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Imagination and Real Quantization

Stanislav Ordin

Published Oct 24, 2024• 0 citations
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Abstract

This chapter shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in quantum theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein’s condition was violated: SOME equations of classical physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the harmonic oscillator, to obtain the Planck-Einstein quantization for da Broglie waves.

DOI
10.1201/9781003570615-18

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This chapter shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in quantum theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein’s condition was violated: SOME equations of classical physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the harmonic oscillator, to obtain the Planck-Einstein quantization for da Broglie waves.
The paper shows that the use of mathematical imaginaries, complex numbers, and even more so quaternions in the Quantum Theory to increase the dimension of solutions of one-dimensional equations is, in principle, incorrect. As a result, Einstein's condition was violated: SOME equations of Classical Physics can be rewritten in operator form. The use of correct multidimensional models makes it possible, using the example of the Harmonic Oscillator, to obtain the Planck-Einstein Quantization for da Broglie waves.
Ideal resonance does not fade, and thus, it is, as it were, out of time. Therefore, the corresponding Planck QUANTUM, which determines the STATIONARY allowed states of energy, is simply a reflection of the ideal resonance. So the corresponding Planck QUANTUM, which determines the STATIONARY Allowed States of Energy, is simply a reflection of the Ideal Resonance. And modern “Quantum Mechanics,” which moved away from both Planck’s Quantization and the correct use of the Mechanical Model, simply used a simplified analysis of the Mechanical Harmonic Oscillator. The fundamental difference of QUANTUMS is only that for frequencies not equal to zero and their minimum energy of the resonance-quantum has a finite, non-zero initial value, which also determines the minimum distance between the allowed energy levels. At the same time, the internal structure of the QUANTUM can not only be electromagnetic, like Planck’s, and not only be acoustic, like Einstein’s, but also be determined by the so-called de Broglie matter waves. However, in order to remove any mysticism from QUANTUMS, one must UNDERSTAND that the DUALISM introduced by them is determined by Pontryagin’s functional connection of the ELEMENTARY PARTICLE and the packet of waves that form it.
In the past few chapters, the Schrödinger equation has been developed and applied to the solution of a number of quantum mechanical problems, mainly to develop experience with the results of simple and common systems. The results from these examples can be summarized relatively simply. In general, quantization enters the problem through the non-commuting nature of conjugate operators such as position and momentum as well as from classical boundary quantization. This has led to the Schrödinger equation itself being the primary equation of motion for the wave function solution to the problem. In essence, duality is incredibly important and is the basic method by which the electron is treated as a wave, rather than as a particle, and the wave equation of interest is the Schrödinger equation. When boundary conditions are applied, either through potential barriers or through the form of the potential (as in the harmonic oscillator), the time-independent Schrödinger equation yields solutions that are often special functions. Examples of this are the sines and cosines in the rectangular-barrier case, the Airy functions in the triangular-well case, and the Hermite polynomials in the harmonic oscillator case.
This chapter presents the origins of quantum mechanics. The story of how people hit on the highly non-intuitive world picture of quantum mechanics, in which the physical state of a system is represented by an element in an abstract linear space and its observable properties by operators in the space, is fascinating and exceedingly complicated. The much greater change from the classical world picture of Newtonian mechanics and general relativity to the quantum world picture came in many steps taken by many people, often against the better judgment of participants. There are three major elements in the story. The first is the experimental evidence that the energy of an isolated system can only assume special discrete or quantized values. The second is the idea that the energy is proportional to the frequency of a wave function associated with the system. The third is the connection between the de Broglie relation and energy quantization through the mathematical result that a wave equation with fixed boundary conditions allows only discrete quantized values of the frequency of oscillation of the wave function (as in the fundamental and harmonics of the vibration of a violin string).
The harmonic oscillator is considered within generalized frameworks using complex and quaternionic numbers. The classical oscillator is considered in terms of a complex position function, and quantum oscillators are examined in terms of complex wave functions, and in terms of quaternionic wave functions as well. Both of the quantum solutions are obtained within the real Hilbert space formalism. The results reveal the complex and quaternionic descriptions as suitable frameworks for non-stationary processes, including damped oscillations, forced oscillations, and additionally self-interacting processes that cannot be appropriately described otherwise.
Abstract Although the classical examples of harmonic oscillators such as springs or the simple pendulum are of limited use in the submicroscopic world of quantum mechanics, there are plenty of other applications that make the oscillator worth studying. Atoms in a crystal, for instance, oscillate about their mean positions and it is their vibrations that carry sound and heat through the material. Indeed, we know from classical theory that by using normal coordinates we may regard any system performing small oscillations about a point of stable equilibrium as a collection of harmonic oscillators. This extends even to the normal modes of vibration of a string or of the electromagnetic field. The harmonic oscillator thus provides the key to understanding how wave phenomena such as light can be quantized, and lies at the heart of Einstein’s idea of the photon and of Planck’s quantum.