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arXiv:physics/0506001

On the Energy-Time Uncertainty Relations: A didactical note for undergraduate students

Gastón E. Giribet

Published May 31, 2005• 0 citations
arXivHTMLPDF

Abstract

The purpose of this brief note is that of discussing the meaning of the uncertainty relations involving energy and time in quantum mechanics by means of a reading of the classical works on the subject. This was written for undergraduate students attending to an introductory course in quantum mechanics.
DOI
10.48550/arxiv.physics/0506001

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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.
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.
Unlike classical mechanics, quantum mechanics assumes the famous Heisenberg uncertainty relations. One of these concerns time: the energy–time uncertainty relation. Unlike the canonical position–momentum uncertainty relation, the energy–time relation is not reflected in the operator formalism of quantum theory. Indeed, it is often said and taken as problematic that there is not a so-called “time operator” in quantum theory. This chapter sheds light on these questions and others, including the absorbing matter of whether quantum mechanics allows for the existence of ideal clocks. The second section notes that quantum mechanics does not involve a special problem for time, and that there is no fundamental asymmetry between space and time in quantum mechanics over and above the asymmetry which already exists in classical physics. The third section studies time operators in detail. The fourth section discusses various uncertainty relations involving time.
Through many years of teaching Quantum Mechanics in Algerian universities, we have noticed that students face great obstacles in understanding the meaning of the basic concepts of this theory, such as the Wave Function, the Uncertainly Principle, the Superposition Principle and the Complementarity Principle. We believe that these obstacles rise due to the influence of basic concepts in Classical Mechanics which students studied for many years before studying Quantum Mechanics. For example, in Classical Mechanics, the concept of wave and the concept of corpuscle refer to different things, while in Quantum Mechanics both concepts refer to the same thing, namely a "quantum particle". In order to study the problem, a questionnaire sheet was presented to concerned students third-year (undergraduate physics students) before they studied the subject and after studying it. We note that these third-year students had had, as an introduction, a deep discussion of the meaning of quantum mechanical concepts for about one hour and a half a week during one month. To compare the results and show the importance of the epistemological and educational aspects, we presented again the same questionnaire to fifth-year physics students (B.Sc degree). We mention that the latter learned Quantum Mechanics without an epistemological introduction of the basic quantum mechanical concepts. Before giving our epistemological and didactical analysis of the results, we give a brief review of a previous didactical study related to the subject made by other authors. We conclude by giving some suggestions which we think are necessary in teaching Quantum Mechanics.