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arXiv:1309.4957

The role of the probability current for time measurements

Nicola Vona, Detlef Dürr

Published Sep 19, 2013• 0 citations
arXivHTMLPDF

Abstract

Time measurements are routinely preformed in laboratories, nevertheless their theoretical account presents some difficulties and for actual experiments an approximate, semiclassical expression is always used. Here, we will discuss their quantum description with particular emphasis on the role of the probability current.
DOI
10.48550/arxiv.1309.4957

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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\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.
Ever since the beginning of quantum mechanics, the measurement problem has been a subject of sometimes discontinued but nevertheless recurrent concern. As students, most physicists have raised questions about it, but the spectacular and renewed successes of quantum theory were a strong incentive for them to work out applications without further worrying about foundations. In each generation, however, some dissatisfaction reemerges, which leads to slow progress, and we may expect that the measurement problem will some day be cleared up rather than forgotten. The situation recalls statistical mechanics, the foundations of which still give rise to scientific discussions, in spite of more than one century’s achievements. Actually, the paradoxes of irreversibility and of Maxwell’s demon share some common features with quantum measurements, to wit, they involve probabilities in a crucial fashion and their explanation requires elucidating the role of the observer.
Darmstadt ννν oscillations in decay of radioactive ion can only come from initial state wave function. Causality forbids any influence on transition probability by detection of ννν or final state interference after decay. Energy-time uncertainty allows two initial state components with different energies to decay into combination of two orthogonal states with same energy, different momenta and different ννν masses. Final amplitudes completely separated at long times have broadened energy spectra overlapping at short times. Their interference produces oscillations between Dicke superradiant and subradiant states having different transition probabilities. Repeated monitoring by interactions with laboratory environment at regular time intervals and same space point in laboratory collapses wave function and destroys entanglement. First-order time dependent perturbation theory gives probability for initial state decay during small interval between two monitoring events. Experiment measures momentum difference between two contributing coherent initial states and obtains information about ννν masses without detecting ννν. Simple model relates observed oscillation to squared ννν mass difference and gives value differing by less than factor of three from values calculated from KAMLAND experiment. Monitoring simply expressed in laboratory frame not easily transformed to other frames and missed in Lorentz-covariant descriptions based on relativistic quantum field theory.