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arXiv:quant-ph/0403038

Uncertainty Relations (UR) Have Nothing to do with Quantum Mechanics (QM)

Vladimir K. Ignatovich

Published Mar 4, 2004• 0 citations
arXivHTMLPDF

Abstract

Uncertainty relations are shown to have nothing specific for quantum mechanics, being the general property valid for arbitrary function. A wave function of a particle having precisely defined position and momentum in QM simultaneously is demonstrated. Interference on two slits in a screen is shown to exist in classical mechanics. A nonlinear classical system of equations replacing QM Schrödinger equation is suggested. This approach is shown to have nothing in common with Bohmian mechanics.

DOI
10.48550/arxiv.quant-ph/0403038

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Uncertainty relations are shown to have nothing specific for quantum mechanics, being the general property valid for arbitrary function. A wave function of a particle having precisely defined position and momentum in QM simultaneously is demonstrated. Interference on two slits in a screen is shown to exist in classical mechanics. A nonlinear classical system of equations replacing QM Schrödinger equation is suggested. This approach is shown to have nothing in common with Bohmian mechanics.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.
We propose that the Schrodinger equation results from applying the classical wave equation to describe the physical system in which subatomic particles play random motion, thereby leading to quantum mechanics. The physical reality described by the wave function is subatomic particle moving randomly. Therefore, the characteristics of quantum mechanics have a dual nature, one of them is the deterministic nature carried on from classical physics, and the other is the probabilistic nature coined by particle's random motion. Based on this model, almost all of open questions in quantum mechanics can be explained consistently, which include the particle-wave duality, the principle of quantum superposition, the interference pattern of double-slit experiments, and the boundary between the classical world and the quantum world. The current quantum mechanics is a mixture of matrix mechanics and wave mechanics, which are sharply conflicting in principle. Matrix mechanics treats quantum particles as classical particles with fixed relation between the particle's position and its momentum. The matrix mechanics, in fact, belongs to the old quantum theory. Both Born's non-commutative relation and Heisenberg uncertainty relation originate from matrix mechanics. However, in wave mechanics, there is no any fixed relation between the particle's position and its momentum, and the particle's position and its momentum belong to immeasurable physical quantities. Therefore, there is no need for non-commutative relation and uncertainty relation in wave mechanics.
Striking parallelism between classical mechanics and quantum mechanics is evidenced for several bound system in a rather general form by introducing reasonably “quantum-mechanical momentum” from the Schrödinger equation. The following are proved: The “quantum-mechanical momentum” which is nearly equal to the corresponding classical momentum in “classical region” can be defined in the whole range of validity of the coordinate for the bound systems. In the same way as in classical mechanics, in quantum mechanics one can construct an associated wave accompanying a motion of a quantum particle and obeying the Schrödinger equation. The uniqueness condition on the associated wave throughout an orbit leads to the boundary condition on the wave function which is composed of the associated wave and to “quantum-mechanical quantization condition”, equivalently. In addition, all these quantum-mechanical quantities are approximately equal to the corresponding classical ones in the “classical region” of the coordinate. In particular it is demonstrated that even for a bound system of a singular (central) potential exists the ground state with the lowest energy eigenvalue. It is proved in one dimension that to any physical system obeying the Schrödinger equation always exists the corresponding classical system.
I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio ∣x∣/∣v∣|x|/|v|∣x∣/∣v∣ between the space coordinate xxx and the velocity vvv. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrödinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function ϕϕϕ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.
I consider in this book a formulation of Quantum Mechanics. Usually QM is formulated based on the notion of time and space, both of which are thought a priori given quantities or notions. However, when we try to define the notion of velocity or momentum, we encounter a difficulty as we will see in chapter 1. The problem is that if the notion of time is given a priori, the velocity is definitely determined when given a position, which contradicts the uncertainty principle of Heisenberg. We then set the basis of QM on the notion of position and momentum operators as in chapter 2. Time of a local system then is defined approximately as a ratio ∣x∣/∣v∣|x|/|v|∣x∣/∣v∣ between the space coordinate xxx and the velocity vvv. In this formulation of QM, we can keep the uncertainty principle, and time is a quantity that does not have precise values unlike the usually supposed notion of time has. The feature of local time is that it is a time proper to each local system, which is defined as a finite set of quantum mechanical particles. We now have an infinite number of local times that are unique and proper to each local system. Based on the notion of local time, the motion inside a local system is described by the usual Schrodinger equation. We investigate such motion in a given local system in part II. This is a usual quantum mechanics. After some excursion of the investigation of local motion, we consider in part III the relative relation or motion between plural local systems. In the final part IV, we will prove that there is at least one Universe wave function ϕ\phiϕ in which all local systems have local motions and thus local times. This concludes our formulation of Quantum Mechanics.