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Quantum Mechanical and Classic Measurement Result Quantities are Equal (Even though their Numerical Values are Not)

Ken Krechmer

Published Jun 8, 2023• 0 citations
Publisher

Abstract

A classic (i.e., physical) measurement result is a quantity (e.g., 6.9 seconds, 13 metres, 0.17 grams, etc.). A quantity is the product of two terms: a numerical value and a unit. The unit when calibrated to a standard has a +/- precision. In quantum mechanics (QM) a wave function produces a distribution of eigenvalues of unity eigenvectors. Each product of one eigenvalue of eigenvectors also represents one measurement result quantity. In a QM measurement process this distribution of quantities appears to collapse into one classic measurement result quantity (one numerical value and a unit). In classic measurement result quantities, (without noise or distortion) the numerical value of an observable appears fixed over repetitive measurements and the units have a precision relative to a standard. However in a QM measurement result quantity, the eigenvalues have Heisenberg's uncertainty and the eigenvectors are unity. This paper formally develops and verifies that both the QM and classic measurement result quantities are equal products and the classic unit precision is equal to the QM eigenvalue uncertainty.

DOI
10.32388/aoxtc5

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A classic (i.e., physical) measurement result is a quantity (e.g., 6.9 seconds, 13 metres, 0.17 grams, etc.). A quantity is the product of two terms: a numerical value and a unit. The unit when calibrated to a standard has a +/- precision. In quantum mechanics (QM) a wave function produces a distribution of eigenvalues of unity eigenvectors. Each product of one eigenvalue of eigenvectors also represents one measurement result quantity. In a QM measurement process this distribution of quantities appears to collapse into one classic measurement result quantity (one numerical value and a unit). In classic measurement result quantities, (without noise or distortion) the numerical value of an observable appears fixed over repetitive measurements and the units have a precision relative to a standard. However in a QM measurement result quantity, the eigenvalues have Heisenberg's uncertainty and the eigenvectors are unity. This paper formally develops and verifies that both the QM and classic measurement result quantities are equal products and the classic unit precision is equal to the QM eigenvalue uncertainty.
This paper is a response to the EPR paper titled: "Can quantum-mechanical description of physical reality be considered complete?", published in Physical Review in 1935. A quantum-mechanical (QM) measurement function describes a distribution of local results and each empirical measurement process produces one result as exact as allowed by a measuring instrument calibrated to a non-local unit standard. Repeating these empirical measurements produces a bell shaped distribution of measurement results. Each of these distributions can be compared to the other. To precisely compare a QM measurement function describing a distribution of eigenvectors to a distribution of repetitive empirical measurement results, it is necessary to determine, by calibration, the precision of the eigenvectors to the same standard as the empirical results, because each eigenvector evidences uncertainty relative to a standard. When the calibration process is recognized as formal as well as empirical, QM measurement function results and metrology measurement process results are unified.
Abstract This paper is a response to the EPR paper titled: "Can quantum-mechanical description of physical reality be considered complete?", published in Physical Review in 1935. A quantum-mechanical (QM) measure function describes a distribution of local results, while each experimental measurement process produces one result as exact as allowed by a measuring apparatus calibrated to a non-local unit standard. Repeating experimental measurements produces a Gaussian distribution of measurement results. These QM and experimental measurement result distributions can be compared. To compare a QM measure function describing a distribution of eigenvectors to a distribution of repetitive experimental measurement results, it is necessary to determine, by calibration, the precision of the eigenvectors to the same standard as the experimental results, because each eigenvector evidences uncertainty relative to a standard. When the calibration process is recognized as formal as well as empirical, QM measure function and experimental measurement process results are unified.
Measurement Mechanics proposes a major modification to the current understanding of measurement. This modification adds a unit standard as axiomatic to measurement theory. Currently, measurement theory (representational) defines the result of a measurement comparison to be a ratio of quantities (numerical value and unit) where the unit terms are equal and cancel. When the units cancel such a ratio is local to one measurement system, i.e., probabilistic. Thus all QM measurement results, based upon current measurement theory are local, as J. S. Bell confirmed. However, the units of each metrology measurement result are correlated by calibration to an independent unit standard. Then measurement result comparisons are a ratio of quantities each correlated to a common unit standard, which allows independent measurement result comparisons that are non-local and deterministic. When unit standards are included in measurement theory all the apparent discrepancies between QM and classic measurements are resolved.
In classic measurement theory each physical measurement result has a precision which becomes infinitesimal as noise and distortion approach zero, i. e., exact repetitive measurement results (± a Planck) are theoretically possible. This classic measurement theory is not well correlated with experimental measurement results. When noise and distortion are minimized, repetitive experimental measurement results always display a Gaussian distribution. This paper first addresses experimental measurements by developing a formal measurement function and related definitions. The formal measurement function identifies that an exact repetitive classic measurement result is not possible in theory as well as in experiments. This new measurement function is then shown to explain the existing quantum measurement problems.
Foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as ccc-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity which persists throughout an experiment and therefore it does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and it does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions like a Bose-Einstein condensate system for which a single measurement yields a real ensemble. We discuss the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from quantum field theory.
In classic measurement theory each physical measurement result has a precision which becomes infinitesimal as noise and distortion approach zero, i. e., exact repetitive measurement results (± a Planck) are theoretically possible. This classic measurement theory is not well correlated with experimental measurement results. When noise and distortion are minimized, repetitive experimental measurement results display a Gaussian distribution. This paper addresses experimental measurements by first developing a formal measurement function and related definitions, which identify that a classic measurement result is not possible in theory as well as in experiments. This new measurement function is then correlated with quantum measurement theory. This correlation explains existing quantum measurement perplexities.