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Journal of Physics Conference Series

A derivation of the time-energy uncertainty relation

J S Briggs

Published Feb 1, 2008• 32 citations
Publisher

Abstract

Some of the difficulties that have occurred in the past associated with the derivation of a time-energy uncertainty relation are discussed. Then it is shown that, beginning with a time-independent quantum composite of system and environment, a time-energy uncertainty relation for the system alone can be derived in the limit that the environment becomes classical, to provide a classical clock with which to monitor the quantum system.

DOI
10.1088/1742-6596/99/1/012002

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Some of the difficulties that have occurred in the past associated with the derivation of a time-energy uncertainty relation are discussed. Then it is shown that, beginning with a time-independent quantum composite of system and environment, a time-energy uncertainty relation for the system alone can be derived in the limit that the environment becomes classical, to provide a classical clock with which to monitor the quantum system.
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 $\mathrm{\ensuremath{\Delta}}t\mathrm{\ensuremath{\Delta}}E\ensuremath{\gtrsim}\ensuremath{\hbar}$, the term $\mathrm{\ensuremath{\Delta}}t$ 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 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.
The meaning and scope of a recent type of uncertainty relation of a very general character are elucidated using the notions of time-indicating dynamical variables (clock variables) and place-indicating dynamical variables (position variables). It is shown that if the total energy (momentum) of a system is certain, all time-indicating (place-indicating) dynamical variables are completely uncertain. The quantum clock is discussed as an illustration of the energy–time uncertainty relation. The relations can be successfully applied to the thought experiments that Einstein introduced into his debate with Bohr about the uncertainty principle(s) and, in particular, to the famous photon-box experiment. It is shown that due to this general relation the photon box can never serve its purpose, independent of the details of the experiment.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.