Abstract In this section, we review the concepts of classical and quantum superposition. Quantum superposition is the framework for understanding all quantum phenomena. As we do not observe quantum phenomena in our everyday lives, it may seem confusing at first. However, as unintuitive as the quantum world may appear, there are a vast number of experiments which conclusively show that the universe really does operate according to the law of quantum superposition at the smallest distances accessible today. Before going into specific details on quantum superposition, it is useful to explain how the term “superposition” is used in different contexts in both classical and quantum physics. At the end of the chapter, we present the related activities and questions. After gaining experience with quantum superposition from working through these problems, it will become more intuitive. The more experience you gain by advancing through this book, the more quantum superposition will make sense.
Abstract In this section, we review the concepts of classical and quantum superposition. Quantum superposition is the framework for understanding all quantum phenomena. As we do not observe quantum phenomena in our everyday lives, it may seem confusing at first. However, as unintuitive as the quantum world may appear, there are a vast number of experiments which conclusively show that the universe really does operate according to the law of quantum superposition at the smallest distances accessible today. Before going into specific details on quantum superposition, it is useful to explain how the term “superposition” is used in different contexts in both classical and quantum physics. At the end of the chapter, we present the related activities and questions. After gaining experience with quantum superposition from working through these problems, it will become more intuitive. The more experience you gain by advancing through this book, the more quantum superposition will make sense.
Quantum mechanics is one of the most challenging subjects to learn. It is challenging because quantum phenomenon is counterintuitive, and the mathematics used to explain such a phenomenon is very abstract, and difficult to grasp. This textbook is an attempt to overcome these challenges. Every chapter presents quantum ideas step- by- step in a structured way with a comparison between quantum and classical concepts. It provides a clear distinction between classical and quantum logic. Conceptual questions are provided after every important section so that the reader can test their understanding at every step. Such an approach aids in preventing misconceptions. Problem solving is not restricted to solving differential equations and integration. But it requires to systematically and creatively analyze a problem, to apply the new and powerful concepts for finding a solution and to understand the physical meaning of the solution. The tutorials on special topics are an effort to teach problem solving by actively engaging the reader in a thinking process, to apply the concepts and to understand the physical meaning of the solution. The simulations are provided for some of the topics. The simulations aid in the visualization of the quantum phenomenon, and for meaningful understanding of the mathematics. This approach may lead to development of "quantum mechanical intuition "as well as learning mathematical techniques for problem solving. Most importantly, the book is not flooded with numerous topics that makes the reader confused and distracted, rather the most important topics are discussed at a deeper level. The understanding of quantum mechanics is incomplete without understanding the early ideas and experiments that lead to the development of the quantum theory. Thus, the first two chapters of the book are dedicated to such topics. The key features of this book are: A simplified, structured, and step-by-step introduction to quantum mechanics. The simplification is attained through use of two-level system, step- by- step discussion of important topics in a simplified language at a deeper level, analogies, and visualization using illustrations and simulations A systematic arrangement of topics, and numerous worked- out examples. The presentation of the structure in the mathematical formalism of quantum mechanics provides clarity in understanding complicated and abstract mathematics. It also helps to understand the distinction between the quantum mechanical and classical approaches Conceptual questions at the end of every important section. The conceptual questions can be used in a classroom as a point of discussion between an instructor and students Tutorials on special topics. Simulations on special topics aid in the visualization of the physical phenomenon, and demonstration of the application of mathematics An in-depth discussion of the wave-particle duality, measurement problem, and their philosophical implications in Chapter 2 provides an understanding of the broader meaning of quantum mechanics
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
Introduction Chapter 1 dealt with one of the deepest mysteries in modern science: how to explain consciousness. This Part deals with another mystery: how to make sense of quantum mechanics. Both involve a problem of reconciliation with the classical worldview, which is one hint that the two mysteries might be related. A second is that, unlike classical physics, which makes no reference to or allowance for consciousness, quantum theory raises the issue of consciousness and its relationship to the physical world in a very direct way. These hints do not necessarily mean there is a connection between the two mysteries, but by the end of this Part I hope to have shown why it would be natural to look for one. My more immediate aim here is to give the social scientific reader an introductory understanding of the experimental findings of quantum theory, its key concepts, and the debates about its interpretation. The discussion assumes no prior knowledge of quantum physics and makes no use of mathematics. Quantum theory without equations might seem like an oxymoron, and certainly reading this Part will not enable anyone to use quantum theory. But to a perhaps surprising degree it is possible to understand quantum theory without being able to use it. Its findings can be communicated in ordinary language, and the same goes for its main concepts and interpretive debates. One consequence of this has been the emergence of a virtual cottage industry of books on quantum theory for a lay audience. Most are written by physicists, and many are quite good. However, it is not just popular treatments that do without equations: much of the professional philosophical literature does as well. (I draw on both kinds of literature below.) Philosophers of physics are trained to understand the math, of course, which is necessary for a full appreciation of the issues.
