Abstract The lattice Boltzmann concepts and applications described so far refer to classical (non-quantum) physics. Whether this limitation is strictly scientific or purely contingental in character, it is difficult to say. However, it appears plausible to speculate that at least part of the LBE assets should be exportable to the quantum world. In principle, this does not even require much of a conceptual jump, since it has been known for a long time that quantum mechanics can be formulated in a fluid dynamic language. This analogy is precisely the stepping stone for this final chapter: stay with us, LBE goes quantum mechanical! Intriguing analogies between quantum mechanics and fluid mechanics have been pointed out since the earliest days of quantum theory [212]. The orthodox tenet is that these analogies are purely formal in character and do not bear upon the basic physics of quantum phenomena. A less orthodox, albeit not minor, stream of thought insists instead that quantum mechanics and notably its philosophical hard-core, Heisenberg’s uncertainty principle, are nothing but a mirror of our ignorance of the underlying (hidden) microscopic physical level. This leads to the puzzling theory of ‘hidden variables’ which can be traced back to Einstein and subsequently to D. Bohm and others [213]. It is not our intention here to enter this fascinating and still open subject [214]. We turn to a practical question instead: what can the analogy do for us in terms of numerical modeling of evolutionary quantum mechanical phenomena? The question is legitimate because, regardless of its philosophical implications, the fluid analogy certainly provides an intuitive and sound basis to develop numerical methods for time-dependent quantum mechanics. In particular, it appears reasonable to ask whether the advantages brought about by LBE in fluid dynamics can (by means of the fluid analogy) be exported to the context of quantum mechanics. Before we put forward our discrete kinetic theory version of the analogy, it is useful to provide a cursory survey of the main ideas behind the analogy itself. To this end, a short recap of basic notions of quantum mechanics is in order.
Abstract The lattice Boltzmann concepts and applications described so far refer to classical (non-quantum) physics. Whether this limitation is strictly scientific or purely contingental in character, it is difficult to say. However, it appears plausible to speculate that at least part of the LBE assets should be exportable to the quantum world. In principle, this does not even require much of a conceptual jump, since it has been known for a long time that quantum mechanics can be formulated in a fluid dynamic language. This analogy is precisely the stepping stone for this final chapter: stay with us, LBE goes quantum mechanical! Intriguing analogies between quantum mechanics and fluid mechanics have been pointed out since the earliest days of quantum theory [212]. The orthodox tenet is that these analogies are purely formal in character and do not bear upon the basic physics of quantum phenomena. A less orthodox, albeit not minor, stream of thought insists instead that quantum mechanics and notably its philosophical hard-core, Heisenberg’s uncertainty principle, are nothing but a mirror of our ignorance of the underlying (hidden) microscopic physical level. This leads to the puzzling theory of ‘hidden variables’ which can be traced back to Einstein and subsequently to D. Bohm and others [213]. It is not our intention here to enter this fascinating and still open subject [214]. We turn to a practical question instead: what can the analogy do for us in terms of numerical modeling of evolutionary quantum mechanical phenomena? The question is legitimate because, regardless of its philosophical implications, the fluid analogy certainly provides an intuitive and sound basis to develop numerical methods for time-dependent quantum mechanics. In particular, it appears reasonable to ask whether the advantages brought about by LBE in fluid dynamics can (by means of the fluid analogy) be exported to the context of quantum mechanics. Before we put forward our discrete kinetic theory version of the analogy, it is useful to provide a cursory survey of the main ideas behind the analogy itself. To this end, a short recap of basic notions of quantum mechanics is in order.
Abstract The Lattice Boltzmann concepts and applications described so far refer to classical, i.e., non-quantum physics. However, the LB formalism is not restricted to classical Newtonian mechanics and indeed an LB formulation of quantum mechanics, going by the name of quantum LB (QLB) has been in existence for more than two decades. Even though it would far-fetched to say that QLB represents a mainstream, in the recent years it has captured some revived interest, mostly on account of recent developments in quantum-computing research. This chapter provides an account of the QLB formulation: stay tuned, LBE goes quantum!
