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arXiv:quant-ph/9906078 · Physical Review A

Measurement as absorption of Feynman trajectories: Collapse of the wave function can be avoided

Avi Marchewka, Z. Schuss

Published Apr 10, 2002• 8 citations
arXivHTMLPDF

Abstract

We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such a detector vanishes on the detector. The trace the absorbed particles leave on the detector is identified as the absorption current density on the detector. This density is calculated from the solution of Schr\"odinger's equation with a reflecting boundary at the detector. This current density is not the usual Schr\"odinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of four postulates. In the resulting theory the quantum-mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two measurement experiments are proposed to measure time of arrival and the probability density function of a freely propagating two-dimensional Gaussian packet from the measurement of the absorption current on two planes.

DOI
10.1103/physreva.65.042112

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We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such a detector vanishes on the detector. The trace the absorbed particles leave on the detector is identified as the absorption current density on the detector. This density is calculated from the solution of Schr"odinger's equation with a reflecting boundary at the detector. This current density is not the usual Schr"odinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of four postulates. In the resulting theory the quantum-mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two measurement experiments are proposed to measure time of arrival and the probability density function of a freely propagating two-dimensional Gaussian packet from the measurement of the absorption current on two planes.
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.
We propose to modify the collapse axiom of quantum measurement theory by replacing the instantaneous with a continuous collapse of the wave function in finite time τττ. We apply it to coordinate measurement of a free quantum particle that is initially confined to a domain $D\subset\rR^d$ and is observed continuously by illuminating $\rR^d-D$. The continuous collapse axiom (CCA) defines the post-measurement wave function (PMWF)in DDD after a negative measurement as the solution of Schrödinger's equation at time τττ with instantaneously collapsed initial condition and homogeneous Dirichlet condition on the boundary of DDD. The CCA applies to all cases that exhibit the Zeno effect. It rids quantum mechanics of the unphysical artifacts caused by instantaneous collapse and introduces no new artifacts.
Abstract Consider a non-relativistic quantum particle with wave function inside a region Ω⊂R3\Omega \subset \mathbb {R}^3Ω⊂R3 Ω ⊂ R 3 , and suppose that detectors are placed along the boundary ∂Ω\partial \Omega ∂Ω ∂ Ω . The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the particle’s wave function ψ\psi ψ ψ expressed by a Schrödinger equation in Ω\Omega Ω Ω together with an “absorbing” boundary condition on ∂Ω\partial \Omega ∂Ω ∂ Ω first considered by Werner in 1987, viz., ∂ψ/∂n=iκψ\partial \psi /\partial n=i\kappa \psi ∂ψ/∂n=iκψ ∂ ψ / ∂ n = i κ ψ with κ>0\kappa >0κ>0 κ > 0 and ∂/∂n\partial /\partial n∂/∂n ∂ / ∂ n the normal derivative. We provide here a discussion of the rigorous mathematical foundation of this rule. First, for the viability of the rule it plays a crucial role that these two equations together uniquely define the time evolution of ψ\psi ψ ψ ; we point out here how, under some technical assumptions on the regularity (i.e., smoothness) of the detecting surface, the Lumer-Phillips theorem implies that the time evolution is well defined and given by a contraction semigroup. Second, we show that the collapse required for the N-particle version of the problem is well defined. We also prove that the joint distribution of the detection times and places, according to the absorbing boundary rule, is governed by a positive-operator-valued measure.
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.
We propose a formulation of an absorbing boundary for a quantum particle. The formulation is based on a Feynman-type integral over trajectories that are confined by an absorbing boundary. Trajectories that reach the absorbing wall are instantaneously terminated and their probability is discounted from the population of the surviving trajectories. This gives rise to a unidirectional absorption current at the boundary. We calculate the survival probability as a function of time. Several applications are given: the calculation of the absorption current in the slit experiment with a totally absorbing screen, the calculation of the survival probability of a particle between two totally absorbing walls, and the calculation of the survival probability and of the reflection coefficient of a Gaussian wave packet incident on a totally absorbing wall. The survival probability of a one-dimensional particle between two totally absorbing walls exhibits decay with beats.
Quantum measurement problem has existed many years and inspired a large of literature in both physics and philosophy, but there is still no conclusion and consensus on it. We show it can be subsumed into the quantum theory if we extend the Feynman path integral by considering the relativistic effect of Feynman paths. According to this extended theory, we deduce not only the Klein-Gordon equation, but also the wave-function-collapse equation. It is shown that the stochastic and instantaneous collapse of the quantum measurement is due to the “potential noise" of the apparatus or environment and “inner correlation" of wave function respectively. Therefore, the definite-status of the macroscopic matter is due to itself and this does not disobey the quantum mechanics. This work will give a new recognition for the measurement problem.