Abstract The time‐dependent Feinberg–Horodecki equation in the presence of a special Mie‐type pseudoharmonic potential was solved and the quantized momentum with the corresponding wave functions were obtained with the supersymmetric approach. The Fisher information for both the time and the momentum space were calculated via expectation values of time and momentum. The effect of the dissociation energy, the equilibrium time point and the quantum number respectively on the Fisher information were studied in detail. The result was applied to five molecules. The results obtained for the time‐dependent Feinberg–Horodecki equation followed the trend of the results obtained for the time‐independent Schrödinger equation.
Abstract The time‐dependent Feinberg–Horodecki equation in the presence of a special Mie‐type pseudoharmonic potential was solved and the quantized momentum with the corresponding wave functions were obtained with the supersymmetric approach. The Fisher information for both the time and the momentum space were calculated via expectation values of time and momentum. The effect of the dissociation energy, the equilibrium time point and the quantum number respectively on the Fisher information were studied in detail. The result was applied to five molecules. The results obtained for the time‐dependent Feinberg–Horodecki equation followed the trend of the results obtained for the time‐independent Schrödinger equation.
Onate and Onyeaju have recently studied the Feinberg–Horodecki equation with a special time-dependent Mie-type pseudoharmonic potential. They claim that the wavefunctions, quantized momentum and Fisher information have been obtained. We show in this comment that the results obtained in Onate and Onyeaju [1] carry several calculation problems that are corrected here. In order to show that the errors made in Onate and Onyeaju [1] are not just due to wrong choices of the constants x and a, we solve the Feinberg–Horodecki equation with a pseudoharmonic-type potential, that is, the Equation (5). After that, we use its solutions to calculate the different properties studied in Onate and Onyeaju [1]. In Figure 1, we show the plots of <t2>, <p2>, I(ρ) and I(γ) as function of te for the ground state of the system. Despite the fact that we have obtained <t2>, <p2> and I(ρ) analytically, we calculated I(γ) numerically. As we can see, <t2> (<p2>) increases (decreases) as te increases. On the other hand, I(ρ) (I(γ)) decreases (increases) with the increasing te. Those behaviors were also observed by the authors in Onate and Onyeaju [1]. However, since the expressions for <t2>, <p2> and I(ρ) presented in Onate and Onyeaju [1] are not correct, the numerical values of the curves and tables presented in Onate and Onyeaju [1] are also incorrect. In Figure 3 we show the plot of the product I(ρ)I(γ) for the ground state of the system as function of te. We observed that I(ρ)I(γ) ≥ 4 = 4D2, where D is the dimension of the problem (D = 1). This is a remarkable result which agree with the work by Sanchez-Moreno et al. [18], valid when either ψ(t) or ϕ(p) is real. In summary, we have shown several mistakes in the article by Onate and Onyeaju [1] in their calculations and showed the correct expressions for the quantized momentum, wavefunctions, mean values (<t2> and <p2>) and Fisher information. We show that Stam inequalities are satisfied (Figure 2) and that the product of Fisher information has a non-trivial lower bound (I(ρ)I(γ) ≥ 4). We hope that our results will be useful in providing new insights in the field of quantum chemistry. The authors are grateful to the National Counsel of Scientific and Technological Development (CNPq) and to the National Council for the Improvement of Higher Education (CAPES) of Brazil for financial support. Raimundo N. Costa Filho is supported by CNPq grant number 312384/2018-1.
The exact solution of the Feinberg-Horodecki equation for time-dependent harmonic vector potential has been investigated under a one-dimensional system. The quantized momentum and its corresponding un-normalized wave functions were explicitly obtained. The Fisher information (for time and momentum) and variance (for time and momentum) were calculated using expectation values of time and momentum via Hellman-Feynman Theory (HFT). The time and momentum Shannon entropy were obtained using an existing formula. Numerical results were for time and momentum Fisher information to confirm the Cramer-Rao inequality. Another numerical results were obtained for time and momentum Shannon entropy to confirm Bialynick-Birula, Mycielski (BBM) inequality. The effects of the potential parameters such as mass of the spring and the frequency on the theoretic quantities were fully examined. The new variance inequality was established using the inequalities of Fisher information. The established inequality was confirmed by numerical results which also satisfied the popular Cramer-Rao inequality. The theoretic impetuses for Fisher information, variance and Shannon entropy respectively, were calculated and their variation with some potentials were studied.
Nascimento et al. Commented and faulted the calculations on the article title “Fisher information of a vector potential for time-dependent Feinberg–Horodecki” by Onate and Onyeaju. The authors claimed that the calculations in the paper of Onate and Onyeaju (Reference [1]) of Nascimento et al carried several problems. Their main claimed is on Equation (3) of their Reference [1]. The authors went further to obtain a quantized momentum equation with their claim as the correct equation. They also have other claims in the calculations. These authors further faulted Reference [1] of their comment for citing wrong reference. Nascimento and co also said that Equation (21) of their Reference [1] is dimensionally wrong, this is not true. It is their Equation (9) that is dimensionally wrong. This also makes other equations that depends on Equation (9) in their claim to be wrong.
We obtain an approximate value of the quantized momentum eigenvalues, Pn, together with the space-like coherent eigenvectors for the space-like counterpart of the Schrödinger equation, the Feinberg-Horodecki equation, with a screened Kratzer-Hellmann potential which is constructed by the temporal counterpart of the spatial form of this potential. In addition, we got exact eigenvalues of the momentum and the eigenstates by solving Feinberg-Horodecki equation with Kratzer potential. The present work is illustrated with three special cases of the screened Kratzer-Hellman potential: the time-dependent screened Kratzer potential, time-dependent Hellmann potential and, the time-dependent screened Coulomb potential.
The solutions of the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potentials has been studied separately in the past. The Kratzer potential despite different reports, the basic theoretic quantity such as Fisher information has not been reported. In this study, the solution of the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential is obtained using the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann-Feynman Theory. The expectation values are used to calculate the Fisher information for both the position and momentum spaces in both the absence and presence of the constant-dependent potential. The obtained results revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. The presence of the constant-dependent potential also increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value along with the product of the Fisher information, satisfies both the Fisher’s inequality and the Cramer-Rao’s inequality.
In this study, we obtained exact solutions of the Feinberg–Horodecki equation for the time-dependent Wei-Hua potential, which is constructed by the temporal counterpart of the spatial form of this potential. We obtained the quantized momentum eigenvalues and the corresponding wave functions. We obtained the partition function for the system and studied other thermodynamic properties that include vibrational mean momentum ( U), vibrational specific heat capacity ( C), vibrational entropy ( s), and vibrational free momentum ( F) as a signature in the momentum space.