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arXiv:2406.16686

Quantum and classical symmetries

A. Anokhina, M. Satleikin, A. Sedova, A. Shukakidze

Published Jun 24, 2024• 0 citations
arXivHTMLPDF

Abstract

We suggest a somewhat non-standard view on a set of curious, paradoxical from the standpoint of simple classical physics and everyday experience phenomena. There are the quantisation (discrete set of values) of the observables (e.g., energy, momentum, angular momentum); forbidden simultaneous measurements of the observables in the most cases (e.g., of a coordinate and momentum, of angular momentum projections on difference axis); counter-intuitive relations on the simultaneously measurable quantities (e.g., the famous expression for the square momentum $l(l+1)$ with the maximal projection $l$). These and other paradoxes are traditionally related to "purely quantum" phenomenon, i.e., having no analogue in the "classical world" ones. However, there are deep analogies between classical and "quantum" worlds, as soon as the quantum technique is applied to the classical phenomenon. We follow these analogies with the examples of relatively simple and well known models of classical physics, such as a simplified model of light transition through the media, a system of electric charges close to each other and far from the observer; the specific of motion in the Coulomb/Newtonian field. This text can be considered as a mini-course addressed to higher school and undergraduate students who are interested in basics of quantum mechanics, but are not yet ready for systematic study of standard courses. The text may be also useful to those who supervise such students.

DOI
10.48550/arxiv.2406.16686

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Namely, the product of two such observables is not a unique new observable but may depend on their order: the products AB and BA may differ.The values that a physical quantity can take constitute a set of ordinary numbers, the eigenvalues a( n) of the observable A. This set may be discrete as a consequence of the algebra. The discreteness of the eigenvalues of the energy of an atom explains the spectrum of its radiation. Any component of the intrinsic angular momentum of an electron can take only two opposite values, baffling our geometric intuition.If two observables commute (i.e., AB = BA), they are compatible: both may take well-defined values chosen among the eigenvalues. If they do not commute (i.e., AB ≠ BA), which occurs for the position and velocity of a particle, for the electric and magnetic field, or for the components of an angular momentum, they are incompatible: their values cannot be measured nor even specified simultaneously. Each of them presents statistical fluctuations which must be larger than some lower bound (Heisenberg’s inequality). The state of a system, prepared under some definite conditions, should be identified as the set of expectation values of all observables.Thus, quantum physics, our most fundamental description of nature, is a theory involving irreducible probabilities and unavoidable uncertainties. Its predictions do not refer to a single system, but to a statistical ensemble, and the observer cannot be completely disregarded. Moreover it may involve aspects which are apparently contradictory (Bohr’s complementarity). An electron, an atom or a molecule may behave in limiting cases either as a particle or as a wave; under intermediate conditions, it behaves as both, but each of the descriptions is blurred with probabilities. As a wave, it can interfere with itself. Likewise, an electromagnetic field has the double nature of a set of photons and of a wave. 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Consider then a statistical ensemble of systems, all prepared identically in such a way that A1 B1 = A2 B2 = A3 B3 = +1. If we measure A1 and B1, we shall find either a1 = b1 = +1 or a1 = b1 = –1, that is always a1 = b1; likewise for a2 = b2 = ±1 and a3 = b1 b2 = ±1. We are tempted to infer that a simultaneous determination of the values a1, a2 and a3 of A1, A2 and A3 (which are compatible) should verify a3 = a1 a2, that is, a1 a2 a3 = 1. However both theory and experiment provide in all cases a1 a2 a3 = –1. In fact the four statements a1 = b1, a2 = b2, a3 = b1 b2 and a1 a2 a3 = –1 which seem contradictory are separately “true”, in the sense that each one can be verified experimentally; but such experiments cannot be performed on a single system. We are not allowed to make a logical reasoning based on a simultaneous consideration of the values of B1 and A2 because these observables do not commute. No calculation, no experiment can simultaneously provide values for the numbers b1 and a2. Hence, we cannot regard as true together the statements a1 = b1 and a2 = b2, although they can be checked by separate experiments. Each one is true, but their union is meaningless.Thus, if theory does not allow us to imagine thought experiments in which several statements on a system might simultaneously be checked, we cannot draw conclusions by putting such statements together. We are led to regard the “truth” of a quantum statement as contextual. The values of physical quantities are not attached only to the system in itself, but also refer to the context in which their relations may be verified. Quantum mechanics does not deal with objects per se, but with what we know and what we can infer about them. It does not provide a direct image of the reality of the object since it involves its observation. 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Elementary particles are found in two different situations: (i) bound to metastable states of matter, for which angular momentum is quantized, and (ii) free, for which, due to their high energy-momentum and leaving aside inner a.m. or spin, the hhh-quantization step is completely harmless. Perhaps Quantum Mechanics can be seen just as the simplest mathematical formalism where angular momentum (the magnitude of each of its three orthogonal projections) is by construction quantized: all possible values are taken from a discrete set. Indeed: (i) This idea finds support in very reasonable, completely classical physical arguments, if we place ourselves in the framework of Stochastic Electrodynamics (SED): there, all sustained periodic movement of a charge must satisfy a power balance that restricts the value of the average angular momentum, on each of its projections. (ii) It gives a natural explanation of the concept of "photon", as a constraint on the observable spectrum of energy-momentum exchanges between metastable physical states, in particular also for its discreteness. QM would be, in this picture, a semi-static theory, transparent to all the (micro)-dynamics taking place between apparently "discrete" events (transitions in the state of the system). For instance, (the magnitude of the projections of) quantum angular momentum would only reflect average values over a (classical) cyclic trajectory, a fact that we regard as almost obvious given the particularity of the corresponding addition rules in QM.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
This book presents the basic concepts and methods of quantum mechanics for upper level undergraduate students, allowing them to master its application to real physical situations. A postulate-based treatment is adopted together with a gradual development of the quantum formalism of wave functions, operators, measurement and temporal evolution. Standard topics of one-dimensional and atomic motion, angular momentum and approximation methods are presented in addition to detailed discussions of many-particle systems, atomic and nuclear radiation. Appropriate mathematical tools and techniques are provided wherever necessary. The core text is supplemented by 77 worked examples, some of which address more complex issues and aspects of present-day research. The aim is to make this textbook a realistic introduction to more advanced and specialized texts. The material provides full coverage of the subject matter, 94 problems with solutions and a further 93 with answers only
This paper concerns the absolute versus relative motion debate. The Barbour and Bertotti 1982 work may be viewed as an indirectly set up relational formulation of a portion of Newtonian mechanics. I consider further direct formulations of this and argue that the portion in question -- universes with zero total angular momentum, that are conservative and with kinetic terms that are (homogeneous) quadratic in their velocities -- is capable of accommodating a wide range of classical physics phenomena. Furthermore, as I develop in Paper II, this relational particle model is a useful toy model for canonical general relativity. I consider what happens if one quantizes relational rather than absolute mechanics, indeed whether the latter is misleading. By exploiting Jacobi coordinates, I show how to access many examples of quantized relational particle models and then interpret these from a relational perspective. By these means, previous suggestions of bad semiclassicality for such models can be eluded. I show how small (particle number) universe relational particle model examples display eigenspectrum truncation, gaps, energy interlocking and counterbalanced total angular momentum. These features mean that these small universe models make interesting toy models for some aspects of closed universe quantum cosmology. While, these features do not compromise the recovery of reality as regards the practicalities of experimentation in a large universe such as our own.