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Fundamental theories of physics

Classical models of quantum mechanics

Klaas Landsman

Published Jan 1, 2017• 0 citations
Publisher

Abstract

This chapter gives an introduction to a chain of results attempting to exclude deeper layers underneath quantum mechanics that restore some form of classical physics: ‘[Such results] more or less illustrate the ways along which some opponents might hope to escape Bohr’s reasonings and von Neumann’s proof and the places where they are dangerously near breaking their necks.’ (Groenewold, 1946, p. 454). In so far as they are mathematically precise, such no-go results have their roots in von Neumann’s 1932 book, which gave rise to two traditions that were often in polemical opposition to each other. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-51777-3_6

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This chapter gives an introduction to a chain of results attempting to exclude deeper layers underneath quantum mechanics that restore some form of classical physics: ‘[Such results] more or less illustrate the ways along which some opponents might hope to escape Bohr’s reasonings and von Neumann’s proof and the places where they are dangerously near breaking their necks.’ (Groenewold, 1946, p. 454). In so far as they are mathematically precise, such no-go results have their roots in von Neumann’s 1932 book, which gave rise to two traditions that were often in polemical opposition to each other. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Passing from finite phase spaces X to infinite ones yields many fascinating new phenomena, some of which even seem genuinely “emergent” in not having any finite dimensional shadow, approximate or otherwise. Nonetheless, practically all results in the previous chapter remain valid, typically after the inclusion of some technical condition(s) that restrict the almost unlimited freedom allowed by infinite sets. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The conclusion of Part I, the non-technical (or somewhat less technical) part of this book, is that a completely convincing model of our universe is lacking. This should be a model that shows how quantum mechanical behaviour, with all its mysteries and paradoxes, can emerge from a local, deterministic theory. Yet we also find that one can assume its existence to arrive at a picture of our quantum mechanical universe that makes perfect sense. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
To begin to describe quantum algorithms, we need an understanding of the physical phenomena that motivate quantum mechanics, a simple mathematical model of a potential quantum storage device and the notation to describe and manipulate such a model. In this chapter we present the context in which we will work, motivate the need for quantum mechanics and provide a quick introduction to the essential mathematical tools and notation to be used throughout the book. As part of the price of brevity, we will work with a static model of quantum mechanics and will defer additional mathematical definitions until they are needed. The reader who is already familiar with this material can easily skip to Chapter 2, giving only a cursory glance at Chapter 1 to identify any idiosyncratic notation. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The authors present briefly the basic notions of the quantum formalism: pure and mixed states; quantum observables; quantum probability; Born’s rule, superposition, and ‘state collapse’; the projection postulate (von Neumann–Lüders postulate); Dirac’s ket and bra-vector notations; elements of quantum information theory and quantum logic; Schrödinger’s and von Neumann’s equations; unitary dynamics; and positive operator valued measures. They also define the basic mathematical notions related to the quantum formalism: Hilbert space; scalar product; norm; Hermitian operator (matrix); projector; and unitary operator and adjoint operator. This chapter may be useful for newcomers to the field, but for those readers who have preliminary knowledge about quantum mechanics (QM) they can proceed directly to advanced chapters. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The founding fathers of quantum mechanics had already perceived the essence of the difficulties of quantum mechanics; today, after almost a century, the discussions are still lively and, if some very interesting new aspects have emerged, at a deeper level the questions have not changed so much. What is more recent, nevertheless, is a general change of attitude among physicists: until about 1970 or 1980, most physicists thought that the essential questions had been settled, and that “Bohr was right and proved his opponents to be wrong”. This was probably a consequence of the famous discussions between Bohr, Einstein, Schrödinger, Heisenberg, Pauli, de Broglie, and others (in particular at the Solvay meetings [1–3], where Bohr's point of view had successfully resisted Einstein's extremely clever attacks). The majority of physicists did not know the details of the arguments. They nevertheless thought that the standard “Copenhagen interpretation” had clearly emerged from the infancy of quantum mechanics as the only sensible attitude for good scientists. This interpretation includes the idea that modern physics must contain indeterminacy as an essential ingredient: it is fundamentally impossible to predict the outcome of single microscopical events; it is impossible to go beyond the formalism of the wave function (or its generalization, the state vector ∣ψ) and complete it.