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Natural space of the micro-object

Alexey Yur'evich Samarin

Published Sep 1, 2011• 4 citations
OpenAlex

Abstract

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

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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
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 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.
We analyze in mathematical detail, within the framework of the QMUPL model of spontaneous wave function collapse, the von Neumann measurement scheme for the measurement of a 1/2 spin particle. We prove that, according to the equation of the model: i) throughout the whole measurement process, the pointer of the measuring device is always perfectly well localized in space; ii) the probabilities for the possible outcomes are distributed in agreement with the Born probability rule; iii) at the end of the measurement the state of the microscopic system has collapsed to the eigenstate corresponding to the measured eigenvalue. This analysis shows rigorously how dynamical reduction models provide a consistent solution to the measurement problem of quantum mechanics.
The change with time of the system consisting of the quantum object and the macroscopic measuring instrument is described on the base of the uniform dynamic law, which is suitable both evolution and reduction processes description. It is the integral wave equation with kernel in the form of a path integral. It is shown, that wave function collapse is the specific transformation which is fundamentally differ from Shrödinger's evolution. Specifically, a formal cause of the collapse is a local time derivative (infinite large) of the potential energy. Such transformation can not be described using mathematical apparatus of conventional quantum mechanics.
The concept of fundamental dynamic uncertainty (multivaluedness) developed in Parts I-III of this work and used to establish the consistent understanding of genuine chaos in Hamiltonian systems provides also causal description of the quantum measurement process. The modified Schroedinger formalism involving multivalued effective dynamical functions reveals the dynamic origin of quantum measurement indeterminacy as the intrinsic instability in the compound system of 'measured object' and (dissipative) 'instrument' with respect to splitting into spatially localised 'realisations'. As a result, the originally wide measured wave catastrophically (and really!) "shrinks" around a random accessible point thus losing all its 'nonlocal properties' with respect to other points/realisations. The dissipativity of one of the interacting objects (serving as 'instrument') is reduced to its (arbitrarily small) openness towards other systems (levels of complexity) and determines the difference between quantum measurement and quantum chaos, the latter corresponding to an effectively isolated system of interacting (micro-) objects. We do not use any assumptions on particular "classical", "macroscopic", "stochastic", etc. nature of the instrument or environment: physical reduction and indeterminacy dynamically appear already in interaction between two microscopic (quantum) deterministic systems, the object and the instrument, possessing just a few degrees of freedom a part of which, belonging to the instrument, should correspond to locally starting, arbitrarily weak excitation. This dynamically indeterminate wave reduction occurs in agreement with the postulates of the conventional quantum mechanics, including the rule of probabilities, which transforms them into consequences of the dynamic uncertainty.
In the classical manner of description, implicitly is accepted the independence of physical processes by the method of observation and the possibility of accurately and completely description of each process. In quantum mechanics such abstractions, are unacceptable. It affirms that the properties of micro-objects are appeared during their interactions with the measurements apparatus, whereas the theoretical predictions are formulated as the expected results of interaction with a certain probability. The set of such interactions leads to the statistical distribution of these probabilities. In the microworld can’t be done without intermediaries. As necessity intermediaries are the measurements apparatus. In this article we will present the principal idea, that the adequate manner of description is determined by chosen method of observation. So, instead of the known concept of relativity to the calculation system (the classical manner of description), is introduced the concept of relativity at the observation tools (quantum manner of description), a more general concept than the first.