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Theoretical Proof of De Broglie’s Wave Function

Sheng Qin

Published Jul 18, 2024• 1 citation
Publisher

Abstract

This paper, based on the "Unified Theory of Forces," rigorously proves the formula of de Broglie's wave function. This helps us better understand the following important issues: 1) Theoretically prove and understand the essence of de Broglie's wave function; 2) Prove that when the particle's momentum P is determined, the greater the mass of the particle, the more concentrated its wave function distribution, and the weaker the interference effect; 3) Correct the spatial distribution of the wave function of free particles, proving that the wave function of free particles is strictly constrained within a finite spatial range, which is consistent with the facts we observe.

DOI
10.20944/preprints202407.1504.v1

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This paper, based on the "Unified Theory of Forces," rigorously proves the formula of de Broglie's wave function. This helps us better understand the following important issues: 1) Theoretically prove and understand the essence of de Broglie's wave function; 2) Prove that when the particle's momentum P is determined, the greater the mass of the particle, the more concentrated its wave function distribution, and the weaker the interference effect; 3) Correct the spatial distribution of the wave function of free particles, proving that the wave function of free particles is strictly constrained within a finite spatial range, which is consistent with the facts we observe.
This paper, based on the "Unified Theory of Forces," rigorously proves the formula of de Broglie's wave function. This helps us better understand the following important issues: 1) Theoretically prove and understand the essence of de Broglie's wave function; 2) Prove that when the particle's momentum P is determined, the greater the mass of the particle, the more concentrated its wave function distribution, and the weaker the interference effect; 3) Correct the spatial distribution of the wave function of free particles, proving that the wave function of free particles is strictly constrained within a finite spatial range, which is consistent with the facts we observe.
The de Broglie wave phenomenon was studied in the framework of Lagrangean formulation of Relativistic Mechanics. The conclusion was made that the de Broglie wave of a particle is a relativistic phenomenon related to the field-dependent proper mass. The wave develops in a process of proper-to-kinetic mass transformation when the particle is given a momentum. The time-part of the 4-wave vector is related to the proper mass oscillation while the spacial part is due to the momentum; both parts can be treated in terms of virtual (longitudinal and scalar) photons characterizing excitation states of a mediating scalar field. It is shown that in Relativistic Mechanics with the variable proper mass both electromagnetic and gravitational force can be treated on the same footing: both are due to the same source that is, the proper mass. Consequently, the scalar mediating field provides a mechanism of unification of gravitational and electromagnetic forces. The concept of the unified divergence-free field is suggested, in which the real (transverse) photon represents a pure space-like particle while the neutrino is a pure time-like particle. In this connection, the parity problem is discussed.
The irrefutable proof of the incorrectness of the de Broglie hypothesis (postulate) and the Schrödinger equation (postulate) is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The unity of formal logic and rational dialectics is the only correct criterion of truth. The proof leads to the following irrefutable conclusion: the de Broglie hypothesis (idea, postulate) and the Schrödinger equation (idea, postulate) are gross errors in mathematics, physics, formal logic and dialectics. This conclusion is based on the following statements: (1) from the point of view of Euler’s formula and the Maclaurin series, the definition of the wave function has neither physical nor mathematical meaning; (2) the substitution of the quantities of energy and momentum, which characterize a quantum (microscopic) particle, into the relationship that describes a macroscopic radiation wave is a gross formal-logical error, because a quantum particle and macroscopic radiation are not identical material objects; (3) in the dialectical and formal-logical points of view, a free classical particle is not identical to a free quantum particle: these particles are non-identical material objects. Therefore, the substitution of the quantities of energy and momentum, characterizing a quantum (microscopic) particle, into the classical relationship that describes the energy and momentum of a classical particle is a gross formal-logical error; (4) in the point of view of formal logic, the concepts of "corpuscular aspect" and "wave aspect" are contradictory (mutually exclusive) concepts. (The concepts of "particle" and "wave" are defined by different essential features of material objects). Therefore, the de Broglie wave function and the Schrödinger wave equation represent a violation of the formal-logical law of lack (absence) of contradiction. This means that the concept of corpuscular-wave dualism is erroneous; (5) The de Broglie wave function and the Schrödinger equation represent a gross dialectical error, because the dimensionless wave function contradicts to the dialectical concept of the measure of a material object (i.e. the dimensionless wave function does not have the qualitative determinacy of a material object, does not characterize the properties of a material object). Therefore, the de Broglie wave function and the Schrödinger equation are meaningless relationships.
