The uncertainty principle for energy and time

Abstract

The meaning and scope of a recent type of uncertainty relation of a very general character are elucidated using the notions of time-indicating dynamical variables ͑clock variables͒ and place-indicating dynamical variables ͑position variables͒. It is shown that if the total energy ͑momentum͒ of a system is certain, all time-indicating ͑place-indicating͒ dynamical variables are completely uncertain. The quantum clock is discussed as an illustration of the energy-time uncertainty relation. The relations can be successfully applied to the thought experiments that Einstein introduced into his debate with Bohr about the uncertainty principle͑s͒ and, in particular, to the famous photon-box experiment. It is shown that due to this general relation the photon box can never serve its purpose, independent of the details of the experiment.

Jump Time and Passage Time: The Duration of a Quantum Transition

“…This makes the jump time a reflection of a kind of time-energy uncertainty relation. As such it is a statement of this relation that is consistent with the views expressed in [19].…”

Evolution time and energy uncertainty

“…The 'exact' results are calculated with the first 700 terms of the series (no changes occur to five significant figures summing the first 1000 terms) and give E exact = 4.714ε 1 . The 'approximate' results ignore square coefficients c (ν) n 2 < 0.02 max c (ν) n 2 , with the maximum taken with respect to n. These give a window W = 14ε 1 (meaning that the sums extend over 111 n 139) and an energy uncertainty E app = 4.654ε 1 . The 'exact' calculation givesĒ = 125.00ε 1 while the 'approximate' calculation givesĒ = 124.74ε 1 .…”

“…The problem most often studied involves the evolution of an initial state under a Hamiltonian which is not an explicit function of time. Hilgevoord [1] emphasizes that this relation is very different from the more familiar positionmomentum relation and Uffink [3] shows that it can be written in terms of the energy width of the distribution and the overlap of the initial and final wavefunctions. Gislason, Sabelli and Wood [2] discuss uncertainty relations and evolution times for several common distributions.…”

“…One such issue is the more conceptually difficult energy-time uncertainty relation, which has been the topic of much recent research [1][2][3]. The problem most often studied involves the evolution of an initial state under a Hamiltonian which is not an explicit function of time.…”

“…The treatment is formulated in terms of explicit states and is suitable for calculations where only numerical data are available. The physics of the problem leads us to a different definition of the evolution time from that employed previously [1][2][3] and we relate this time to the energy uncertainty.…”