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arXiv:1809.06539

Comment on Energy-Time Uncertainty Relations in Quantum Measurements

Yasuda, Koji

Published Sep 18, 2018• 0 citations
arXivHTMLPDF

Abstract

The measurement apparatus proposed in the titled paper, "Energy-Time Uncertainty Relations in Quantum Measurements" [Found. Phys. 46, 1522-1550 (2016)] was examined. A simple proof was presented for the non-existence of the apparatus.
DOI
10.48550/arxiv.1809.06539

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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 physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time-energy relation is much more difficult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schro¨dinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schro¨dinger’s equation.
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