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.
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.
The Energy‐Time Uncertainty (ETU) has always been a problem‐ridden relation, its problems stemming uniquely from the perplexing question of how to understand this mysterious Δt. On the face of it (and, indeed, far deeper than that), we always know what time it is. Few theorists were ignorant of the fact that time in quantum mechanics is exogenously defined, in no ways intrinsically related to the system. Time in quantum theory is an independent parameter, which simply means independently known. In the early 1960s Aharonov (1961–64) and Bohm (1961–64) mounted a spirited attack against the ETU, which sealed its fate to the present date. By emphasising that time is always “well‐defined” in quantum theory, they were led to the conclusion that no ETU should exist, a view shared by many in the 1990s, if Busch (1990) is to be believed. In a similar vein, I emphasize that (a) physical systems occupy a particular energy state at a particular instant of time, if at all; (b) even in absence of all time‐measuring instruments, it is still trivially warranted that one can measure a system's energy as accurately as one pleases, and simply announce “The system's energy is exactly E NOW!”, a possibility which no quantum mechanics of any sort, or any physical theory whatsoever, can afford to tamper with or change, except circularly. One never loses one's own perception of time, when one measures the energy, a fact which no measurement conceivable can interfere with or affect. Both (a) and (b) uniquely entail that energy and time are compatible, if not indeed intimately interconnected, contrary to what the relevant uncertainty seems to affirm. In response to Aharonov's and Bohm's initial problem, I reinterpret ΔEΔt ≥ h, as directly derived from authentic quantum principles, without however having to assume a direct incompatibility between its related concepts, attributing their complementarity to conditions other than ordinarily assumed.
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.
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.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality,
ΔpΔq>2πh
(1)
This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3 have been derived, where ψ denotes the initial state, H the Hamiltonian, R a positive constant, and C is explicitly known. As intermediate step, bounds on the derivatives of the S-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3, and the constants Cn,K explicitly known.
The problem of measuring the energy of a quantum system, the process of emission of an electromagnetic field quantum, and their connection with the provisions of the Heisenberg uncertainty principle are considered. It is shown that the processes of measuring the energy of a quantum system and the time of transition between levels are performed not only simultaneously, but in a single act of measurement, and cannot be fulfilled in another way. This contradicts the physical ideas embedded in the uncertainty principle. As an alternative, the concept of "determinacy relations" is proposed.