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The Wave-Mechanical Propagation Way
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Abstract
An exact description of point-particle dynamics is shown to be allowed by the Hamiltonian ray-tracing system associated with the time-independent Schrödinger equation, starting from any wave-front assignment. Matter waves are seen to be a general property of moving particles.
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Both classical and wave-mechanical monochromatic waves may be treated in terms of exact ray-trajectories (encoded in the structure itself of Helmholtz-like equations) whose mutual coupling is the one and only cause of any diffraction and interference process. In the case of Wave Mechanics, de Broglie’s merging of Maupertuis’s and Fermat’s principles provides, without resorting to the probability-based guidance-laws and flow-lines of the Bohmian theory, the simple law addressing particles along the Helmholtz rays of the relevant matter waves. A numerical treatment not substantially less manageable than its classical counterpart shows, in a number of examples, that each particle "dances a wave-mechanical dance” around its classical trajectory.
In this paper we attempt to analyze the physical and philosophical meaning of quantum contextuality. We will argue that there exists a general confusion within the foundational literature arising from the improper “scrambling” of two different meanings of quantum contextuality. While the first one, introduced by Bohr, is related to an epistemic interpretation of contextuality which stresses the incompatibility (or complementarity) of certain measurement situations described in classical terms; the second meaning of contextuality is related to a purely formal understanding of contextuality as exposed by the Kochen- Specker (KS) theorem which focuses instead on the constraints of the orthodox quantum formalism in order to interpret projection operators as preexistent or actual (definite valued) properties. We will show how these two notions have been scrambled together creating an “omelette of contextuality” which has been fully widespread through a popularized “epistemic explanation” of the KS theorem according to which: The measurement outcome of the observable A when measured together with B or together with C will necessarily differ in case [A, B] = [A, C] = 0, and [B, C] ≠ 0. We will show why this statement is not only improperly scrambling epistemic and formal perspectives, but is also physically and philosophically meaningless. Finally, we analyze the consequences of such widespread epistemic reading of KS theorem as related to statistical statements of measurement outcomes.
The behavior of classical monochromatic waves in sta-
tionary media is shown to be ruled by a novel, frequency-dependent
function which we call Wave Potential, and which we show to be enco-
ded in the structure of the Helmholtz equation. An exact, Hamiltonian,
ray-based kinematical treatment, reducing to the usual eikonal approxi-
mation in the absence of Wave Potential, shows that its presence in-
duces a mutual, perpendicular ray-coupling, which is the one and only
cause of wave-like phenomena such as diffraction and interference. The
"piloting" role of the Wave Potential, whose discovery does already
constitute a striking novelty in the case of classical waves, turns out
to play an even more important role in the quantum case. Recalling,
indeed, that the time-independent Schrödinger equation (associating
the motion of mono-energetic particles with stationary monochromatic
matter waves) is itself a Helmholtz-like equation, the exact, ray-based
treatment developed in the classical case is extended - without resorting
to statistical concepts - to the exact, trajectory-based Hamiltonian dy-
namics of mono-energetic point-like particles. Exact, classical-looking
particle trajectories may be defined, contrary to common belief, and
turn out to be perpendicularly coupled and piloted by an exact, energy-
dependent Wave Potential, similar in the form, but not in the physical
meaning, to the statistical, energy-independent "Quantum Potential"
of Bohm’s theory, which is affected, as is well known, by the prac-
tical necessity of representing particles by means of statistical wave
packets, moving along probability flux lines. This result, together with
the connection shown to exist between Wave Potential and Uncertainty
Principle, allows a novel, non-probabilistic interpretation of Wave Me-
chanics, in the original spirit both of de Broglie and Schrödinger.
FULL TEXT AVAILABLE AT:
http://aflb.ensmp.fr/AFLB-381/aflb381m753.pdf
The analysis of the Helmholtz equation is shown to lead to an exact Hamiltonian system of equations describing in terms of ray trajectories a very wide family of wave-like phenomena (including diffraction and interference) going much beyond the limits of the ordinary geometrical optics approximation, which is contained as a simple limiting case. Due to the fact that the time independent Schroedinger equation is itself a Helmoltz-like equation, the same mathematical solutions holding for a classical optical beam turn out to apply to a quantum particle beam, leading to a complete system of Hamiltonian equations which provide a set of particle trajectories and motion laws containing as a limiting case the ones encountered in classical Mechanics.
that we can never identify a reality independent of our experimental activity, then we must be prepared for that, too. 1 Chris Fuchs, previously the Lee DuBridge Prize Postdoctoral Fellow at Caltech, is now a DirectorFunded Fellow at Los Alamos National Laboratory. His daytime research focuses on quantum information theory and quantum computation. Asher Peres is the Gerard Swope Distinguished Professor of Physics at Technion|Israel Institute of Technology, Haifa, Israel. He is author of the book, Quantum Theory: Concepts and Methods (Kluwer, Dordrecht, 1995). 1 The thread common to all the nonstandard interpretations" is the desire to create a new theory with features that correspond to some reality independent of our potential experiments. But, trying to fulll a classical worldview by encumbering quantum mechanics with hidden variables, multiple worlds, consistency rules, or spontaneous collapse, with
Contrary to a wide-spread commonplace, an exact, ray-based treatment holding
for any kind of monochromatic wave-like features (such as diffraction and
interference) is provided by the structure itself of the Helmholtz equation.
