Classical Propagation in the Quantum Inverted Oscillator

TL;DR

This paper demonstrates that quantum state evolution in the inverted oscillator can be described using classical equations of motion, with the Wigner and Ambiguity functions revealing classical propagation characteristics and degenerate energy states.

Contribution

It introduces the use of the Ambiguity function in Reciprocal phase space to analyze classical propagation in the quantum inverted oscillator, highlighting classical dynamics in quantum evolution.

Findings

Ambiguity function exhibits classical propagation behavior.

Quantum states in IO have doubly degenerate energy states.

Classical equations of motion suffice for describing IO evolution.

Abstract

We emphasize the fact the evolution of quantum states in the inverted oscillator (IO) is reduced to classical equations of motion, stressing that the corresponding tunnelling and reflexion coefficients addressed in the literature are calculated by considering only classically trajectories. The Wigner function formalism is employed to describe the IO classical dynamics, subsequently leading to the introduction of the Ambiguity function lying in the so-called Reciprocal phase space. Our findings, show that the Ambiguity function behavior, subjected to the IO, allude a classical propagation with an associated integral of motion, and complex conjugated doubly degenerate energy states.