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Physical Review A

Finite-time measurement of quantum particle’s mean position

Y. Liu, D. Sokolovski

Published Dec 13, 2000• 12 citations
Publisher

Abstract

We analyze nonrelativistic quantum measurement of the time average of the particle's coordinate, $X\ensuremath{\equiv}{t}^{\ensuremath{-}1}{\ensuremath{\int}}_{0}^{t}{x(t}^{\ensuremath{'}}{)dt}^{\ensuremath{'}}.$ The measurement amplitude is constructed by restricting the Feynman path integral to paths with the required value of X. The resulting decomposition of the Schr\"odinger wave function determines the type of meter needed to measure X. We show that such meter can be realized as a magnetic moment traveling with the particle in a magnetic field whose magnitude linearly changes with x. Weak and strong measurement regimes are discussed.

DOI
10.1103/physreva.63.014102

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We analyze nonrelativistic quantum measurement of the time average of the particle's coordinate, X\ensuremath≡t\ensuremath−1\ensuremath∫0tx(t\ensuremath′)dt\ensuremath′.X\ensuremath{\equiv}{t}^{\ensuremath{-}1}{\ensuremath{\int}}_{0}^{t}{x(t}^{\ensuremath{'}}{)dt}^{\ensuremath{'}}.X\ensuremath≡t\ensuremath−1\ensuremath∫0t​x(t\ensuremath′)dt\ensuremath′. The measurement amplitude is constructed by restricting the Feynman path integral to paths with the required value of X. The resulting decomposition of the Schr"odinger wave function determines the type of meter needed to measure X. We show that such meter can be realized as a magnetic moment traveling with the particle in a magnetic field whose magnitude linearly changes with x. Weak and strong measurement regimes are discussed.
We analyze the connection between the path integral and operator approaches to the quantum measurement problem. In general, an act of measurement is shown to destroy interference between components of the wave function related to the particle's histories. Classically, there exist a class of meters suitable for measuring the value of a given dynamical variable F. Quantally, different meters produce different results. The standard von Neumann measurement is one particular case. Rearranging Feynman paths according to the value of (time average of) F defines a different type of meter. The two methods disagree if the duration of the measurement is very short. Possible ways to measure the particle's momentum are studied in detail. The semiclassical limit of a measurement is analyzed.
We analyze a quantum observable fff given by the time average of a dynamical variable F along a Feynman path. We show that fff can be described by a (generalized) wave function satisfying the Schr\"odinger-like equation and measured by a meter similar to that of von Neumann. A connection with operators is established. Examples include the particle's momentum, energy and the fractional traversal time.
Background and Objectives: There are a few different ways to change the quantum mechanical wave function. One is the continuous evolution according to the Schrodinger equation; the others are the collapse provided by strong measurements and the weak continuous measurement. The question arises: Can the strong measurement be considered as a dynamical process. Methods: To answer the question, a dynamic model is proposed to simulate numerically the measurement of the coordinates of quantum particles. By measuring the coordinate, we mean the capture of a particle in a narrow box and the excitation of the stationary wave function in it. We consider the amplitude of the probability of transition to a stationary state as a wave function, and the non-stationary Schrodinger equation is solved using this initial function. The transition to a stationary state was modeled as a gated process. Results and Conclusion: It is shown that when measuring coordinates, along with the instant collapse, there are cases when the measurement takes a short but finite time. In particular, the projective measurement of the coordinate of one of the entangled particles takes some time and is accompanied by the correlation in the probability density of the particles coordinates. Moreover, the correlation is lost over time.
It has been stated in the physics literature that the probability amplitude for a quantum system after a continuous position measurement can be expressed by the restricted or the weighted Feynman path integral containing information about the result and the resolution of its measurement. In the present paper it is proved that the restricted Feynman path integrals can be defined in a rigorous sense. More specifically, we consider a quantum system consisting of some particles and simultaneously perform a continuous measurement of the positions of all particles in the time interval [0,T][0,T][0,T]. Then, it is shown rigorously that the restricted Feynman path integral after the measurement can be defined in the L2L^2L2 space and satisfies the corresponding non-self-adjoint Schroeodinger equation. In addition, our results are generalized to a quantum spin system where a continuous position measurement for all spin components of all particles is simultaneously performed, though the results obtained are inadequate for general use.
We propose a general theoretical approach to quantum measurements based on the path (histories) summation technique. For a given dynamical variable A, the Schrödinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time or continuous type. Transformations between the histories obtained for non-commuting variables and construction of simultaneous histories for non-commuting observables are discussed. Important cases of a particle described by Feynman paths in the coordinate space and a qubit in a two dimensional Hilbert space are studied in some detail.
Consider measurements that provide information about the position of a nonrelativistic, one-dimensional, quantum-mechanical system. An outstanding question in quantum mechanics asks how to analyze measurements distributed in time---i.e., measurements that provide information about the position at more than one time. I develop a formulation in terms of a path integral and show that it applies to a large class of measurements distributed in time. For measurements in this class, the path-integral formulation provides the joint statistics of a sequence of measurements. Specialized to the case of instantaneous position measurements, the path-integral formulation breaks down into the conventional machinery of nonrelativistic quantum mechanics: a system quantum state evolving in time according to two rules---between measurements, unitary evolution, and at each measurement, ``collapse of the wave function'' (``reduction of the state vector''). For measurements distributed in time, the path-integral formulation has no similar decomposition; the notion of a system quantum state evolving in time has no place.