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arXiv:0911.2980

From a 1D completed scattering and double slit diffraction to the quantum-classical problem: A new approach

Nikolay Leonidovich Chuprikov

Published Nov 16, 2009• 0 citations
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Abstract

We present a new approach to the quantum-classical problem, which treats it as the problem of modelling the quantum phenomenon described by a coherent superposition of microscopically distinct substates (CSMDS) as a compound one consisting of alternative subprocesses creating unremovable contexts for each other, or as that of reducing a non-Kolmogorovian quantum probability space to underlie a CSMDS to the sum of Kolmogorovian ones. We develop such models for a 1D completed scattering and double slit diffraction. The quantum-classical problem disappears when, in quantum theory with its integral superposition principle, CSMDSs obey the "either-or" rule to guide alternative random events. There is no observable which could be associated with the whole ensemble of statistical data described by a CSMDS, because such data are incompatible -- in the case of a CSMDS, any observable splits into noncommuting observables associated with the substates. To calculate the average value of any observable as well as to introduce characteristic times is meaningful only for the substates of a CSMDS. Ignoring this feature in the conventional description of CSMDSs just leads to paradoxical results (e.g., to the Hartman effect and passing a particle through two slits in the screen simultaneously).

DOI
10.48550/arxiv.0911.2980

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We present a new approach to the quantum-classical problem, which treats it as the problem of modelling the quantum phenomenon described by a coherent superposition of microscopically distinct substates (CSMDS) as a compound one consisting of alternative subprocesses creating unremovable contexts for each other, or as that of reducing a non-Kolmogorovian quantum probability space to underlie a CSMDS to the sum of Kolmogorovian ones. We develop such models for a 1D completed scattering and double slit diffraction. The quantum-classical problem disappears when, in quantum theory with its integral superposition principle, CSMDSs obey the "either-or" rule to guide alternative random events. There is no observable which could be associated with the whole ensemble of statistical data described by a CSMDS, because such data are incompatible -- in the case of a CSMDS, any observable splits into noncommuting observables associated with the substates. To calculate the average value of any observable as well as to introduce characteristic times is meaningful only for the substates of a CSMDS. Ignoring this feature in the conventional description of CSMDSs just leads to paradoxical results (e.g., to the Hartman effect and passing a particle through two slits in the screen simultaneously).
A new quantum-mechanical model of the two-slit experiment is presented, in which the wave properties of a particle are compatible with its corpuscular ones. The model is based on the idea which also underlies our previous model of a one-dimensional completed scattering. Its essence is that quantum processes, in which the systems state represents a coherent superposition of microscopically distinct substates (CSMDS), must be considered as complex random processes consisting of alternative subprocesses. Only with a new approach to CSMDSs quantum theory admits a consistent statistical interpretation and becomes free from the paradoxes associated at present with CSMDSs.
A new quantum-mechanical model of the two-slit experiment is presented, in which the wave properties of a particle are compatible with its corpuscular ones. The model is based on the idea which also underlies our previous model of a one-dimensional completed scattering. Its essence is that quantum processes, in which the systems state represents a coherent superposition of microscopically distinct substates (CSMDS), must be considered as complex random processes consisting of alternative subprocesses. Only with a new approach to CSMDSs quantum theory admits a consistent statistical interpretation and becomes free from the paradoxes associated at present with CSMDSs.
The origin of nonlocality in quantum mechanics (QM) is analyzed from the viewpoint of our new model of a one-dimensional (1D) completed scattering. Our study of quantum nonlocality complements those carried out by Volovich and Khrennikov. They pointed to an unphysical character of nonlocality in Bell's theorem whose context does not contain the very structure of the space-time. However, there is another reason leading to nonlocality in QM. The existing model of a 1D completed scattering evidences that QM, as it stands, even with a proper space-time context, contradicts special relativity. By our model this scattering process represents an entanglement of two coherently evolved alternative sub-processes, transmission and reflection; whose characteristics are measured well after the scattering event. Quantum nonlocality appears in this problem due to the inconsistency of the superposition principle with the corpuscular properties of a particle. It can take part only in one of the sub-processes. However the superposition principle allows introducing observables common for them. In the fresh wording, this principle must forbid introducing observables for entangled states.
