# Information-Entropic Measures in Confined Isotropic Harmonic Oscillator

## TL;DR

This paper investigates how radial confinement affects uncertainty measures like Shannon and Rényi entropies in isotropic harmonic oscillators, revealing new insights into their behavior across different confinement regimes.

## Contribution

It provides the first detailed analytical study of information entropic measures in confined isotropic harmonic oscillators, connecting particle-in-box and free oscillator limits.

## Key findings

- Entropic measures vary systematically with confinement radius and quantum state.
- Confined harmonic oscillator acts as a bridge between particle in a box and free oscillator.
- New features of entropic measures under radial confinement are reported for the first time.

## 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.