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The Explanatory Context of Energy-Time Uncertainty Relation

Zhao Da

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

There are many controversies on energy—time uncertainty relation. Because of the confused role time plays in quantum mechanics, the references and meanings of energy—time uncertainty relation are different in different explanatory contexts. But what do time and energy really mean? What is the philosophical foundation of this relationship? The article bases on the three different meanings of time in quantum mechanics, discusses the references and meanings of energy—time uncertainty relation in various contexts, and aims to clarify the confusion on the references and the meanings of the uncertainty relation between energy and time.

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There are many controversies on energy—time uncertainty relation. Because of the confused role time plays in quantum mechanics, the references and meanings of energy—time uncertainty relation are different in different explanatory contexts. But what do time and energy really mean? What is the philosophical foundation of this relationship? The article bases on the three different meanings of time in quantum mechanics, discusses the references and meanings of energy—time uncertainty relation in various contexts, and aims to clarify the confusion on the references and the meanings of the uncertainty relation between energy and time.
The time energy uncertainty relation has been a controversial issue since the advent of quantum theory, with respect to appropriate formalisation, validity and possible meanings. A comprehensive account of the development of this subject up to the 1980s is provided by a combination of the reviews of Jammer (1974), Bauer and Mello (1978), and Busch (1990). More recent reviews are concerned with different specific aspects of the subject. The purpose of this chapter is to show that different types of time energy uncertainty relation can indeed be deduced in specific contexts, but that there is no unique universal relation that could stand on equal footing with the position-momentum uncertainty relation. To this end, we will survey the various formulations of a time energy uncertainty relation, with a brief assessment of their validity, and along the way we will indicate some new developments that emerged since the 1990s.
On the basis of the principles proposed by the first author in 1970 and 1979 the time-energy uncertainty relationship in quantum mechanics is constructed explicitly for transitions between adequately prescribed states of a simple system of a nonrelativistic particle moving in a constant uniform field of force. The implications of the uncertainties in time and energy contemplated for transitions are clarified by contrasting them with ordinary uncertainties of various dynamical variables evaluated in one single state of the system. Decay process and the mean lifetime related to it are discussed too in a particular case when a pair of states, initial and final, for a transition happen to be identical.
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.
Energy-time uncertainty plays an important role in quantum foundations and technologies, and it was even discussed by the founders of quantum mechanics. However, standard approaches (e.g., Robertson's uncertainty relation) do not apply to energy-time uncertainty because, in general, there is no Hermitian operator associated with time. Following previous approaches, we quantify time uncertainty by how well one can read off the time from a quantum clock. We then use entropy to quantify the information-theoretic distinguishability of the various time states of the clock. Our main result is an entropic energy-time uncertainty relation for general time-independent Hamiltonians, stated for both the discrete-time and continuous-time cases. Our uncertainty relation is strong, in the sense that it allows for a quantum memory to help reduce the uncertainty, and this formulation leads us to reinterpret it as a bound on the relative entropy of asymmetry. Because of the operational relevance of entropy, we anticipate that our uncertainty relation will have information-processing applications.
This brief note has the didactical purpose of discussing the meaning of the uncertainty principle involving energy and time in quantum mechanics within the context of the seminal works on this subject. The importance of this topic is, indeed, frecuently neglected in several textbooks on quantum mechanics. Then, these pages are addressed to students attending to an undergraduate course on quantum mechanics; and the original aim is to present a list of references of the points which take part in the principal discussions on the subject. Principal attention is devoted to the discussion of the fallacies, which are indeed persistent in the literature.