Quantum inverted harmonic potential

TL;DR

This paper explores the quantum properties of particles in an inverted harmonic potential, revealing unique states, zero-position uncertainty solutions, and potential applications in optical beam focusing.

Contribution

It introduces a new set of orthogonal states, discusses infinite entropy at zero temperature, and proposes a novel wave packet with self-focusing capabilities for optical systems.

Findings

Existence of infinitely degenerate non-stationary orthogonal states

Zero uncertainty in particle position with Dirac delta asymptote

A new self-focusing wave packet for optical applications

Abstract

We consider a non-interacting gas under the inverted harmonic potential and present infinitely degenerate non-stationary orthogonal states. We discuss that it has an infinite entropy at the absolute zero temperature. We show that uncertainty in position of a particle under the inverted harmonic potential can be zero as there exists a solution which asymptotes to a Dirac delta function. We obtain a new free particle wave packet using the eigenstates for the inverted harmonic potential. It has unique self-focusing feature and can be used as focusing beam without a lens in optical systems where paraxial approximation is used.