Shell-confined atom and plasma: incidental degeneracy, metallic character and information entropy

Shell confined atom can serve as a generalized model to explain both \emph{free} and \emph{confined} condition. In this scenario, an atom is trapped inside two concentric spheres of inner $(R_{a})$ and outer $(R_{b})$ radius. The choice of $R_{a}, R_{b}$ renders four different quantum mechanical systems. In hydrogenic atom, they are termed as (a) free hydrogen atom (FHA) (b) confined hydrogen atom (CHA) (c) shell-confined hydrogen atom (SCHA) (d) left-confined hydrogen atom (LCHA). By placing $R_{a}, R_{b}$ at the location of radial nodes of respective \emph{free} $n,\ell$ states, a new kind of degeneracy may arise. At a given $n$ of FHA, there exists $\frac{n(n+1)(n+2)}{6}$ number of iso-energic states with energy $-\frac{Z^{2}}{2n^{2}}$. Furthermore, within a given $n$, the individual contribution of each of these four potentials has also been enumerated. This incidental degeneracy concept is further explored and analyzed in certain well-known \emph{plasma} (Debye and exponential cosine screened) systems. Multipole oscillator strength, $f^{(k)}$, and polarizability, $\alpha^{(k)}$, are evaluated for (a)-(d) in some low-lying states $(k=1-4)$. In excited states, \emph{negative} polarizability is also observed. In this context, metallic behavior of H-like systems in SCHA is discussed and demonstrated. Additionally analytical closed-form expression of $f^{(k)}$ and $\alpha^{(k)}$ are reported for $1s,2s,2p,3d,4f,5g$ states of FHA. Finally, Shannon entropy and Onicescu {\color{red}information} energies are investigated in ground state in SCHA and LCHA in both position and momentum spaces. Much of the results are reported here for first time.