Classical and statistical limits of the quantum singular oscillator

TL;DR

This paper explores the classical and quantum boundaries of the singular oscillator using phase-space and Bohmian mechanics, analyzing quantum fluctuations, trajectories, and thermal effects to understand quantum-classical transition and system equivalence.

Contribution

It provides analytical Bohmian trajectories for the singular oscillator and compares quantum and classical limits, highlighting the system's statistical equivalence with the harmonic oscillator at thermal equilibrium.

Findings

Bohmian trajectories show how energy and anharmonicity influence quantum behavior.

Thermal effects alter local quantum fluctuations and quantum purity.

Singular and harmonic oscillators are statistically equivalent at thermal equilibrium.

Abstract

The classical boundaries of the quantum singular oscillator (SO) is addressed under Weyl-Wigner phase-space and Bohmian mechanics frameworks as to comparatively evaluate phase-space and configuration space quantum trajectories as well as to compute distorting quantum fluctuations. For an engendered pure state \textit{quasi}-gaussian Wigner function that recovers the classical time evolution (at phase and configuration spaces), Bohmian trajectories are analytically obtained as to show how the SO energy and anharmonicity parameters drive the quantum regime through the so-called quantum force, which quantitatively distorts the recovered classical behavior. Extending the discussion of classical-quantum limits to a quantum statistical ensemble, the thermalized Wigner function and the corresponding Wigner currents are computed as to show how the temperature dependence affects the local quantum fluctuations. Considering that the level of quantum mixing is quantified by the quantum purity, the loss of information is quantified in terms of the temperature effects. Despite having contrasting phase-space flow profiles, two inequivalent quantum systems, namely the singular and the harmonic oscillators, besides reproducing stable classical limits, are shown to be statistically equivalent at thermal equilibrium, a fact that raises the SO non-linear system to a very particular category of quantum systems.