# Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

## TL;DR

This paper investigates how symmetric and asymmetric confinement affect a quantum harmonic oscillator using information entropy measures, providing accurate eigenvalues and analyzing localization and energy behaviors.

## Contribution

It introduces a variation-induced diagonalization method for asymmetric confinement and explores the impact of confinement asymmetry on uncertainty measures and energy spectra.

## Key findings

- Uncertainty measures converge to free oscillator values at large confinement lengths.
- Asymmetric confinement causes excited-state energies to pass through a maximum.
- Localization increases with asymmetry parameter.

## Abstract

Information-based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of \emph{symmetric and asymmetric} confinement in a quantum harmonic oscillator. Also, the transformation of Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ($x_c$). In symmetric case, a wide range of $x_{c}$ has been taken up, whereas asymmetric confinement is dealt by shifting the minimum of potential from origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary time propagation scheme. For asymmetric confinement, a variation-induced exact diagonalization procedure is also introduced, which produces very high-quality results. One finds that, in symmetric confinement, after a certain characteristic $x_{c}$, all these properties converge to respective values of free harmonic oscillator. In asymmetric situation, excited-state energies always pass through a maximum. For this potential, the classical turning-point decreases, whereas well depth increases with the strength of asymmetry. Study of these uncertainty measures reveals that, localization increases with an increase of asymmetric parameter.