Inverted oscillator: pseudo hermiticity and coherent states

TL;DR

This paper explores the mathematical structure of the inverted harmonic oscillator, establishing its pseudo-Hermiticity and constructing coherent states that minimize uncertainty, thus providing new insights into its quantum properties.

Contribution

It introduces ladder operators for the inverted oscillator and constructs coherent states, linking pseudo-Hermiticity to the inverted oscillator's analytical solutions.

Findings

Ladder operators for the inverted harmonic oscillator are derived.

Coherent states minimizing uncertainty are constructed.

Pseudo-Hermiticity relates the anti-PT-symmetric Hamiltonian to the inverted oscillator.

Abstract

It is known that the standard and the inverted harmonic oscillator are different. Replacing thus of {\omega} by i{\omega} in the regular oscillator is necessary going to give the inverted oscillator H^{r}. This replacement would lead to anti- PT-symmetric harmonic oscillator Hamiltonian (iH^{os}). The pseudo-hermiticity relation has been used here to relate the anti-PT-symmetric harmonic Hamiltonian to the inverted oscillator. By using a simple algebra, we introduce the ladder operators describing the inverted harmonic oscillator to reproduce the analytical solutions.We construct the inverted coherent states which minimize the quantum mechanical uncertainty between the position and the momentum.