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arXiv:quant-ph/0207048 · Reviews in Mathematical Physics

REMARKS ON TIME-ENERGY UNCERTAINTY RELATIONS

ROMEO BRUNETTI, Klaus Fredenhagen

Published Jul 1, 2002• 18 citations
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Abstract

Using a recent construction of observables characterizing the time of occurence of an effect in quantum theory, we present a rigorous derivation of the standard time-energy uncertainty relation. In addition, we prove an uncertainty relation for time measurements alone.
DOI
10.1142/s0129055x02001417

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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.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
Our formalism for describing quantum measurements is generalized to the multitime case and used to derive rigorous time-energy uncertainty relations. For a free particle, we find that $T\ensuremath{\delta}H\ensuremath{\ge}\frac{1}{2}$, where $\ensuremath{\delta}H$ is the dispersion in energy and TTT is the measurement duration as given by an external clock of arbitrarily high accuracy. Moreover, any system used as a clock obeys $\ensuremath{\delta}t\ensuremath{\delta}H\ensuremath{\ge}\frac{1}{2}$, where $\ensuremath{\delta}t$ is the dispersion in the values of time as measured by the system.