The time-energy uncertainty relation for quantum events

TL;DR

This paper derives true quantum uncertainty relations linking the measurement of event timing to energy uncertainty, addressing the classical treatment of time in quantum mechanics.

Contribution

It introduces a framework using quantum clocks to establish genuine time-energy uncertainty relations, moving beyond classical parameter interpretations.

Findings

Derived two true time-energy uncertainty relations.

Connected event timing uncertainty with energy uncertainty.

Provided a quantum clock-based measurement framework.

Abstract

Textbook quantum mechanics treats time as a classical parameter, and not as a quantum observable with an associated Hermitian operator. For this reason, to make sense of usual time-energy uncertainty relations such as $\Delta {t}\Delta E\gtrsim\hbar$, the term $\Delta t$ must be interpreted as a time interval, and not as a time measurement uncertainty due to quantum noise. However, quantum clocks allow for a measurement of the "time at which an event happens" by conditioning the system's evolution on an additional quantum degree of freedom. Within this framework, we derive here two {\em true} uncertainty relations that relate the uncertainty in the quantum measurement of the time at which a quantum event happens on a system to its energy uncertainty.