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.
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.
In this chapter we introduce the Hilbert space which plays the central role in quantum mechanics. For a more detailed discussion of this subject we refer to the books of Balakrishnan [5], Richtmyer [137], Sewell [146], Stakgold [152], Steeb [163], Weidmann [181], Yosida [185]. Moreover the proofs of the theorems given in this chapter can be found in these books. We assume that the reader is familiar with the notation of a linear space. First we introduce the pre-Hilbert space. 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.
Mathematical formalism of quantum mechanics: brief introduction In this chapter, we present basic notions of quantum mechanics, and we emphasize “quantum mathematics.” To posit that the mathematical formalism of quantum mechanics can be used outside of quantum physics (as in sociology, economics, or finance) may be of interest. If one can steer away from merely positing and veer instead into convincing the reader that such a position is tenable, then some achievement has been booked. But the price to pay is that the formalism of quantum mechanics needs to be uncovered: how else can one judge whether what we posit is senseful? Thus, a reader can either (i) formally accept our viewpoint about the quantum-like paradigm (see Khrennikov [1]), or (ii) ask for a detailed analysis of the quantum foundations which would justify this paradigm. In the first case, it may well be sufficient to skip this chapter (but we should suggest to attempt to include Chapter 3). In the second case, we invite especially those readers who are concerned about the fact that the formal usage of the mathematical formalism of quantum mechanics may transfer major problems of quantum foundations to social science, economics, and finance, to read the present chapter carefully.
The fundamental laws of Nature that govern the physics of atoms, elementary particles, light quanta, and everything else that is small and light, have been found to fit in one grandiose framework called Quantum Mechanics. The internal logic of this framework appears to be counter intuitive and mysterious, while extremely coherent and effective. In this general introduction the author explains why he feels the need for expressing his views on how to understand these facts, even if many known scientists have already given extensive reviews of numerous ideas and approaches. 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.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.
The objective of Chapter 4 is to give an overview of the basic concepts of quantum mechanics and their interpretation. Thus this chapter functions as a theoretical preparation for a better understanding of the quantum world and of the knowledge measurement process. First, the terminology of the word “quantum” and its impact on the derivation of the term “knowledge quantum” is explained in Chapter 4.1. Then, the discussion moves to the historical development of quantum mechanics. The key issues and current research field for quantum theory will be given in an overview. The following chapters show some insights into key quantum mechanical concepts such as slit-experiments, wave-particle duality, the Copenhagener interpretation, and quantum probability. Chapter 4 closes with an explanation of the Uncertainty Principle of Werner Heisenberg, especially since the Uncertainty Principle is used to derive the knowledge measurement process through the method of analogical reasoning. 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.