Information-Entropic Measures in Confined Isotropic Harmonic Oscillator

Abstract

Information based uncertainty measures like R{\'e}nyi entropy (R), Shannon entropy (S) and Onicescu energy (E) (in both position and momentum space) are employed to understand the influence of radial confinement in isotropic harmonic oscillator. The transformation of Hamiltonian in to a dimensionless form gives an idea of the composite effect of oscillation frequency ($\omega$) and confinement radius ($r_{c}$). For a given quantum state, accurate results are provided by applying respective \emph{exact} analytical wave function in $r$ space. The $p$-space wave functions are produced from Fourier transforms of radial functions. 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 confined harmonic oscillator (CHO) with respect to state indices $n_{r},l$, and $r_c$. It has been found that, CHO acts as a bridge between particle in a spherical box (PISB) and free isotropic harmonic oscillator (IHO). At smaller $r_c$, $E_{\rvec}$ increases and $R_{\rvec}^{\alpha}, S_{\rvec}$ decrease with rise of $n_{r}$. At moderate $r_{c}$, there exists an interaction between two competing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of $n_{r}$ (delocalization). Most of these results are reported here for the first time, revealing many new interesting features.