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arXiv:1703.03578

Fisher information for quasi-one-dimensional hydrogen atom

Saha, Aparna, B. Talukdar, Supriya Chatterjee

Published Mar 10, 2017• 3 citations
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Abstract

The coordinate-space wave function $ψ(x)$ of quasi-one-dimensional atoms is defined in the $x\geq 0$ region only. This poses a typical problem to write a physically acceptable momentum-space wave function $ϕ(p)$ from the Fourier transform of $ψ(x)$. We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function $ϕ(p)$ as a function of $p$ and confirm that $ϕ_i(p)$ (the imaginary part of $ϕ(p)$) represents the correct momentum-space wave function. We make use of the results for $ψ(x)$ and $ϕ_i(p)$ to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.

DOI
10.48550/arxiv.1703.03578

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The coordinate-space wave function ψ(x)ψ(x)ψ(x) of quasi-one-dimensional atoms is defined in the x≥0x\geq 0x≥0 region only. This poses a typical problem to write a physically acceptable momentum-space wave function ϕ(p)ϕ(p)ϕ(p) from the Fourier transform of ψ(x)ψ(x)ψ(x). We resolve the problem with special attention to the behavior of real and imaginary parts of the complex-valued function ϕ(p)ϕ(p)ϕ(p) as a function of ppp and confirm that ϕi(p)ϕ_i(p)ϕi​(p) (the imaginary part of ϕ(p)ϕ(p)ϕ(p)) represents the correct momentum-space wave function. We make use of the results for ψ(x)ψ(x)ψ(x) and ϕi(p)ϕ_i(p)ϕi​(p) to express the position- and momentum-space Fisher information in terms of the principal quantum number and energy eigen value of the system and provide some useful checks on the result presented with particular attention on the information theoretic uncertainty relation.
Abstract With special attention to the role of information theory in physical sciences we present analytical results for the coordinate- and momentum-space Fisher information of some important one-dimensional quantum systems which differ in spacing of their energy levels. The studies envisaged allow us to relate the coordinate-space information ( I ρ ) with the familiar energy levels of the quantum system. The corresponding momentum-space information ( I γ ) does not obey such a simple relationship with the energy spectrum. Our results for the product ( I ρ I γ ) depend quadratically on the principal quantum number n and satisfy an appropriate uncertainty relation derived by Dehesa et al (2007 J. Phys. A: Math. Theor. 40 1845 )
The article intertwines the study of wave packet dynamics with information-theoretic measurements in one dimensional (D =1) system. A localized wave packet at time t=0 has been considered here and its change at later instant of time t is calculated for the position space wave function. The momentum space wave function is obtained by taking the Fourier transform of the position space wave function at time t= 0 . These wave functions are then employed to construct the wave packet's respective probability densities in position and momentum space, and later have been used to compute the corresponding position and momentum space Shannon (S) and Fisher information (I) entropies. It has been observed that although the Shannon and Fisher information entropies explicitly depend on the standard deviation, neither the Shannon entropy sum nor the product of Fisher information entropies is. Moreover, for D dimensional systems, the computed values of the Shannon and Fisher entropies are found to satisfy the lower bounds of the Bialynicki-Birula and Myceilski (BBM) inequality relation and the Stam-Cramer-Rao inequalities better known as the Fisher based uncertainty relation. Thus, our theoretical study explores the validation of information-theoretic measurements for wave packet dynamics using the basic formulations of information theory.
The Fisher information functional occupies a very special position among the uncertainty measures of quantum systems of any dimensionality. In contrast with (global) entropic measures of Shannon, Renyi or Tsallis types, it contitutes a local measure in the sense that it is sensitive to the oscillatory character of the probability density distribution. Moreover, the Fisher information is closely related to various macroscopic properties of the system described by different density functionals. Here, we review its main analytical properties such as, inequality-based relationships with some radial expectation values and its uncertainty relation. Tighter versions of these inequalities are shown for those systems subject to an arbitrary central potential. Moreover, the Fisher information is computed for D-dimensional hydrogenic systems explicitly in terms of the radial and angular hyperquantum numbers characterizing the quantum states of the system in both position and momentum spaces. Finally, the utility of this quantity is discussed for various physico-chemical processes of recent interest: Abstraction and nucleophilic substitution reactions.
It is a commonplace that eigenfunctions in momentum space are the Fourier transform of eigenfunctions in position space. However, if we choose that space for which the relevant variables are chosen canonically conjugate to the position space variables, then the transform is not Fourier, but must involve the Jacobian of transformation between Cartesian coordinates and the curvilinear position space coordinates. (For Cartesian space the transform is Fourier.) In this work we find the correct form of the Hamiltonian in momentum space for the Yukawa potential. It is shown that the potential function in momentum space becomes an operator. The first term is shown to be the Coulomb operator in momentum space, while the remaining terms just mix other hydrogen atom momentum space functions. The resulting eigenvalues are in agreement with the comparable position space results. The analytic behavior of the eigenfunctions is examined. It is shown that all the singularities are poles (the function is meromorphic) and that all the poles lie along the imaginary axis no farther from the origin than p0 (the inverse of the Bohr radius). This result should be of considerable value in simplifying calculations with these functions.
In the context of informational analysis of atomic and molecular systems, the connection between local quantum observables and information measures is of interest. In this paper, analytical relationships for the imaginary part of the total local momentum (coordinate) in terms of information theoretic measures have been established. Moreover, on the basis of another scheme in which the relationship among densities of information energy, Shannon entropy and Fisher information has been proposed [M. Alipour and A. Mohajeri, Mol. Phys. 110, 403 (2012)], the general formulae for the imaginary part of the total local momentum and the corresponding variance are expressed. The presented proof may be viewed in light of the relation between the local wave vector and the information energy density. Through this study, another noteworthy application of information theory is highlighted.