Characterizing Multiple Solutions to the Time-Energy Canonical Commutation Relation via Quantum Dynamics

TL;DR

This paper investigates multiple solutions to the time-energy canonical commutation relation in quantum mechanics, focusing on how different self-adjoint time operators relate to quantum dynamics in confined systems.

Contribution

It characterizes distinct self-adjoint time operators for a confined particle and links their eigenvector dynamics to specific quantum time properties.

Findings

Existence of two distinct self-adjoint time operators

Eigenvector dynamics differentiate between time operators

Distinct properties of time operators relate to quantum time aspects

Abstract

We address the multiplicity of solutions to the time-energy canonical commutation relation for a given Hamiltonian. Specifically, we consider a particle spatially confined in a potential free interval, where it is known that two distinct self-adjoint and compact time operators conjugate to the system Hamiltonian exist. The dynamics of the eigenvectors of these operators indicate that different time operators posses distinguishing properties that can unambiguously associate them to specific aspects of the quantum time problem.