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Foundations of Physics

The estimation of the mean value of copies of uncertain positions and its relation to the basic structure of quantum mechanics

Zinoviy M. Landsman, Tomer Shushi

Published Jun 1, 2025• 0 citations
Publisher

Abstract

Abstract In this short paper, we propose a new framework for obtaining basic aspects of quantum mechanics that originate from estimating the mean value of the position of a statistical system based on the generalized Bayes estimators. We show that while the first-order estimation leads to a classical system, the second-order estimation produces the time-independent Schrödinger equation. The Born rule describes the probabilistic nature of quantum particles, and Max Born postulated it independently from the Schrödinger equation. We show that under the proposed model, both the Schrödinger equation and the Born rule are captured organically; particularly, we show that the Born rule leads to the Schrödinger equation. Finally, we show how the proposed model deals with the transition from quantum mechanics into classical mechanics when dealing with macroscopic objects without external assumptions.

DOI
10.1007/s10701-025-00863-0

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Abstract In this short paper, we propose a new framework for obtaining basic aspects of quantum mechanics that originate from estimating the mean value of the position of a statistical system based on the generalized Bayes estimators. We show that while the first-order estimation leads to a classical system, the second-order estimation produces the time-independent Schrödinger equation. The Born rule describes the probabilistic nature of quantum particles, and Max Born postulated it independently from the Schrödinger equation. We show that under the proposed model, both the Schrödinger equation and the Born rule are captured organically; particularly, we show that the Born rule leads to the Schrödinger equation. Finally, we show how the proposed model deals with the transition from quantum mechanics into classical mechanics when dealing with macroscopic objects without external assumptions.
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.
It is shown that Schrödinger's equation may be derived from three postulates. The first is a kind of statistical metamorphosis of classical mechanics, a set of two relations which are obtained from the canonical equations of particle mechanics by replacing all observables by statistical averages. The second is a local conservation law of probability with a probability current which takes the form of a gradient. The third is a principle of maximal disorder as realized by the requirement of minimal Fisher information. The rule for calculating expectation values is obtained from a fourth postulate, the requirement of energy conservation in the mean. The fact that all these basic relations of quantum theory may be derived from premises which are statistical in character is interpreted as a strong argument in favor of the statistical interpretation of quantum mechanics. The structures of quantum theory and classical statistical theories are compared, and some fundamental differences are identified.
A physical experiment comprises along the time trajectory a start, a time evolution (duration), and an end, which is the measurement. In non relativistic quantum mechanics the start of the experiment is defined by the wave function at time 0 taking into account the starting conditions, the evolution is described by the wave function following the Schrödinger equation and the measurement by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that makes quantum mechanics statistical with all its consequences. The nature of the Born rule is thereby unknown albeit necessary since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is necessary in order to eliminate the unknown initial and end time coordinate of the experiment in reference to the beginning of the universe.
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, probability density current, momentum operator, rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrodinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of the delta-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many particle systems and the postulates of quantum mechanics are also discussed.
Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, the Born rule, commutation and uncertainty relations, probability density current, momentum operator, and rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, the Schrödinger equation, and the Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of th e δ-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Manyparticle systems and the postulates of quantum mechanics are also discussed.
Abstract In terms of its time trajectory, a physical experiment is comprised of a preparation time, a start time, a time evolution (duration), and an end time (point at which the measurement actually occurs). In non-relativistic quantum mechanics the start of the experiment is described by the wave function at time t = 0 taking into account the starting conditions, the evolution is described by the wave function according to the Schrödinger equation and the measurement is described by the Born rule. While the Schrödinger equation is deterministic, it is the Born rule that renders quantum mechanics stochastic with all its consequences it entails. Still, the nature of the Born rule is unknown yet it is useful since it produces the correct ensemble averaged measures of the experiment. Here, it is demonstrated that the origin of the Born rule is the projection from the quantum frame (i.e. wave description) to the classical mechanics frame (i.e. particle description) described by a Ehrenfest theorem-oriented Fourier transformation. The statistical averaging over many measurements is therefore only necessary in order to account for the unknown preparation time and thus starting time of the experiment because only the experimental duration (time between start time and end time) can be known.