The difficulties of the old quantum theory were largely dispelled by the development of quantum mechanics in the 1920s. First Heisenberg (1925) developed the rather abstract matrix mechanics and soon after Schro¨dinger (1926; see also Eckart, 1926) developed the physically more appealing wave mechanics. Subsequently he showed them to be mathematically equivalent. The physicists were greatly relieved; now they had once again a mathematical formalism that enabled the results of experiments to be calculated. It was possible to move forward. It is certainly a very elegant and indeed beautiful formalism, and does enable the results of many physical measurements to be calculated with great accuracy. It is used daily by atomic and nuclear physicists; there is no other way to make calculations. From the earliest days there has been intense debate about the interpretation of quantum mechanics, and it shows no signs of being settled; if anything it is becoming more intense over the years. As Penrose has remarked, no one understands quantum mechanics. Feynman (1967; see also Sykes, 1994) also admitted that ‘nobody understands quantum mechanics’, and Einstein once remarked: ‘I have thought a hundred times as much about the quantum problems as I have about general relativity theory’ (Pais, 1994, p. 57). It should therefore be evident that the most that can be done here is to try to show where the real difficulties lie, and how the failure to recognize that quantum mechanics is an essentially statistical theory has led to a series of confusing quantum paradoxes. It is just these paradoxes that have often been used to support theological speculations. It is useful to ask what ‘understanding quantum mechanics’ actually means. Certainly it means being able to calculate the results of many measurements. The same applies to Newton’s theory of gravitation, which we understand in the sense that we can use it as a basis of calculation, although we do not really understand the gravitational force itself. Very often, especially in modern physics, the mathematical formalism does not give us the sort of intuitive physical feeling for what is going on that we often obtain from classical mechanics. Perhaps we can never achieve more than this for quantum mechanics. In classical mechanics we can assume that a property that is measured objectively exists prior to the interaction of the measuring apparatus with the observed system. Quantum mechanics, however, ‘is incompatible with the proposition that measurement discovers some unknown but pre-existing property’ (Peres, 1997, p. 14). There is an important distinction between mysteries and absurdities. We may be forced to live with mysteries, but there is no reason why we should put up with absurdities such as the wave-particle duality, tunnelling through potential barriers, acausal events and the collapse of the wave function. These are the quantum paradoxes that will be discussed later on. They arise within the Copenhagen interpretation of quantum mechanics and are solved by other interpretations. These alternative interpretations enable us to make conjectures about what is going on, even though we cannot understand it in detail. This is acceptable and a stimulus to further research. It is certainly difficult enough to understand quantum mechanics, but there is no need to try to visualize logical absurdities in the effort to do so. Quantum mechanics is a mathematical formalism that makes it possible to calculate the results of some experiments. Anything more than this, such as questions about the interpretation of what is measured and whether it tells us what is actually going on in the system being studied, belongs to philosophy. Thus we are concerned here with what is essentially a philosophical debate; if it were just a matter of physics the debate would have been settled long ago. One of the difficulties is that it invokes the results of experiments and uses the language of quantum mechanics. Inevitably this language uses philosophical terms and then it is easy to give the impression that the philosophical propositions are in some way entailed by the physical results. These in turn can influence theological thinking. In order to be quite clear which arguments are justified and which are not, it is essential to separate the three levels of discourse: physical, philosophical and theological. Our task is to relate these together, and this immediately raises the traditional problems of the philosophy of science, including the relation of experiment to theory, and of theory to explanation. We begin by discussing experiments, distinguishing between those that are real and those that are imaginary.
Quantum fluids are fluids, indeed. We have seen in the previous chapter what it means from a classical physics point of view. But quantum fluids are by definition ruled by quantum physics, as well. The goal of this chapter is to introduce some key concepts of quantum physics necessary to understand what makes a fluid a quantum one or rather what quantum property makes an ensemble of particles a quantum fluid. We are here precisely at the edge of our basic intuition of quantum physics. Indeed, one usually associates the notion of quantum object to a microscopic nature and it may thus a priori seem paradoxical to link the word quantum to a macroscopic object, such as the notion of fluid. Thinking it over a bit deeper, quantum physics is certainly present at all scales and we are confronted with quantum mechanics through the many electronic devices which have invaded our everyday life. And it actually turns out that, as we will see later on, most of these quantum devices are directly connected to the notion of quantum fluids, the case of electron fluids being certainly the most common one in such devices. Quantum physics has for sure deeply modified our everyday life. But it has also changed our perception of the world, and as we shall see in this chapter, it is precisely this new vision of the world which allows us to understand the concept of quantum fluids underlying so many physical systems.
