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arXiv:quant-ph/9704006

Measurement processes in quantum physics: a new theory of measurements in terms of statistical ensembles

Werner A. Hofer

Published Apr 3, 1997• 0 citations
arXivHTMLPDF

Abstract

Considering the recently established arbitrariness the Schroedinger equation has to be interpreted as an equation of motion for a statistical ensemble of particles. The statistical qualities of individual particles derive from the unknown intrinsic energy components, they depend on the physical environment by way of external potentials. Due to these statistical qualities and wave function normalization, non-locality is inherent to the fundamental relations of Planck, de Broglie and Schroedinger. A local formulation of these statements is introduced and briefly assessed, the modified and local Schroedinger equation is non-linear. Quantum measurements are analyzed in detail, the exact interplay between causal and statistical reasons in a measurement process can be accounted for. Examples of individual measurement effects in quantum theory are given, the treatment of diffraction experiments, neutron interferences, quantum erasers, the quantum Zeno effect, and interaction-free measurements can be described consistent with the suggested framework. The paper additionally provides a strictly local and deterministic calculation of interactions in a magnetic field. The results suggest that quantum theory is a statistical formalism which derives its validity in measurements from considering every possible measurement of a given system. It can equally be established, that the framework of quantum physics is theoretically incomplete, because a justification of ensemble qualities is not provided.

DOI
10.48550/arxiv.quant-ph/9704006

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Considering the recently established arbitrariness the Schroedinger equation has to be interpreted as an equation of motion for a statistical ensemble of particles. The statistical qualities of individual particles derive from the unknown intrinsic energy components, they depend on the physical environment by way of external potentials. Due to these statistical qualities and wave function normalization, non-locality is inherent to the fundamental relations of Planck, de Broglie and Schroedinger. A local formulation of these statements is introduced and briefly assessed, the modified and local Schroedinger equation is non-linear. Quantum measurements are analyzed in detail, the exact interplay between causal and statistical reasons in a measurement process can be accounted for. Examples of individual measurement effects in quantum theory are given, the treatment of diffraction experiments, neutron interferences, quantum erasers, the quantum Zeno effect, and interaction-free measurements can be described consistent with the suggested framework. The paper additionally provides a strictly local and deterministic calculation of interactions in a magnetic field. The results suggest that quantum theory is a statistical formalism which derives its validity in measurements from considering every possible measurement of a given system. It can equally be established, that the framework of quantum physics is theoretically incomplete, because a justification of ensemble qualities is not provided.
A previous derivation of the single-particle Schrödinger equation from statistical assumptions is generalized to an arbitrary number NNN of particles moving in three-dimensional space. Spin and gauge fields are also taken into account. It is found that the same statistical assumptions that imply Schrödinger's equation determine also the form of the gauge coupling terms, and the form of the corresponding local (Lorentz) forces. An explanation for the role of the electrodynamic potentials, as statistical representatives of the Lorentz force, is given. For a single particle, spin one-half is introduced as the property of a statistical ensemble to respond to an external gauge field in two different ways. A generalized calculation, using the twofold number of variables, leads to Pauli's equation. The new spin term is again the statistical representative of the corresponding local force. For a NNN-particle system, spin is introduced as a consequence of gauge coupling.The classical limit ℏ→0\hbar \to 0ℏ→0 of Schrödinger's equation and closely related questions of interpretation of the quantum mechanical formalism are discussed.
The wave-structure of moving electrons is analyzed on a fundamental level by employing a modified de Broglie relation. Formalizing the wave-function ψψψ in real notation yields internal energy components due to mass oscillations. The wave-features can then be referred to physical waves of discrete frequency ννν and the classical dispersion relation $λν= u $, complying with the classical wave equation. Including external potentials yields the Schrödinger equation, which, in this context, is arbitrary due to the internal energy components. It can be established that the uncertainty relations are an expression of this, fundamental, arbitrariness. Electrons and photons can be described by an identical formalism, providing formulations equivalent to the Maxwell equations. The wave equations of intrinsic particle properties are Lorentz invariant considering total energy of particles, although transformations into a moving reference frame lead to an increase of intrinsic potentials. Interactions of photons and electrons are treated extensively, the results achieved are equivalent to the results in quantum theory. Electrostatic interactions provide, a posteriori, a justification for the initial assumption of electron-wave stability: the stability of electron waves can be referred to vanishing intrinsic fields of interaction. The concept finally allows the conclusion that a significant correlation for a pair of spin particles in EPR--like measurements is likely to violate the uncertainty relations.
Theory of measurement is presented which is based on the statistical interpretation of quantum mechanics. The reduction of wave packet is ascribed to undue comparison of the state of an ensemble before measurement and the state of an individual system after measurement. By studying the evolution of statistical operators for subsystems and composite systems comprising an object, a measuring apparatus and an observer, one can definitely locate where the measurement is done, thus negating the principle of psycho-physical parallelism due to von Neumann. Time needed for measurement is also discussed. The Einstein-Podolsky-Rosen state is described as an eigenstate of correlation. The argument lends support to the standpoint that the physical state of a system is to be represented by the statistical operator, not by the wave function, in the conceptual foundation of quantum mechanics.
We present here the exact, trajectory based wave-mechanical description of particle dynamics provided by the time-independent Schroedinger equation, of which the time-dependent equation allows a statistical treatment. Let us remind, for comparison, that Bohmian mechanics is largely recognized to provide a basically hydrodynamic formulation of the quantum theory, describing the evolution of physical systems in terms of statistical streamlines which indicate the regions of space where the moving particles preferentially gather: the very concept of true particle path is avoided, in Bohmian theory, by a weak, non-perturbing measurement. In the present paper the wave-dynamical Hamiltonian equations providing the exact particle trajectories are directly obtained from the time-independent Schroedinger equation, and numerical solutions are presented for a number of different external potentials acting on the moving particles. A Wave-Potential function (holding for any - even classical - wave process described by a Helmholtz-like equation) is shown to be the one and only cause of wave-like features such as diffraction and interference, and to act, for matter waves, without any wave-particle energy exchange.
The present contribution is based on the assumption that the probabilistic character of quantum mechanics does not originate from uncertainties caused by the process of measurement or observation, but rather reflects the presence of objectively existing vacuum fluctuations whose action on massive particles is calibrated by Planck's constant and effects an additional irregular motion. We describe a one-particle system by introducing two equally large sub-ensembles for each of which one obtains an equation of motion that has the form of a Navier-Stokes- or ''anti''-Navier-Stokes-type equation, respectively. By averaging over the total ensemble one obtains a new equation of motion which can be converted into the time-dependent Schroedinger equation. The concept allows the inclusion of electromagnetic fields and can be extended to interacting N-particle systems. We analyze the problem of how an experimental setup can consistently be decomposed into the quantum system under study and the residual quantum system ''apparatus''. The irregular extra motion of the particle under study allows a decomposition of the associated ensemble into two sub-ensembles the members of which perform, respectively, a right-handed or left-handed irregular circular motion about a given axis which becomes physically relevant in the presence of a magnetic field. We demonstrate that this orientation-decomposed ''Zitterbewegung'' behaves - in accordance with Schroedinger's original idea - as a spin-type angular momentum which appears in addition to a possible orbital angular moment of the particle. We derive the non-relativistic time-dependent Pauli equation and propose a theory of the Stern-Gerlach experiment. The Dirac equation can be derived similarly.