This observation allows to dispel - in apparent violation of the Uncertainty
Principle - another commonplace, forbidding an exact, trajectory-based approach
to Wave Mechanics.
The time-independent Schroedinger and Klein-Gordon equations - as well as any other Helmholtz-like equation - turn out to be associated with exact sets of Hamiltonian ray-trajectories (coupled by a "Wave Potential" function, encoded in their structure itself) describing any kind of wave-like features, such as diffraction and interference. This property suggests to view Wave Mechanics as a direct, causal and realistic, extension of Classical Mechanics, based on exact trajectories and motion laws of point-like particles "piloted" by de Broglie's mono-energetic matter waves and avoiding the probabilistic content and the wave-packets both of the standard Copenhagen interpretation and of Bohm's theory.
FULL TEXT AVAILABLE AT
http://aflb.ensmp.fr/AFLB-401/aflb401m806.pdf
The usual interpretation of the quantum theory is self-consistent, but it involves an assumption that cannot be tested experimentally, viz., that the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes. The only way of investigating the truth of this assumption is by trying to find some other interpretation of the quantum theory in terms of at present "hidden" variables, which in principle determine the precise behavior of an individual system, but which are in practice averaged over in measurements of the types that can now be carried out. In this paper and in a subsequent paper, an interpretation of the quantum theory in terms of just such "hidden" variables is suggested. It is shown that as long as the mathematical theory retains its present general form, this suggested interpretation leads to precisely the same results for all physical processes as does the usual interpretation. Nevertheless, the suggested interpretation provides a broader conceptual framework than the usual interpretation, because it makes possible a precise and continuous description of all processes, even at the quantum level. This broader conceptual framework allows more general mathematical formulations of the theory than those allowed by the usual interpretation. Now, the usual mathematical formulation seems to lead to insoluble difficulties when it is extrapolated into the domain of distances of the order of 10-13 cm or less. It is therefore entirely possible that the interpretation suggested here may be needed for the resolution of these difficulties. In any case, the mere possibility of such an interpretation proves that it is not necessary for us to give up a precise, rational, and objective description of individual systems at a quantum level of accuracy.
The main effect of the Gaussian behavior of an electromagnetic beam consists in a waist formation in focal regions, where the ordinary geometric optics would collapse. The characteristic features of Gaussian beams, both when they interact with the components of transmission lines and when they propagate through inhomogeneous and anisotropic media (as happens, for instance, in the case of diagnostic or heating experiments in magnetoactive plasmas of fusion interest), are of crucial relevance for many technical and scientific purposes. The present paper is devoted to the analysis of the propagation of Gaussian beams, showing in particular that a properly formulated eikonal equation contains all the elements required by a correct ray tracing procedure, basically amounting to a first‐order description of the beam diffraction. Simple methods are proposed, apt to follow numerically the beam evolution for a quite general choice of refractive media and of wave launching conditions. Numerical results are presented for Gaussian beam propagation in vacuum, in isotropic and anisotropic media, and compared (evidencing significant deviations) to the corresponding ones in the optical case.
IN a series of experiments now in progress, we are directing a narrow beam of electrons normally against a target cut from a single crystal of nickel, and are measuring the intensity of scattering (number of electrons per unit solid angle with speeds near that of the bombarding electrons) in various directions in front of the target. The experimental arrangement is such that the intensity of scattering can be measured in any latitude from the equator (plane of the target) to within 20° of the pole (incident beam) and in any azimuth.
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Restatement of the interpretation of the wave mechanic by the double-solution theory, wich the author, beginning again his attempts of 1924-1927, tried to develop with the collaboration of several young scientists. Some results recently obtained are specially made obvious, in connection with new points of wiew, which commanded the author's attention since his statements of the years 1954-1955. L'auteur fait une mise au point de la réinterprétation de la Mécanique ondulatoire par la théorie de la double solution que, reprenant ses tentatives de 1924-1927, il a tenté, depuis 1951 de développer avec la collaboration d'un certain nombre de jeunes chercheurs. Il a particulièrement insisté sur des résultats récemment obtenus ainsi que sur des points de vue nouveaux qui se sont imposés à son esprit depuis ses exposés de 1954-1955.
This paper is a review of the theory-of laser beams and resonators. It is meant to be tutorial in nature and useful in scope. No attempt is made to be exhaustive in the treatment. Rather, emphasis is placed on formulations and derivations which lead to basic understanding and on results which bear practical significance.
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