Entanglement is usually associated with compound systems. We first show that a one-dimensional (1D) completed scattering of a particle on a static potential barrier represents an entanglement of two alternative one-particle sub-processes, transmission and reflection, macroscopically distinct at the final stage of scattering. The wave function for the whole ensemble of scattering particles can be uniquely presented as the sum of two isometrically evolved wave packets to describe the (to-be-)transmitted and (to-be-)reflected subensembles of particles at all stages of scattering. A noninvasive Larmor-clock timing procedure adapted to either subensemble shows that namely the dwell time gives the time spent, on the average, by a particle in the barrier region, and it denies the Hartman effect. As regards the group time, it cannot be measured and hence it cannot be accepted as a measure of the tunneling time. We argue that nonlocality of entangled states appears in quantum mechanics due to inconsistency of its superposition principle with the corpuscular properties of a particle. For example, this principle associates a 1D completed scattering with a single (one-way) process, while a particle, as an indivisible object, cannot take part in transmission and reflection, simultaneously.
Inter-relation between quantum and classical probability models is one of the most fundamental problems of quantum foundations. Nowadays this problem also plays an important role in quantum technologies, in quantum cryptography and the theory of quantum random generators. In this letter, we compare the viewpoint of Richard Feynman that the behavior of quantum particles cannot be described by classical probability theory with the viewpoint that quantum–classical inter-relation is more complicated (cf, in particular, with the tomographic model of quantum mechanics developed in detail by Vladimir Man'ko). As a basic example, we consider the two-slit experiment, which played a crucial role in quantum foundational debates at the beginning of quantum mechanics (QM). In particular, its analysis led Niels Bohr to the formulation of the principle of complementarity. First, we demonstrate that in complete accordance with Feynman's viewpoint, the probabilities for the two-slit experiment have the non-Kolmogorovian structure, since they violate one of basic laws of classical probability theory, the law of total probability (the heart of the Bayesian analysis). However, then we show that these probabilities can be embedded in a natural way into the classical (Kolmogorov, 1933) probability model. To do this, one has to take into account the randomness of selection of different experimental contexts, the joint consideration of which led Feynman to a conclusion about the non-classicality of quantum probability. We compare this embedding of non-Kolmogorovian quantum probabilities into the Kolmogorov model with well-known embeddings of non-Euclidean geometries into Euclidean space (e.g., the Poincaré disk model for the Lobachvesky plane).
It is shown that in the state space of a quantum particle involved in a one-dimensional completed scattering (OCS) there is a superselection rule (SSR) induced by the dichotomous physical context that determines the properties of a quantum ensemble of scattering particles in the disjoint spatial regions located on the different sides of the potential barrier at asymptotically large distances from it. At the initial stage of the OCS this context is associated, in the general case, with two sources of particles located on the different sides of the barrier; and at the final stage it is associated, in the general case, with two particle detectors located on different sides of the barrier. The role of a superselection operator is played by the Pauli matrix σ3σ_3σ3​ that divides the space of in- and out-asymptotes into two coherent sectors. In the course of the OCS, its unitary quantum dynamics crosses these sectors. In particular, in the scattering problem with one source and two detectors, a pure initial state of a particle is converted into a mixed final state. According to this rule, the OCS is a complex quantum process consisting of two coherent subprocesses -- transmission and reflection; the average value of any observable as well as characteristic times can be defined for the subprocesses only. It is shown that the quantum dynamics of both subprocesses at all stages of scattering is uniquely determined by the initial state of the whole ensemble of particles and by the final states of transmitted and reflected particles.