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London journal of physics.

The Uncertainty Relation and the Determinacy Relation

Nikolay Chavarga

Published Jul 28, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.69710/ljp.v3i2.17423

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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.
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.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
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.
Abstract One of the foundational principles of quantum mechanics is Heisenberg’s uncertainty relation. According to this relation, no one can measure both the position of a particle and how fast it is moving simultaneously no matter how precise the measuring instruments. Heisenberg’s uncertainty relation and the associated Bohr’s principle of complementarity are explained using simple examples. The idea that measurement disturbs the system is a source of the uncertainty. The philosophical question whether the uncertainty represents only our inability to measure the properties precisely or it has much deeper consequences related to the question of reality is also discussed.
Heisenberg's uncertainty principle, which imposes intrinsic restrictions on our ability to predict the outcomes of incompatible quantum measurements to arbitrary precision, demonstrates one of the key differences between classical and quantum mechanics.The physical systems considered in the uncertainty principle are static in nature and described mathematically with a quantum state in a Hilbert space.However, many physical systems are dynamic in nature and described with the formalism of a quantum channel.In this paper, we show that the uncertainty principle can be reformulated to include process measurements that are performed on quantum channels.Since both the preparation of quantum states and the implementation of quantum measurements are themselves special cases of quantum channels, our formalism encapsulates the uncertainty principle in its utmost generality.More specifically, we obtain expressions that generalize the Maassen-Uffink uncertainty relation and the universal uncertainty relations from quantum states to quantum channels.
Abstract Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays an important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is well known that the case of energy and time principally differs from more familiar examples of two non‐commuting observables. Since quantum theory was originated, many approaches to energy–time uncertainties have been proposed. Entropic way to formulate the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, uncertainty relations of the “energy–time” type are obtained in terms of Rényi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state‐dependent and state‐independent formulations are of interest. Some of the derived state‐independent bounds are similar to the results obtained within a more general approach on the basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.