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arXiv:2111.06221 · Physics of Particles and Nuclei Letters

Restrictions Imposed by the Wave Function on the Results of Particle Momentum Measurements

Nikolay Leonidovich Chuprikov

Published Dec 1, 2022• 0 citations
arXivHTMLPDF

Abstract

Abstract It is demonstrated with a particle in the one-dimensional configuration space (OCS) that knowledge of the wave function supposes more than statistical restrictions on the results of measurements. In particular, apart from the probability (density) field in the OCS, the wave function also suggests the existence of two fields that predict two (equiprobable) particle momentum values for each OCS point, and the average of these two momenta at each point is related only to the wave function phase, while their difference (coinciding with the Bohm quantum-mechanical potential) is related only to the wave function amplitude. An analogue of the Heisenberg uncertainty relation is obtained for both fields.

DOI
10.1134/s1547477122060073

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Abstract It is demonstrated with a particle in the one-dimensional configuration space (OCS) that knowledge of the wave function supposes more than statistical restrictions on the results of measurements. In particular, apart from the probability (density) field in the OCS, the wave function also suggests the existence of two fields that predict two (equiprobable) particle momentum values for each OCS point, and the average of these two momenta at each point is related only to the wave function phase, while their difference (coinciding with the Bohm quantum-mechanical potential) is related only to the wave function amplitude. An analogue of the Heisenberg uncertainty relation is obtained for both fields.
Using the one-dimensional Schr\{o}dinger equation as an example, it is shown that for any self-adjoint operator it is possible to uniquely predict the value of the corresponding observable (including momentum, kinetic and total energy of the particle) at that spatial point where the particle will be accidentally detected at a given instant of time. It is shown that the Schr\{o}dinger formalism ensures the fulfillment of the relations of wave-particle duality, which connect the energy and momentum of a particle with the frequency and wave number of the wave function, for any spatial point and any instant of time.
Two distinct measures of information, connected respectively to the amplitude and phase of the wave function of a particle, are proposed. There are relations between the time derivatives of these two measures and their gradients on the configuration space, which are equivalent to the wave equation. The information related to the amplitude measures the strength of the potential coupling of the particle (which is itself aspatial) with each volume of its configuration space, i.e. its tendency to participate in an interaction localized in a region of ordinary physical space corresponding to that volume. The information connected to the phase is that required to obtain the time evolution of the particle as a persistent entity starting from a random succession of bits. It can be considered as the information provided by conservation principles. The meaning of the so-called "quantum potential" in this context is briefly discussed.
According to the postulates of quantum mechanics, the state of a system is associated with a wave function that contains any measurable information on the system at any time. In this chapter we become familiar with wave functions and how they represent the position of particles within the system. Within the realm of quantum mechanics, the position of particles is not deterministic. It is defined by a probability distribution. The wave function is a position-dependent complex-valued amplitude, whose absolute value squared is identified with the probability density for locating the particle in the position space. This identification of the wave function with a probability amplitude imposes some limitation. Particularly, for a closed system in which the particles are bound, the wave function must be proper (square integrable) and normalizable. These properties are discussed and demonstrated for different coordinate systems.
Here the probability density of relativistic particles coordinates, satisfying the formal conditions of the quantum mechanics and the special relativity, is determined (under textbooks view, such density does not exist). It is specified for the boson, neutral and charged, scalar and vector, including the photon, the electron. Uniqueness of this design has been proved. This is a relativistic invariant, describing the probability of the particle observation in points of space-time. The time is deprived of his role of a dynamic parameter of distribution and formally equalized with other coordinates. Also the meaning of the energy-momentum distribution is clarified. Conditions of such distributions observation are described. As applied to the quantum field these structures are transformed into new characteristics of particles distribution in space-time in addition to those for momentum states. Their operators for free fields are obtained. Under such model the Heisenberg's uncertainty relations are grounded for the relativistic particle; both particles and antiparticles turn out to be different states of a single particle; vacuum state of boson quantum field has zero energy.
The motion of a particle in quantum mechanics is described by a complex wave function that gives the probability amplitude for finding the particle at point at time. Corresponding to every physical quantity is an operator. A particle whose state is an eigenstate of the operator has a definite value of the physical quantity. This value is just the eigenvalue belonging to the eigenstate. The relations between the spatial extents of the wave packet, that is, the uncertainty in the position of the particle, and the uncertainty in the momentum of the particle, are known as the Heisenberg uncertainty relations. They form the cornerstone on which the logical consistency of quantum mechanics rests.
Quantum mechanics relates probability of an observable event to the absolute square of the corresponding probability amplitude. It may, therefore, seem that the information about the amplitudes' phases must be irretrievably lost in the experimental data. Yet, there are experiments which report measurements of wave functions, and closely related quantities such as Bohm's velocities and positions of bohmian particles. We invert the question, and ask under which conditions the values of quantum amplitudes can be recovered from observed probability distributions and averages.