenever possible.
In this paper, a three parameter (α, λ, μ) ‘Hartree and Ingman (1933)’ type correlated wave function has been used to derive an analytical model to quantify the two electron systems. The model is used to construct the expressions for the single particle wave functions [ 𝜓(𝑟) , 𝜙(𝑝)] in coordinate space and momentum space respectively with the subscripts marked as correlated (c) and uncorrelated (uc) ones. Further, the correlated and uncorrelated wave functions are used to form the single particle charge densities [𝜌(𝑟) , 𝛾(𝑝)] in coordinate space and momentum space. These wave functions and the single particle charge densities are used to compute the numerical values of the Shannon (S) and Fisher (I) information entropies for both the correlated and uncorrelated systems in coordinate space and momentum spaces and furthermore to examine their correlation effects in coordinate and momentum spaces. The uncorrelated and correlated values of Shannon information entropies indicate the reciprocity between the coordinate and momentum spaces such that low values of the Shannon information entropy in coordinate space ( 𝑆𝜌), are associated with the high values of Shannon information entropy in momentum space (𝑆𝛾) . The uncorrelated and correlated values of Shannon information entropies (S) also validate the entropic uncertainty relations Bialynicki-Birula and Mycielski (BBM) inequality which reads as (𝑆𝜌+𝑆𝛾)≥3(1+𝑙𝑛𝜋), as well as another stronger version of the uncertainty relation that is (𝑆𝜌𝑐+𝑆𝛾𝑐)>(𝑆𝜌𝑢𝑐+𝑆𝛾𝑢𝑐). Further, both the products of uncorrelated and correlated values of Fisher information entropies 𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 and 𝐼𝜌𝑐𝐼𝛾𝑐 satisfy the inequality condition 𝐼𝜌𝑐𝐼𝛾𝑐>𝐼𝜌𝑢𝑐𝐼𝛾𝑢𝑐 in coordinate and momentum spaces. At the same time these products also satisfy the Fisher based uncertainty relation, 𝐼𝜌𝐼𝛾≥36. The consistency of the results satisfying the uncertainty relations can be checked from the data demonstrated in the Table 1 and Table 2.
We solve the Schrödinger equation in the spherical or Hylleraas-coordinate systems, and within the framework of the Ritz's variational principle. The eigenvalues, and the eigenfunctions ψ ( r ) in r - or Hylleraas-space for the 1s 2 -state of the He-like atoms as a function of two variational parameters are calculated. Using a simple scaling procedure, we calculate the scaled wavefunction as a function of the nuclear charge Z . Given the density of states, ρ ( r ), the scaling behavior of the information entropies, e.g., Fisher, Shannon and Rényi's entropies, with their powers and products, as functions of Z are calculated. Scaled wavefunctions for the 1s 2 -state of the He-like atoms, with exchange, have been used to study the scaling behavior. Our results agree with the published results. Furthermore, we present a simple logarithmic equation that shows the dependence of information entropies on Z for He-like atoms. The formulation enhances the computational efficiency of the entropies and other related quantities.