It is regrettable that the quantum length of an object is rarely if ever discussed, because it provides an ideal pedagogical paradigm for understanding how a physicist uses classical intuition to define quantum properties and how such quantum properties behave as one would expect in the classical limit. It also provides for a way to understand many-particle states, and leads to interesting quantum behavior that challenges our intuition about measurement. This exercise parallels the ways in which theories are developed, giving the student a concrete example of the thought process involved.
It is regrettable that the quantum length of an object is rarely if ever discussed, because it provides an ideal pedagogical paradigm for understanding how a physicist uses classical intuition to define quantum properties and how such quantum properties behave as one would expect in the classical limit. It also provides for a way to understand many-particle states, and leads to interesting quantum behavior that challenges our intuition about measurement. This exercise parallels the ways in which theories are developed, giving the student a concrete example of the thought process involved.
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.
Although it is 70 years old, there are still areas of quantum theory which are open to interpretation. This book deals with such interpretational problems, and in particular with the issue of the measurement process. The main theme is whether quantum mechanics can be viewed as a (semantically) complete theory, namely as a theory capable of describing not only the physical object under observation but also the measuring instruments used to detect the properties of that physical object. The problematic aspects arise from the fact that the measuring instruments themselves behave as classical physical systems. Thus, we are faced with the open problem of the transition from the quantum to the classical, or more explicitly, the problem of why the composition of many quantum micro-objects gives rise to classical macro-objects. This book goes carefully, exhaustively, and deeply into the interpretational problems of quantum mechanics induced by the measurement process. The solution of many relevant issues, as well as the problems that remain unsolved, emerge clearly. The discussion itself is self-contained and the technical parts are reader-friendly. This book is of value to graduate students and researchers who have an interest in the physical, mathematical, and philosophical foundations of quantum mechanics.
In quantum physics lectures, half-integer spins are generally introduced as “objects that do not come back to their original state after one full turn but that do after two.” As a consequence, students often consider this behavior to be purely quantum mechanical. However, spin-1/2 is above all a geometrical property of the rotations group and can, therefore, also have practical consequences at the macroscopic scale. To illustrate this, we introduce and describe in this work a new pedagogical tool named the spinorial ball. It allows students to concretely manipulate a macroscopic 1/2-spin, which helps them to build intuition as to how the latter behaves under rotations. This object can also be used to introduce several general concepts from the theory of Lie groups, such as group homomorphism and homotopy classes of loops through the example of the groups SU(2) and SO(3). The spinorial ball provides a macroscopic visualization of all these concepts, which are ubiquitous in quantum physics.
WHAT TO LOOK FOR Although the classical theory developed in Chapter 2 can be used to describe a great many phenomena in the real world, there is little doubt that the proper description of the world is quantum. To a large measure we perceive the world classically and we must find ways to uncover the underlying quantum theory. The process of “quantization” normally consists of turning a classical theory into a corresponding quantum theory. In turn, the “classical limit” is the process by which a quantum theory is brought to its associated classical theory. The parameter ħ = 1.0545 × 10 −27 (≃ 10 −27 ) erg-seconds (cgs units), referred to as Planck's constant, sets the scale of quantum phenomena. In many applications it will prove useful to use “natural” units in which ħ = 1 just so that formulas are less cumbersome. For most typical formulations of quantum mechanics the classical limit means the limit that ħ → 0, and in this text we shall have many occasions to use this definition as well. In addition, we shall also learn to speak about the classical and quantum theories simultaneously coexisting – as they do in the real world – without the need to take the limit ħ → 0. There are three generally accepted “royal” routes to quantization, due principally to Heisenberg, Schrödinger, and Feynman, respectively, and we shall exploit them all. In each case we begin with a single degree of freedom ( N = 1).
Spin, still a concept or property not fully exploited, has already impacted the world in many ways with its many applications. The history of how spin came to light and how it was interpreted and developed as investigations were conducted is of particular interest with the many odd outcomes of experiments concerning this intrinsic property. This article will sort out the historical outline of how spin was first indicated in theories and how it was eventually embraced as a fundamental property of quantum particles in an attempt to holistically review the process of recognition of this remarkable concept. Through this review, it becomes apparent how the recognition and development of a concept involves the intellect of a broad range of physicists who are willing to view things from both classical and quantum perspectives particularly when it comes to new concepts as spin. How elements like luck and chance play a role will also be revealed. As it turns out, our intuitions about the world around us may not be the ultimate truth, especially when it comes to Quantum Mechanics. Space Quantization was once thought of as no more than a theoretical outcome, however, experiments anyway impose this reality on physicists requiring them to abandon stubborn biases. Admittedly, not every speculation was correct, the analogy of “Spin” was an unfortunate one. But this article shows that this is just how understanding the mysteries of the quantum world really unfolds.