Information-entropic measures in free and confined hydrogen atom

Abstract

Shannon entropy ($S$), R{é}nyi entropy ($R$), Tsallis entropy ($T$), Fisher information ($I$) and Onicescu energy ($E$) have been explored extensively in both \emph{free} H atom (FHA) and \emph{confined} H atom (CHA). For a given quantum state, accurate results are presented by employing respective \emph{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. \emph{Exact} mathematical expressions of $R_r^{\alpha },R_p^{\beta }$, $T_r^{\alpha },T_p^{\beta },E_r,E_p$ are derived for \emph{circular} states of a FHA. Pilot calculations are done taking order of entropic moments ($\alpha ,\beta$) as $(\frac{3}{5},3)$ 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 ($r_c$) for the latter. In a CHA, at small $r_c$, kinetic energy increases, whereas $S_{\rvec}, R^α_{\rvec}$ decrease with growth of $n$, signifying greater localization in high-lying states. At moderate $r_c$, 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.

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