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arXiv:quant-ph/0412079 · Physical Review A

Measurement of the total energy of an isolated system by an internal observer

Serge Massar, Sandu Popescu

Published Apr 20, 2005• 3 citations
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Abstract

We consider the situation in which an observer internal to an isolated system wants to measure the total energy of the isolated system (this includes his own energy, that of the measuring device and clocks used, etc.). We show that he can do this in an arbitrarily short time, as measured by his own clock. This measurement is not subjected to a time-energy uncertainty relation. The properties of such measurements are discussed in detail with particular emphasis on the relation between the duration of the measurement as measured by internal clocks versus external clocks.

DOI
10.1103/physreva.71.042106

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We consider the situation in which an observer internal to an isolated system wants to measure the total energy of the isolated system (this includes his own energy, that of the measuring device and clocks used, etc.). We show that he can do this in an arbitrarily short time, as measured by his own clock. This measurement is not subjected to a time-energy uncertainty relation. The properties of such measurements are discussed in detail with particular emphasis on the relation between the duration of the measurement as measured by internal clocks versus external clocks.
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.
Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time $\ensuremath{\Delta}t$ that obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}H\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can, in principle, be measured with arbitrarily large precision in an arbitrarily short time. On the other hand, we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\ensuremath{\Delta}t$ which obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}E\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.
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.
This paper focuses on "time-extended" properties that quantum mechanics represents as existing at a given moment, but which cannot be measured at a given moment. Several examples are presented in which the energy cannot be measured in a short time. The essence of these examples is an attempt to measure the momenta (or the energy) of an ideal quantum clock by having an interaction that lasts only a short time, where this short time is defined with respect to the internal time which is conjugate to this momenta. However, this momenta and this time cannot both be definite at once, even though quantum mechanics claims that a definite energy should be definite at a definite time. However, the momenta can be definite at a definite external parameter time, rather than this internal time. From the internal perspective, however, it is shown that the energy cannot be defined at a given internal time and therefore, this aspect of "time-extension" is completely quantum in origin, unlike the classical aspect of "energy is frequency." An additional consequence of time extended properties is that the connection between the internal time before and after the energy measurement is made uncertain. This suggests that the complementarity between energy and time is deeper than the notion that precise measurements of energy take a long time. Going beyond the "negative" statements about what cannot be measured (which characterize most of the discussions of the energy-time uncertainty relation), a positive aspect of ΔEΔt > 1 is demonstrated in a closed system based on causality. In this example, if one were to argue that energy really existed at a definite moment in time, then causality could be called into question. The standard understanding of the relationship between energy and time is that if the energy is conserved then we can calculate what the energy is at any point in time and thus we should be able to speak about energy as actually existing at that definite moment in time. This section challenges that assertion and is motivated by the question: "is there an example in physics of a property that the formalism tells us exists at a given moment, but which cannot be checked at a given time?"
A gedanken experiment is proposed for "weighing" the total mass of a closed system from within the system. We prove that for an internal observer the time tau, required to measure the total energy with accuracy DeltaE, is bounded according to tauDeltaE>Planck's over 2pi. This time-energy uncertainty principle for a closed system follows from the measurement backreaction on the system. We generally examine what other conserved observables are in principle measurable within a closed system and what are the corresponding uncertainty relations.
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.