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Oxford University Press eBooks

Time in Quantum Mechanics

Jan Hilgevoord, David Atkinson

Published Sep 2, 2011• 37 citations
Publisher

Abstract

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.

DOI
10.1093/oxfordhb/9780199298204.003.0023

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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.
Abstract The problem of time in quantum mechanics (QM) concerns the fact that in the Schrödinger equation time is a parameter, not an operator. Pauli's objection to a time–energy uncertainty relation analogue to the position–momentum one, conjectured by Heisenberg early on, seemed to exclude the existence of such an operator. However Dirac's formulation of an electron's relativistic QM does allow the introduction of a dynamical time operator that is self-adjoint. Consequently, it can be considered as the generator of a unitary transformation of the system, as well as an additional system observable subject to uncertainty. In the present paper these aspects are examined within the standard framework of relativistic QM.
It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.
Heisenberg guessed, after he established the matrix quantum mechanics, that the non-commutativity of the matrices of position and momentum implied that the position and momentum of a particle could not be precisely simultaneously determined. He consequently conjectured that time and energy should also have a similar relationship. Soon after, Robertson derived an inequality concerning the space coordinate and momentum, which was thought to be the mathematical expression of the uncertainty relation guessed by Heisenberg. Since then, people have tried various devices to prove the correctness of these two relations. However, no one conducted a careful analysis of Heisenberg’s primary paper. In this work, we point out some serious problems in Heisenberg’s paper and the literature talking about the uncertainty relationships: the physical concepts involved in the uncertainty relations are not clear; one physical concept had more than one explanation, i.e., switching concepts; there has never been measurement experiment to support the relations. The conclusions are that the so-called coordinate–momentum uncertainty relation has never been related to actual measurement and there does not exist a time–energy uncertainty relation.
Since the very begining of quantum theory there started a debate on the proper role of space and time in it. Some authors assumed that space and time have their own algebraic operators. On that basis they supposed that quantum particles had “coordinates of position”, even though those coordinates were not possible to determine with infinite precision. Furthermore, time in quantum physics was taken to be on an equal foot, by means of a so-called “Heisenberg’s fourth relation of indeterminacy” concerning time and energy. In this paper, the proper role of space and time in the core of non-relativistic quantum phsysics is analyzed. We will find that, rigorously, that relation for time and energy shows a different root. For the role of space, it will be discussed that there is no “coordinate of position” in the conceptual structure of quantum physics because quantum particles are not point-like objects.
Abstract: Since the very begining of quantum theory there started a debate on the proper role of space and time in it. Some authors assumed that space and time have their own algebraic operators. On that basis they supposed that quantum particles had “coordinates of position”, even though those coordinates were not possible to determine with infinite precision. Furthermore, time in quantum physics was taken to be on an equal foot, by means of a so-called “Heisenberg’s fourth relation of indeterminacy ” concerning time and energy. In this paper, the proper role of space and time in the core of non-relativistic quantum phsysics is analyzed. We will find that, rigorously, that relation for time and energy shows a different root. For the role of space, it will be discussed that there is no “coordinate of position ” in the conceptual structure of quantum physics because quantum particles are not point-like objects. 1.
Since the very begining of quantum theory there started a debate on the proper role of space and time in it. Some authors assumed that space and time have their own algebraic operators. On that basis they supposed that quantum particles had “coordinates of position”, even though those coordinates were not possible to determine with infinite precision. Furthermore, time in quantum physics was taken to be on an equal foot, by means of a so-called “Heisenberg’s fourth relation of indeterminacy” concerning time and energy. In this paper, the proper role of space and time in the core of non-relativistic quantum phsysics is analyzed. We will find that, rigorously, that relation for time and energy shows a different root. For the role of space, it will be discussed that there is no “coordinate of position” in the conceptual structure of quantum physics because quantum particles are not point-like objects.