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Science

Measurement of Time Correlations in Quantum-Mechanical Systems

M. L. Goldberger

Published May 1, 1964• 1 citation
Publisher

Abstract

Measurements that are of limited accuracy, are incomplete, or require a finite time to make do not generally permit one to construct a wave function for describing a physical system. The use of such partial information to predict the results of subsequent measurements is studied here. There are several practical applications of this problem, including the use of the autocorrelation function for a particle counter in a scattering experiment.

DOI
10.1126/science.144.3618.563-a

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Measurements that are of limited accuracy, are incomplete, or require a finite time to make do not generally permit one to construct a wave function for describing a physical system. The use of such partial information to predict the results of subsequent measurements is studied here. There are several practical applications of this problem, including the use of the autocorrelation function for a particle counter in a scattering experiment.
It is cu~tomary in the pedagogica.ldevelopment of quantum mechanics • ...and field theory to mention'the limitations on correlated measurements of observables at different space-time points.Little attention has .•8been given, however, to actual experiments for making such observations, or their usefulness.In this paper and in a subsequent one, we shall •,• discuss both of these subjects from a general point of view and w1 th •.particular applications to scattering processes. :! .This work is an outgrowth from a recent paper on the correlated :1 .counting rate of two detectors recording particles scattered from a 1 . .
A long-standing quantum-mechanical puzzle is whether the collapse of the wave function is a real physical process or simply an epiphenomenon. This puzzle lies at the heart of the measurement problem. One way to choose between the alternatives is to assume that one or the other is correct and attempt to draw physical, observable consequences which then could be empirically verified or ruled out. As a working hypothesis, we propose simple models of collapse as a real physical process for direct binary symmetric measurements made on one particle. This allows one to construct irreversible unstable Schrödinger equations capable of describing continuously the process of collapse induced by the interaction of the quantum system with the measuring device. Due to unknown initial conditions the collapse outcome remains unpredictable so no contradictions with quantum mechanics arise. Our theoretical framework predicts a finite time-scale of the collapse and links with experiment. Sensitive probes of the collapse dynamics could be done using Bose-Einstein condensates, ultracold neutrons or ultrafast optics. If confirmed, the formulation could be relevant to the transition from quantum fluctuations to classical inhomogeneities in early cosmology and to establishing the ultimate limits on the speed of quantum computation and information processing.
The concept of position and its measurement are fundamental to modern navigation technology as well as to theoretical physics. Nevertheless, a consistent notion of position has not been found in modern physics because of apparent inconsistencies between quantum mechanics and relativity. In quantum mechanics the probability distribution for a particle's position is determined by its wave function. A wave function strictly localized in space describes a particle within that locality, and nowhere else. But the quantum equations of motion predict the function will disperse, after any length of time at all, to occupy all space. Therefore the theoretical prediction is that, shortly after detection in a laboratory, a particle may be found anywhere in the universe, including regions requiring faster-than-light travel to reach. This contradicts the theory of relativity, which precludes a body traveling faster than the speed of light. The conflict arises from unexamined assumptions regarding the time of the measurement event and the time of the system under measurement. The prevailing view in physics is that time is only a parameter, having a definite value, describing dynamic changes in physical systems. This view endures even though quantum mechanics recognizes position as a quantum variable with inherent uncertainty, and relativity indicates that time should be placed on the same footing as position. But if time is recognized as a quantum variable with inherent uncertainty, the conceptual problems with position are resolved. The particle has finite probability of being in the past (or future) of the observer, while the quantum mechanical operator describing the position measurement contains a term which extrapolates from the intrinsic time of the particle to the time of the observer. Being potentially well in the past of the measurement event, the particle has time to travel to remote regions without exceeding the speed of light, and the conflict with relativity is resolved. A relativistic quantum theory can therefore be constructed incorporating a well-defined concept of position.
In the past few years experimenters have learned how to determine the complete quantum state of an ensemble of particles or fields which have been prepared according to some unknown procedure. Through these experiments they have answered a question posed by W. Pauli in the 1930s. The methods used involve measuring statistical distributions of a well chosen set of physical observables and using a tomographic inversion algorithm to reconstruct the Wigner function and its corresponding wave function or density matrix. Recontructions have been successfully carried out in atomic, molecular, and optical physics. The development of these procedures helps to firm up the interpretation of the Schrodinger wave function.