A Symmetrical Theory of Nonrelativistic Quantum Mechanics

Penn State preprint, 1985

1999

The objective of this paper is to axiomatically derive quantum mechanics from three basic axioms. In this paper, the Schr odinger equation for a characteristic function is ÿrst obtained and from it the Schr odinger equation for the probability amplitudes is also derived. The momentum and position operators acting upon the characteristic function are deÿned and it is then demonstrated that they do commute, while those acting upon the probability amplitudes obey the usual commutation relation. We also show that, for dispersion free ensembles, the Schr odinger equation for the characteristic function is equivalent to Newton's equations, thus providing us with a correspondence between both theories. As an application of the method, we show how it can be used to make quantization in generalized coordinates.

2021

Wave-Particle Duality, Spin, Mass and Energy1 Damon Wai Kwan So2 Oxford Centre for Mission Studies, Woodstock Road, Oxford OX2 6HR. Abstract A previous paper [1] proposed an alternative interpretation of quantum mechanics which is distinct from the Copenhagen Interpretation and the pilot wave theory. This alternative interpretation and its associated nondeterministic surfing velocity on the S (phase) surface are applied here to model point-like entities (particles and photons) with non-deterministic trajectories in the one-slit experiment and the two-slit experiment. The diffraction integral for these point-like entities, as a solution to the Helmholtz equation, is in the same form as the diffraction integral found in optical studies so that the mathematical results in optical studies regarding diffraction and interference patterns can be readily applied to the point-like entities modelled here. The nonrelativistic model for slowly moving particles and the relativistic model for fas...

Does the measurement of a quantum system necessarily break Lorentz invariance? We present a simple model of a detector that measures the spacetime localization of a relativistic particle in a Lorentz invariant manner. The detector does not select a preferred Lorentz frame as a Newton-Wigner measurement would do. The result indicates that there exists a Lorentz invariant notion of quantum measurement and sheds light on the issue of the localization of a relativistic particle. The framework considered is that of single-particle mechanics as opposed to field theory. The result may be taken as support for the interpretation postulate of the spacetime-states formulation of single-particle quantum theory. * We follow tradition and use such language, though it is not necessarily our intention to endorse a Copenhagen interpretation of quantum mechanics.

It has been debated whether quantum mechanics and special relativity are incompatible and whether there is a preferred Lorentz frame if they are incompatible. Bell's theorem is an important cornerstone, but it does not give us a definite positive answer due to the existence of supplementary assumptions or theoretical loopholes; there are unitary quantum theories which evade Bell's theorem and claim that they are compatible with special relativity. In this paper, I address the important issue of whether unitary quantum theories are compatible with special relativity. I propose a new Gedankenexperiment, a variant of the EPR-Bohm experiment with a superobserver who can undo a measurement. In this experiment, there is a stronger correlation (between the results of two spacelike separated measurements) than the correlation investigated in Bell's theorem. Based on an analysis of the correlations in different Lorentz frames, I prove that unitary single-world theories are incompatible with special relativity, and in order to avoid the incompatibility, there must exist a preferred Lorentz frame in these theories. Moreover, I argue that the incompatibility proof also applies to a proper version of the many-worlds interpretation of quantum mechanics. This closes the major theoretical loopholes of Bell's theorem, including relationalism, retrocausality, and superdeterminism. Finally, I argue that the stronger correlation found in the Gedanken-experiment cannot be explained by retrocausal processes or even the common causes in the past, but only be explained by nonlocal processes or actions at a distance. This provides a test of unitary quantum theories , as well as a further support for the new incompatibility proof beyond Bell's theorem.

Revista Mexicana de Física E, 2022

In this article we explain in a new light two fundamental concepts ofquantum optics, the quantum beam splitter and the quantum interferometer, in termsof two state quantum wave functions. This method is consistent with the concept ofentanglement, and hence the algebra needed to describe them is reduced to additionsand products of the components of the quantum states. Furthermore, under thepremises of this method it is possible to study quantum states of greater complexity,like those arising from the addition and products of single photon states.

2008

Despite the widespread assumptions on the compatibility between nonrelativistic quantum mechanics and special relativity, there still remains a considerable amount of unresolved problems to which few authors explicitly pay attention. Most of them involve the aim of coherently achieving a relativistic description of quantum collapses and quantum entanglements. These processes seem to challenge our present picture of the physical world in terms of spacetime structures.

Physics Essays , 2022

The unknown mechanism of wave-function collapse is called the measurement problem. The problem is best portrayed by a beam-split coincidence test, usually performed with visible light. The notion that energy conservation requires quantization is challenged by considering new beam-split tests and a threshold model (TM). An analysis of pulse heights in detectors for visible light concludes that their pulse height distribution is too broad to make the quantum/threshold dis- tinction. This is because TM recognizes a preloaded state, understood in the loading theories of Planck, Debye, and Millikan, but usually unrecognized. The narrow pulse height distribution of gamma-ray detectors overcomes this detector problem. In addition, a source of singly emitted radi- ation is required for these beam-split tests. To assure a singly emitted source, the well-known true- coincidence test from nuclear physics is far more reliable than any test with visible light. One of my many successful beam-split coincidence tests with gamma-rays is described revealing the fail- ure of quantum mechanics. After plotting the times between photoelectric effect pulses from the two detectors and comparing to accidental chance, I report a seemingly two-for-one effect that contradicts a photon kind of energy conservation. My similar tests performed with alpha-rays also contradict quantum mechanics. To explain how matter can load up, I hypothesize that our electron constants h, e, and m are maxima. Simple conserved ratios of these constants h/m, e/m, h/e, seen in equations involving electron beams, can explain how charge waves can spread, yet accumulate to measurable threshold values h, e, m, upon absorption to convey particle-like effects.

2012

The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schroendiger’s equation for the density matrix is fist obtained and from it Schroedinger’s equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli’s equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm’s hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free ensembles, Schroedinger’s equation for the density matrix is equivalent to Newton’s equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which a...

Mathematics, 2021

A new formulation of relativistic quantum mechanics is presented and applied to a free, massive, and spin-zero elementary particle in the Minkowski spacetime. The reformulation requires that time and space, as well as the timelike and spacelike intervals, are treated equally, which makes the new theory fully symmetric and consistent with the special theory of relativity. The theory correctly reproduces the classical action of a relativistic particle in the path integral formalism, and allows for the introduction of a new quantity called vector-mass, whose physical implications for nonlocality, the uncertainty principle, and quantum vacuum are described and discussed.