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Physical Review A

New form of the time-energy uncertainty relation

Eric A. Gislason, Nora H. Sabelli, John W. Wood

Published Apr 1, 1985• 58 citations
Publisher

Abstract

A new form of the time-energy uncertainty principle is presented. It has the form \ensuremath{\tau} \ensuremath{\Delta}E\ensuremath{\ge}3\ensuremath{\pi}${5}^{1/2}$/25, where \ensuremath{\tau} is the average lifetime of a decaying state, and \ensuremath{\Delta}E is the energy spread of the state computed from \ensuremath{\Delta}${E}^{2}$=〈${E}^{2}$〉-〈E${〉}^{2}$. The particular state for which the above relationship becomes an equality is identified, and it is proved that all other states give a greater \ensuremath{\tau} \ensuremath{\Delta}E product. This general result should have a wide range of applications.

DOI
10.1103/physreva.31.2078

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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.
Contrary to a wide-spread impression, the possibility of measuring an energy in a finite time without changing its initial value (E' =E0) is not in contradiction with the principles of quantum mechanics. The relation Δ(E'-E0)Δt≥ holds only in the case when the energy of interaction between the quantum system in question and the apparatus is a function of a coordinate of the system. The condition for a nonperturbing energy measurement is that the interaction energy H1, of the system and the apparatus depend on the energy operator E and that the operators Ĥ and E commute. It is also possible to have a nonperturbing measurement in which the error in measuring the energy is so small that ΔE/Δt. Measurement of the energy of a given system is accompanied by an increase in the uncertainty Δe of the energy of the apparatus. The error ΔE in the measurement of the system's energy and the perturbation Δeof the energy of the apparatus are connected by the relations (ΔE + Δe) Δt≥ and ΔE Δe≥(/2Δt)2.
A derivation from first principles is given of the energy-time uncertainty relation in quantum mechanics. A canonical transformation is made in classical mechanics to a new canonical momentum, which is energy E, and a new canonical coordinate T, which is called tempus, conjugate to the energy. Tempus T, the canonical coordinate conjugate to the energy, is conceptually different from the time t in which the system evolves. The Poisson bracket is a canonical invariant, so that energy and tempus satisfy the same Poisson bracket as do p and q. When the system is quantized, we find the energy-time uncertainty relation \ensuremath{\Delta}E\ensuremath{\Delta}T\ensuremath{\ge}\ensuremath{\Elzxh}/2. For a conservative system the average of the tempus operator T^ is the time t plus a constant. For a free particle and a particle acted on by a constant force, the tempus operators are constructed explicitly, and the energy-time uncertainty relation is explicitly verified.
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.