The time of arrival in quantum mechanics III. The measurement ensemble

Abstract

This paper contains further discussion of the phenomena and problems of interpretation which arise when a nonrelativistic quantum particle falls upon an apparatus designed to measure the time of its arrival at a fixed spatial point. Full allowance is made for the finite time resolution of the apparatus. The probabilities elicited in the various time-labelled output channels of the apparatus are studied to see whether it is possible to represent them in terms of the classical concept of an apparatus-independent arrival probability per unit time as input. It is proved that a classical interpretation of the type required can only be given at an approximate level. Two distinct types of uncertainty appear in every attempted inference from the observed output probabilities to the hypothetical input probabilities. The first type of uncertainty applies to the magnitudes of the inferred input probabilities. It is found that the output probabilities are never completely consistent with the idea that the action of the apparatus is merely to process an incoming and apparatus-independent arrival probability distribution. The apparatus is in fact processing an incoming and apparatus-independent wave, and this gives rise to superimposed and temporally correlated apparatus-dependent quantum deviations in the magnitude of the output. Because of these, the output cannot be accurately described in any scheme which uses only the classical particle concept of arrival probability and the measurement concepts of classical physics, and therefore exact inference is not possible. There are however certain aspects of the apparatus response which can be interpreted in a classical way to within a small and effectively random error. The search for these aspects discloses that the times to be associated with the inferred input probabilities are also uncertain. This is the second type of uncertainty. The input probabilities for arrival which can be approximately inferred from experiment correspond to weighted averages over time intervals having various independently assigned end points. Neither the beginning, the duration, nor the end of a time interval can be specified with complete precision if one wishes to use the measured output to infer an input probability falling within the interval. The inferential uncertainty in the arrival probability and the uncertainty in the definition of the associated time interval are subject to a mutual relationship not unlike the relationships of uncertainty which apply in problems of measurement of complementary pairs of dynamical variables. If the end points of a time interval are defined rather sharply, and more particularly if they are also rather close together, then the input arrival probability which can be inferred for that interval becomes correspondingly uncertain, and vice versa. The relationship of complementarity between the magnitudes of the uncertainties in the probability and in the time is found to be mathematically the same as that which applies to individual arrival-time measurements, although the uncertainties themselves have a somewhat different physical meaning in the two cases. (In an individual measurement the relevant time uncertainty does not refer to the temporal location of a partial probability, but arises directly for the single event concerned because the time resolution of the apparatus is finite. Likewise the probability uncertainty does not refer to the magnitude of a partial probability, but rather to the chance that the attempt to observe the impending individual arrival event will fail due to reflection of the incident wave by the apparatus). From this mathematical equivalence, which holds good in spite of the obvious contextual differences, it follows that the concept of arrival time is effectively subject to one and the same irreducible uncertainty, irrespective of whether the concept be applied to the analysis of an individual event or to the analysis of the ensemble. The approximately measurable arrival probability distribution which emerges from the analysis of the ensemble is quantitatively different from the well known probability current. Good reasons are found to explain why this difference occurs. The uncertainty relation between energy and time is written in a new and modified form, in order to take account of the extra uncertainty inherent in the statistical arrival-time concept. The statistical concept of time delay in scattering processes is also examined. It is shown that the statistical concept of time delay can be measured with complete precision in an ensemble of arrival-time experiments, even though the corresponding arrival-time probability distributions can only be inferred approximately and with finite time uncertainty.