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Механизм возникновения стохастичности в квантовой механике

Alexey Yur'evich Samarin

Published Dec 15, 2012• 0 citations
OpenAlex

Abstract

The technique for the dynamic description of interaction between the quantum particle and the measuring instrument is offered. This description allows to determine, that the statistic dispersion of measuring instrument characteristics is the cause of the results randomness of the quantum particle space localization.Space-time consideration of the macroscopic meter evolution, initiated by the quantum particle, allows to represent the mechanism of the appearance of probabilistic measure, expressed by the wave function modulus square.

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The technique for the dynamic description of interaction between the quantum particle and the measuring instrument is offered. This description allows to determine, that the statistic dispersion of measuring instrument characteristics is the cause of the results randomness of the quantum particle space localization.Space-time consideration of the macroscopic meter evolution, initiated by the quantum particle, allows to represent the mechanism of the appearance of probabilistic measure, expressed by the wave function modulus square.
It is shown, that in addition to the evolution quantum object motion wave equation in the integral form can describe the wave function reduction as a physical process. Such description is represented for the space localization process, taking place when the space coordinate is measured, and it is shown, that collapse arises, as the result of the quantum particle and corresponding measuring instrument interaction. This physical phenomenon mathematical image looks like the instantaneous transformation of the virtual paths set to the subset, determined by the measuring process conditions, when the macroscopic changes appears in the measuring instrument.In conventional quantum mechanics such Hilbert space collapse itself corresponds to such reduction phenomenon.
The immutability of classical dynamical laws for the microscopic object in the space which coordinate axes are the transition matrix elements of the corresponding coordinates is proved. It is stated that measurement is a micro-object localization process in the classical space when it interacts with the device. The description of the wave function reduction is obtained using the path integrals. The mechanism of the probability arising on measurement is offered, where the hidden parameter that is the cause of the measurement randomness of microscopic characteristics relates to the interaction process of classical instrument with micro-object. Both types of quantum mechanics processes - evolution and reduction of the wave functions - are described in a unified approach
In the present paper we consider the problem of the description of an arbitrary generalized quantum measurement with outcomes in a measurable space. Analysing the unitary invariants of a separable statistical realization of a quantum instrument, we present the most general form of a possible integral representation of an instrument. We introduce the notion of a stochastic realization of an instrument and establish a one-to-one correspondence between the class of unitarily and phase equivalent separable statistical realizations and the equivalence class of stochastic realizations of an instrument. We further single out the invariant class of unitarily and phase equivalent separable statistical realizations for which the integral representation of an instrument is the same for all statistical realizations from this class and is wholly determined by the invariants of this class. We call the special form of this integral representation the quantum stochastic representation of an instrument. We show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach based on the notion of a family of quantum stochastic evolution operators satisfying the orthonormality relation. This approach gives not only the complete statistical description of any generalized direct quantum measurement but the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a generalized direct measurement in the sense of specifying the probabilistic transition law governing the change from the initial state to a final one under a single measurement. Under this approach a unitary evolution of an isolated quantum system is included as a special case. In the frame of the proposed approach, which we call quantum stochastic approach, all possible schemes of measurements upon a quantum system can be considered. In the case of repeated or continuous in time measurements the quantum stochastic approach allows to define, in the most general case, the notion of the family of posterior pure state trajectories (quantum trajectories in discrete or continuous time) in the Hilbert space of a quantum system and to give their probabilistic treatment.
Characteristic features of the stochastic models used in the classical mechanics, statistical physics and quantum theory are discussed. The arguments are given in favor of point of view that Nature, in itself, does not include probabilities. Researcher is forced to introduce the probabilistic concepts and corresponding models in conditions providing only partial predictability of a studied system behavior. This approach allows us to achieve a consistent interpretation of some important physical phenomena, in particular, the relationship between instability and irreversibility in time, the stochastic evolution of systems in the deterministic chaos theory, the H-theorem of Boltzmann, and paradoxes of quantum mechanics.
A quantum-mechanical measurement process is analyzed in terms of a model Hamiltonian describing the interaction between a quantum system (a ``particle'') and a macroscopic apparatus (a ``detector''), which is assumed to be made up of N two-level elementary constituents (``molecules''). The description of the molecule locations introduces an effective fluctuating coupling constant, and this provokes a loss of quantum-mechanical coherence in the limit of large N. It is argued that coherence is lost statistically, as a result of the interaction: The collapse of the wave function is indeed obtained when the same experiment is performed many times, as a result of the microscopic differences among macroscopically identical initial states of the detector. In this way, insight is obtained into the mechanism engendering the loss of coherence suffered by a quantum-mechanical system when interacting with a macroscopic apparatus, and the concept of ``wave-function collapse'' is replaced by that of a statistically defined dephasing process. No classical behavior of the detection system is postulated and the presence of no external observer is required.
MEASUREMENT INTERACTIONS In the present interpretation terms such as ‘measurement,’ ‘observation,’ ‘find,’ ‘classical,’ and ‘apparatus’ do not appear in the formulation of quantum mechanics. Yet it must of course be true that quantum mechanics can be used to predict features of the results of measurements – in particular their statistics. What is required, then, is an account in quantum mechanical terms of those physical interactions capable of being used to perform measurements of quantum dynamical variables, from which it follows that the predicted probabilities for specified behavior of certain quantum systems in these interactions coincide with what, on a more conventional interpretation, are simply postulated as the quantum mechanical probabilities for various measurement results on a system with a given quantum state. It is necessary to see how this consequence may be derived, not only in order to explicate the Born rules, but also to explain the role of quantum states within the present interpretation. The first step toward the required account is to analyze the nature of an interaction capable of being used to perform a measurement of the value of a quantum dynamical variable in a quantum system. Any such interaction will be called an M -type interaction, whether or not it is in fact used to perform a measurement. The basic characteristic of an M -type interaction is that it establishes a correlation between the initial dynamical state of one quantum system σ (the object system) and the final dynamical state of another quantum system α (the apparatus system).