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Physical Review A

Improvement of the Eberly-Singh time-energy inequality by combination with the Mandelstam-Tamm approach

C. Leubner, Christoph Alexander Kiener

Published Jan 1, 1985• 15 citations
Publisher

Abstract

It is demonstrated that Eberly and Singh's notion of partial stationarity of a quantum system can be reconciled with Mandelstam and Tamm's approach to deriving energy-time inequalities. The result of this merger is an improved stationarity time, which provides a more precise measure of the degree of stationarity of the system under study, and which permits a sharper statement of the energy-time uncertainty relation than that given by Eberly and Singh.

DOI
10.1103/physreva.31.483

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It is demonstrated that Eberly and Singh's notion of partial stationarity of a quantum system can be reconciled with Mandelstam and Tamm's approach to deriving energy-time inequalities. The result of this merger is an improved stationarity time, which provides a more precise measure of the degree of stationarity of the system under study, and which permits a sharper statement of the energy-time uncertainty relation than that given by Eberly and Singh.
We give a short review of known exact inequalities that can be interpreted as ‘energy–time’ and ‘frequency–time’ uncertainty relations. In particular we discuss a precise form of signals minimizing the physical frequency–time uncertainty product. Also, we calculate the ‘stationarity time’ for mixed Gaussian states of a quantum harmonic oscillator, showing explicitly that pure quantum states are ‘more fragile’ than mixed ones with the same value of the energy dispersion. The problems of quantum evolution speed limits, time operators and measurements of energy and time are briefly discussed, too.