Almost-periodic time observables for bound quantum systems

TL;DR

This paper introduces a general framework for defining time observables in bound quantum systems with discrete energy spectra, extending previous results to almost-periodic cases and deriving a new entropic uncertainty relation.

Contribution

It generalizes the concept of time observables to almost-periodic systems using POMs and establishes an entropic uncertainty relation for energy and time.

Findings

Time observables can be defined for any quantum system with discrete energy levels.

Almost-periodic probability operator measures enable expectation calculations for almost-periodic functions.

An entropic uncertainty relation for energy and time is derived, extending known relations.

Abstract

It is shown that a canonical time observable may be defined for any quantum system having a discrete set of energy eigenvalues, thus significantly generalising the known case of time observables for periodic quantum systems (such as the harmonic oscillator). The general case requires the introduction of almost-periodic probability operator measures (POMs), which allow the expectation value of any almost-periodic function to be calculated. An entropic uncertainty relation for energy and time is obtained which generalises the known uncertainty relation for periodic quantum systems. While non-periodic quantum systems with discrete energy spectra, such as hydrogen atoms, typically make poor clocks that yield no more than 1 bit of time information, the anisotropic oscillator provides an interesting exception. More generally, a canonically conjugate observable may be defined for any Hermitian operator having a discrete spectrum.