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arXiv:quant-ph/0504105

Wave function collapse implies divergence of average displacement

Avi Marchewka, Z. Schuss

Published Apr 14, 2005• 0 citations
arXivHTMLPDF

Abstract

We show that propagating a truncated discontinuous wave function by Schrödinger's equation, as asserted by the collapse axiom, gives rise to non-existence of the average displacement of the particle on the line. It also implies that there is no Zeno effect. On the other hand, if the truncation is done so that the reduced wave function is continuous, the average coordinate is finite and there is a Zeno effect. Therefore the collapse axiom of measurement needs to be revised.

DOI
10.48550/arxiv.quant-ph/0504105

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We show that propagating a truncated discontinuous wave function by Schrödinger's equation, as asserted by the collapse axiom, gives rise to non-existence of the average displacement of the particle on the line. It also implies that there is no Zeno effect. On the other hand, if the truncation is done so that the reduced wave function is continuous, the average coordinate is finite and there is a Zeno effect. Therefore the collapse axiom of measurement needs to be revised.
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.
The quantum Zeno effect is not a general characteristic of continuous measurements. In a recently reported experiment [Itano et al., Phys. Rev. A 41, 2295 (1990)], the inhibition of atomic excitation and deexcitation is not due to any ``collapse of the wave function,'' but instead is caused by a very strong perturbation due to the optical pulses and the coupling to the radiation field. The experiment should not be cited as providing empirical evidence in favor of the notion of ``wave-function collapse.''
We present a theory of discontinuous motion of particles in continuous space-time. We show that the simplest nonrelativistic evolution equation of such motion is just the Schrodinger equation in quantum mechanics. This strongly implies what quantum mechanics describes is discontinuous motion of particles. Considering the fact that space-time may be essentially discrete when considering gravity, we further present a theory of discontinuous motion of particles in discrete space-time. We show that its evolution will naturally result in the dynamical collapse process of the wave function, and this collapse will bring about the appearance of continuous motion of objects in the macroscopic world.
The problem of measurement in quantum mechanics is that the quantum particle in the course of evolution, as described by the linear Schrodinger equation, exists in all of its possible states, but in measuring, the particle is always detected in only one of its states. This property is called the "collapse of the wave function" and was formulated by Von Neumann as one of the postulates of quantum mechanics. However, it remains unclear at what point in time and under what laws this transition occurs. This article demonstrates that the collapse of the wave function may be due to the creation or annihilation of particles (quasi-particles). The processes of the creation or annihilation of particles play a key role in the measurements and are described on the basis of quantum field theory. The system of equations of quantum field theory of particles and fields is non-linear; as a result, the principle of superposition does not hold for the theory. The collapse of the wave function is a consequence of this non-linearity and occurs at the moment of creation (annihilation) of a particle. This result demonstrates that the wave function collapse can occur in both microscopic and macroscopic systems. Understanding the mechanisms of the collapse of the wave function can lead to the creation of microscopic devices involved in the calculations based on quantum computing.
We present a theory of discontinuous motion of particles in continuous space-time. We show that the simplest nonrelativistic evolution equation of such motion is just the Schroedinger equation in quantum mechanics. This strongly implies what quantum mechanics describes is discontinuous motion of particles. Considering the fact that space-time may be essentially discrete when considering gravity, we further present a theory of discontinuous motion of particles in discrete space-time. We show that its evolution will naturally result in the dynamical collapse process of the wave function, and this collapse will bring about the appearance of continuous motion of objects in the macroscopic world.
We present a theory of discontinuous motion of particles in continuous space-time. We show that the simplest nonrelativistic evolution equation of such motion is just the Schroedinger equation in quantum mechanics. This strongly implies what quantum mechanics describes is discontinuous motion of particles. Considering the fact that space-time may be essentially discrete when considering gravity, we further present a theory of discontinuous motion of particles in discrete space-time. We show that its evolution may naturally result in the dynamical collapse process of the wave function, and this collapse will bring about the appearance of continuous motion of objects in the macroscopic world.