Information‐entropic measures in free and confined hydrogen atom

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 Rrnormalα,Rpnormalβ, Trnormalα,Tpnormalβ,Er,Ep are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as (35,3) 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 Sboldr,Rboldrnormalα 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.

Quantum information entropies of multiple quantum well systems in fractional Schrödinger equations

“…potentials, [27] D-dimensional harmonic oscillator and hydrogen atom, [28,29] for Morse and Pöschl-Teller potentials, [30] the Rydberg-like harmonic states, [31] infinite potential well, [32] double square well potential, [33] infinite circular and spherical wells, [34,35] an electron in one-dimensional nonuniform systems, [36] one-dimensional Anderson model, [37] two-electron atoms, [38] hydrogen atom under soft spherical confinement, [39] the information-entropic measures in free and confined hydrogen atom, [40] information entropy for Eckart potential, [41] modified Hylleraas plus exponential Rosen-Morse potential, [42] a squared tangent potential well, [43] a parity-restricted harmonic oscillator, [44] the Fisher entropy for infinite circular and spherical wells, [45,46] and so on. The quantum information theory plays an important role in the measurement of uncertainty and other related parameters of an assumed quantum system.…”

Fisher information in confined isotropic harmonic oscillator

“…Here, we have adopted the first route which eliminates the necessity to do numerical differentiation in either spaces (all integrands are available analytically). This has been discussed in some more detail in a recent communication …”

“…In reality, information measures like Rènyi ( R ) and Shannon ( S ) entropy, Fisher information ( I ), Onicescu energy ( E ), quantify the spatial delocalization of ρ ( τ ), and hence can be explicitly employed in numerous interesting occurrences in quantum mechanics . In a recent work present authors have thoroughly investigated all these information theoretic measures ( R , S , E , I and complexity) for confined hydrogen atom (CHA) . Fundamentally, I quantifies expected error in a given measurement.…”

“…[2][3][4] In a recent work present authors have thoroughly investigated all these information theoretic measures (R, S, E, I and complexity) for confined hydrogen atom (CHA). [28][29][30] Fundamentally, I quantifies expected error in a given measurement. By definition, it is a gradient functional of density.…”

Study of the kinetic energy densities of electrons as applied to quantum dots in a magnetic field

“…Yet, another area where our work will be of value is one in which information‐entropic measures are employed within the framework of quantum mechanics to define various properties of systems. For example, in the work of Ghosh et al and Ghosh and Parr, density functional theory is described in terms of a local temperature T (r).…”