The uncertainty principle is the most important for the foundations of quantum mechanics but it still remains failed to reach a consensus of its interpretation, which gives rise to debates upon its physical nature. In this work, we address the problem of its foundation from a different aspect to present an alternative formulation for proving the uncertainty relations in a general way in terms of commutation relations and action function. The relationship between the de Broglie relation and the uncertainty principle is studied from a new angle. As a result, it is demonstrated that the de Broglie relation is the foundation of the uncertainty principle. Starting with the origin of the problem, we show that the de Broglie relation provides the form of the wave function and the determined form of the wave function in turn leads to the conception of operators for quantum mechanics, and thus it is possible to provide with the help of the operators and wave function a generalized proof of the uncertainty principle as the law governing ensemble of states. As a decisive solution to the problem, the interpretation of the uncertainty principle in terms of the action function is offered that gives one self-consistent explanation in agreement with the known physical phenomena. Eventually, we show the necessity and possibility of reassessing and improving the foundation and interpretation of the uncertainty principle as a leading principle of quantum mechanics.
Abstract We present a unified field theory of wave and particle in quantum mechanics. This emerges from an investigation of three weaknesses in the de Broglie–Bohm theory: its reliance on the quantum probability formula to justify the particle-guidance equation; its insouciance regarding the absence of reciprocal action of the particle on the guiding wavefunction; and its lack of a unified model to represent its inseparable components. Following the author’s previous work, these problems are examined within an analytical framework by requiring that the wave–particle composite exhibits no observable differences with a quantum system. This scheme is implemented by appealing to symmetries (global gauge and spacetime translations) and imposing equality of the corresponding conserved Noether densities (matter, energy, and momentum) with their Schrödinger counterparts. In conjunction with the condition of time-reversal covariance, this implies the de Broglie–Bohm law for the particle where the quantum potential mediates the wave–particle interaction (we also show how the time-reversal assumption may be replaced by a statistical condition). The method clarifies the nature of the composite’s mass, and its energy and momentum conservation laws. Our principal result is the unification of the Schrödinger equation and the de Broglie–Bohm law in a single inhomogeneous equation whose solution amalgamates the wavefunction and a singular soliton model of the particle in a unified spacetime field. The wavefunction suffers no reaction from the particle since it is the homogeneous part of the unified field to whose source the particle contributes via the quantum potential. The theory is extended to many-body systems. We review de Broglie’s objections to the pilot-wave theory and suggest that our field-theoretic description provides a realization of his hitherto unfulfilled ‘double solution’ programme. A revised set of postulates for the de Broglie–Bohm theory is proposed in which the unified field is taken as the basic descriptive element of a physical system.
Louis de Broglie has the essential result of interpreting the physical optics based on diffraction phenomena without making any reference to the idea of harmonic oscillator and its classical dynamics in order to calculate the intensity of light. This chapter presents an equation, which shows that the density of the light particles classically conceived as material points or localized quanta, is proportional with the intensity of the classical theory of light. The effectively inadequate point of the de Broglie’s argument is in fact only the mathematical expression of his way to conclusions. de Broglie added an important attribute to the very classical notion of a light ray. However, his theory means in fact much more, far and beyond the physical theory of light. The de Broglie’s physical ray allows us to construct an interpretation based on the idea of adiabaticity.