  Shannon entropy ($S$), R{\'e}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
$S_{\rvec}, R^{\alpha}_{\rvec}$ 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.