This chapter begins by explaining why physics is the fundamental of all natural sciences. Throughout most its history, atoms were hardly the center of attention in physics. Once called natural philosophy, physics originated in the study of the motion of inanimate objects. From those studies emerged a body of knowledge, including various natural laws, that today we call ‘classical’ physics. Newton's laws of motion, the basis of the science of mechanics, come to mind, as does his law of universal gravitation. These laws are obeyed by objects that in some sense are ‘large’ and, in fact, the laws were discovered after careful observation of the behavior of such objects. The discussion then turns to the focus of the book—the quantum world—where the laws of physics turn out to be different from the laws of classical physics used to explain phenomena in the macroscopic world of everyday life. The book discusses a number of mostly recent experiments on quantum mechanical foundations, most of them involving light at the level of a few photons—sometimes only one.
From this chapter we start our exposition of quantum mechanics. First, we lay down the typical setting of a quantum mechanical problem. The fundamental feature of quantum indeterminism and the probabilistic nature of quantum mechanics is emphasized. We proceed by constructing, from the single notion of measurement, all the essential conceptual ingredients of quantum mechanics (i.e., the concepts of observable, state and system). Then we formulate the target questions that quantum mechanics aims to solve. Next, we proceed to state the specific laws that constitute the logical premise of the subject. Finally, we demonstrate how the laws, most dominantly, the superposition principle (the fact that quantum probabilities interfere) provides an improved recipe for obtaining the answers to the target questions of quantum mechanics. This also leads to a more refined definition of the quantum state. In order to be able to quickly introduce the basic framework of quantum mechanics, we restrict the discussion to a simple scenario of “instantaneously subsequent measurements” (where there is no time evolution), and where states can be prepared by a single measurement (in the standard jargon, this would be called, “quantum mechanics without degeneracy”; these are descriptions involving CSCOs comprising just one observable).
Lying in the core of the conceptual scheme of current quantum theory, the notion of quantum superposition is a consequence of the quantum-mechanical superposition principle. However, relying on precise but not attainable space or time coordinates, quantum superposition makes little sense physically if we use it to describe or explain experimental results obtained by measurement. Space or time is usually modeled by a metric space endowed with the usual distance function. Irrelevant to instrument precision and measurement accuracy, unattainability of precise coordinates is a well-established mathematical fact following from properties of the corresponding metric topology, and hence rules out completely, in principle, the possibility for us to obtain precise coordinates by measurement in actually performed experiments. In classical mechanics, unattainability of precise coordinates is hardly noticeable, and measurement results obtained in classical physics can still be reasonably explained even though precise coordinates are unattainable. However, when we explain measurement results obtained in quantum physics, unattainability of precise coordinates should not be omitted. In the conceptual scheme of current quantum theory, omission of unattainability of precise coordinates is an unfortunate flaw, which is largely responsible not only for the invalidity of quantum superposition in description of the physical world but also for various ineligible applications of quantum mechanics. Nevertheless, in no sense does the flaw imply quantum mechanics as a successful theory failing to be correct. As shown in this article, the flaw is repairable. By taking into account unattainability of precise coordinates, we can reinterpret the meaning of quantum superposition to repair the flaw without affecting eligible applications of quantum mechanics while avoiding all the ineligible applications.