Quantum Mechanics is an essential topic in today's physics curriculum at both the undergraduate and graduate levels. Quantum mechanics can explain the microscopic world with fantastic accuracy; the fruits from its insights have created technologies that have revolutionized the world. Computers, lasers, mobile telephones, optical communications are but a few examples. The language of quantum mechanics is now an accepted part of the language of physics and day-to-day usage of this language provides physicists with the intuition that is essential for achieving meaningful results. Nevertheless, most physicists acknowledge that, at least once in their scientific career, they have had difficulties understanding the foundations of quantum theory, perhaps even the impression that a really satisfactory and convincing formulation of the theory is still lacking. Numerous quantum mechanics textbooks are available for explaining quantum formalism and applying it to understand problems such as the properties of atoms, molecules, liquids, and solids; the interactions between matter and radiation; and more generally to understand the physical world that surrounds us. Other texts are available for elucidating the historical development of this discipline and describing the steps through which it went before quantum mechanics reached its modern formulation. In contrast, books are rare that review the conceptual difficulties of the theory and then provide a comprehensive overview of the various attempts to reformulate quantum mechanics in order to solve these difficulties. The present text by Franck Laloë does precisely this.
This essay examines some of the arguments in David Deutsch's book The Fabric of Reality , chief among them its case for the so-called many-universe interpretation of quantum mechanics (QM), presented as the only physically and logically consistent solution to the QM paradoxes of wave/particle dualism, remote simultaneous interaction, the observer-induced 'collapse of the wave-packet', etc. The hypothesis assumes that all possible outcomes are realized in every such momentary 'collapse', since the observer splits off into so many parallel, coexisting, but epistemically non-interaccessible 'worlds' whose subsequent branchings constitute the lifeline-or experiential world-series- for each of those proliferating centres of consciousness. Although Deutsch concedes that his 'multiverse' theory is counter-intuitive, he none the less takes it to be borne out beyond question by the sheer observational/predictive success of QM and the conceptual dilemmas that supposedly arise with alternative (single-universe) accounts. Moreover, he claims the theory resolves a range of longstanding philosophical problems, notably those of mind/body dualism, the various traditional paradoxes of time, and the freewill/determinism issue. The essay suggests on the contrary, that Deutsch unwittingly transposes into the framework of presentday quantum debate speculative themes from the history of rationalist metaphysics, often with bizarre or philosophically dubious results, and that he rules out at least one promising rival account, namely Bohm's 'hidden variables' theory. It goes on to consider reasons for resistance to that theory among proponents of the 'orthodox' (Copenhagen) doctrine, and for the strong anti-realist, at times even irrationalist bias that has characterized much of this discussion since Bohr's debates with Einstein about quantum non-locality, observer-intervention, and the limits of precise measurement. Finally, the contrast is pointed out between Deutsch's ontologically extravagant use of the many-worlds hypothesis (akin to certain ideas advanced by speculative metaphysicians from Leibniz down) and those realist modes of counterfactual reasoning- e.g. in Kripke and the early Putnam- which deploy similar arguments to very different causal-explanatory ends.
Since its very beginning, quantum mechanics has been developed to deal with systems on the atomic or sub-atomic scale. For many decades, there has been no reason to think about its application to macroscopic systems. Actually, macroscopic objects have even been used to show how bizarre quantum effects would appear if quantum mechanics were applied beyond its original realm. This is, for example, the essence of the so-called Schrödinger cat paradox (Schrödinger, 1935). How-ever, due to recent advances in the design of systems on the meso- and nanoscopic scales, as well as in cryogenic techniques, this situation has changed drastically. It is now quite natural to ask whether a specific quantum effect, collectively involving a macroscopic number of particles, could occur in these systems. In this book it is our intention to address the quantum mechanical effects that take place in properly chosen or built “macroscopic” systems. Starting from a very naïve point of view, we could always ask what happens to systems whose classical dynamics can be described by equations of motion equivalent to those of particles (or fields) in a given potential (or potential energy density). These can be represented by a generalized “coordinate” φ( r , t ) which could either describe a field variable or a “point particle” if it is not position dependent, φ( r , t ) = φ( t ).