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Physical Review Letters

Time in Quantum Measurements

M. Hossein Partovi, R. Blankenbecler

Published Dec 8, 1986• 21 citations
Publisher

Abstract

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 $T$ 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.

DOI
10.1103/physrevlett.57.2887

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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\ensuremathδH\ensuremath≥12T\ensuremath{\delta}H\ensuremath{\ge}\frac{1}{2}T\ensuremathδH\ensuremath≥21​, where \ensuremathδH\ensuremath{\delta}H\ensuremathδH 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 \ensuremathδt\ensuremathδH\ensuremath≥12\ensuremath{\delta}t\ensuremath{\delta}H\ensuremath{\ge}\frac{1}{2}\ensuremathδt\ensuremathδH\ensuremath≥21​, where \ensuremathδt\ensuremath{\delta}t\ensuremathδt is the dispersion in the values of time as measured by the system.
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.
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.
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.
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.
The physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time-energy relation is much more difficult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schro¨dinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schro¨dinger’s equation.
A clock is a dynamical system which passes through a succession of states at constant time intervals. If coupled to another system, it can measure the duration of a physical process and even keep a permanent record of it, such as in a time-of-flight experiment or in observing the lifetime of an unstable atom. A clock can also be used to control the duration of a process, e.g., the precession of a spin in a magnetic field which is turned on and off at prescribed times. This article shows how to construct time-independent Hamiltonians describing these possible uses of a quantum clock. As expected, a good time resolution entails a large energy exchange between the clock and the other system, thereby modifying the evolution of the latter. This evolution may even be halted by using a clock which is too precise (this is the quantum analog of Zeno’s paradox).