A note on a classical dynamical system and its quantization

TL;DR

This paper examines a modified classical system related to the damped harmonic oscillator and explores its quantum behavior, revealing that it can exhibit either standard or inverted oscillator dynamics depending on parameter relations.

Contribution

It demonstrates that the quantized version of the modified system can display different behaviors, including standard and inverted oscillators, based on parameter conditions.

Findings

Quantized system can behave as a standard harmonic oscillator.

Quantized system can behave as an inverted oscillator.

Different behaviors depend on parameter relations.

Abstract

In a recent paper a slightly modified version of the Bateman system, originally proposed to describe a damped harmonic oscillator, was proposed. This system is really different from the Bateman's one, in the sense that this latter cannot be recovered for any choice of its parameters. In this paper we consider this system and we show that, at a quantum level, it is not necessarily dissipative. In particular we show that the Hamiltonian of the system, when quantized, produces different behaviors, depending on some relations between its parameters. In fact, it gives rise to either a two dimensional (standard) harmonic oscillator, or to two independent oscillators, one of which is again {\em standard}, and a second one which is an inverted oscillator. The two cases are analyzed in terms of bosonic or pseudo-bosonic ladder operators, and the appearance of distributions for the inverted oscillator is commented.