Quantum technology is seeing a remarkable explosion in interest due to a wave of successful commercial technology. As a wider array of engineers and scientists are needed, it is time we rethink quantum educational paradigms. Current approaches often start from classical physics, linear algebra, or differential equations. This chapter advocates for beginning with probability theory. In the approach outlined in this chapter, there is less in the way of explicit axioms of quantum mechanics. Instead the historically problematic measurement axiom is inherited from probability theory where many philosophical debates remain. Although not a typical route in introductory material, this route is nonetheless a standard vantage on quantum mechanics. This chapter outlines an elementary route to arrive at the Schrödinger equation by considering allowable transformations of quantum probability functions (density matrices). The central tenet of this chapter is that probability theory provides the best conceptual and mathematical foundations for introducing the quantum sciences.
Quantum technology is seeing a remarkable explosion in interest due to a wave of successful commercial technology. As a wider array of engineers and scientists are needed, it is time we rethink quantum educational paradigms. Current approaches often start from classical physics, linear algebra, or differential equations. This chapter advocates for beginning with probability theory. In the approach outlined in this chapter, there is less in the way of explicit axioms of quantum mechanics. Instead the historically problematic measurement axiom is inherited from probability theory where many philosophical debates remain. Although not a typical route in introductory material, this route is nonetheless a standard vantage on quantum mechanics. This chapter outlines an elementary route to arrive at the Schrödinger equation by considering allowable transformations of quantum probability functions (density matrices). The central tenet of this chapter is that probability theory provides the best conceptual and mathematical foundations for introducing the quantum sciences.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
In this introductory chapter, which actually is the core of the entire book, we make a serious effort to bring together a variety of ideas from philosophy and physics. We first lay the epistemological foundations for our approach to particle physics. These philosophical foundations, combined with tools borrowed from the amazingly sophisticated, but still not entirely satisfactory, mathematical apparatus of present-day quantum field theory and augmented by some new ideas, are then employed to develop a mathematical representation of fundamental particles and their interactions. In this process, also the relation between “fundamental particle physics” and “quantum field theory” is going to be clarified. This first chapter consists of two voluminous sections: a number of philosophical contemplations followed by a discussion of mathematical and physical elements . The reader might wonder why these massive sections are not presented as two separate chapters. The reason for keeping them together is the desire to emphasize the intimate relation between these two sections: the selection and construction of the mathematical and physical elements is directly based on our philosophical contemplations. The beauty of the entire approach is a fruit of this intimate relationship. The presentation of the material in this chapter is based on the assumption that the reader has a basic working knowledge of linear algebra and quantum mechanics. If that is not the case, the reader might want to consult the equally entertaining and serious introduction Quantum Mechanics: The Theoretical Minimum by Susskind and Friedman [4]. The book Quantum Mechanics and Experience by Albert [5] offers a fascinating introduction with a strong focus on the measurement problem. An almost equation-free discussion of the history and foundations of quantum mechanics can be found in the book Einstein, Bohr and the Quantum Dilemma: From Quantum Theory to Quantum Information by Whitaker [6]. Complex vector spaces, Hilbert space vectors and density matrices for describing states of quantum systems, bosons and fermions, canonical commutation relations, Heisenberg's uncertainty relation, the Schrodinger and Heisenberg pictures for the time evolution of quantum systems, as well as a basic idea of the measurement process are all referred or alluded to in Section 1.1; these basics of quantum mechanics are recapped only very briefly in Section 1.2. The crucial construct of Fock spaces is explained in a loose way in Section 1.1.3 and later elaborated in full detail in Section 1.2.1.
This chapter begins by developing the relation between the quantum theory and the linear algebra. It discusses the most fundamental postulates of the theory and provides a phenomenological development of the Schrödinger wave equation (SWE) although the results might be gleaned from the classical Lagrangian and Hamiltonian with the Poisson bracket relations. Afterward some simple examples for the infinitely and finitely deep wells help clarify the basic postulates. The simple harmonic oscillator has perhaps the most sweeping implications and applications for the quantum theory. For this reason, we present the most applicable formalism for the Harmonic oscillator that uses the operator approach rather than the classical method of partial differential equations. The angular momentum and spin are presented for the study of the atom and for applications for nanodevices, and quantum computing. Next the representation formalism clarifies the distinction between the dynamical operators and the wave functions and their interrelations. The chapter discusses both timeindependent and time-dependent perturbation theories. The density operator combines the quantum theory with the classical theory for finding the behavior of a system. The density operator represents one of the basic concepts in the quantum theory. The remainder of the chapter introduces the more advanced material meant to introduce the quantum field theory starting with the second quantization and the propagator.
Introduction to the Maths and Physics of Quantum Mechanics details the mathematics and physics that are needed to learn the principles of quantum mechanics. It provides an accessible treatment of how to use quantum mechanics and why it is so successful in explaining natural phenomena. This book clarifies various aspects of quantum physics such as ‘why quantum mechanics equations contain “I”, the imaginary number?’, ‘Is it possible to make a transition from classical mechanics to quantum physics without using postulates?’ and ‘What is the origin of the uncertainty principle?’. A significant proportion of discussion is dedicated to the issue of why the wave function must be complex to properly describe our “real” world. The book also addresses the different formulations of quantum mechanics. A relatively simple introductory treatment is given for the “standard” Heisenberg matrix formulation and Schrodinger wave-function formulation and Feynman path integrals and second quantization are then discussed. This book will appeal to first- and second-year university students in physics, mathematics, engineering and other sciences studying quantum mechanics who will find material and clarifications not easily found in other textbooks. It will also appeal to self-taught readers with a genuine interest in modern physics who are willing to examine the mathematics and physics in a simple but rigorous way. Key Features: Written in an engaging and approachable manner, with fully explained mathematics and physics concepts Suitable as a companion to all introductory quantum mechanics textbooks Accessible to a general audience
This chapter begins by exploiting the Einstein–de Broglie relation and the notion of symbol. The following steps are the local and global conservation laws associated with the Schrödinger equation, probabilistic interpretation of the wave function and probability distribution, the spreading of wave packets and transformation properties of wave functions. In the last part, the Heisenberg picture is outlined at an introductory level. From classical to wave mechanics In the previous chapter we have seen how to go from wave equations on space–time to Hamilton equations on phase space by means of an appropriate approximation requirement on wave solutions. In more formal language, we could say we have considered a procedure to associate a Hamiltonian ordinary differential equation on phase space starting with a partial differential equation on configuration space. This association produces a correspondence between differential operators and polynomials on phase space, which relies on the notion of symbol (see below). Now we are aiming to pass from a function on phase space, i.e. the classical Hamiltonian, to an operator which, by analogy, is called the Hamiltonian operator. Such a transition is physically non-trivial because it amounts to building a wave equation from the equation on rays. From the formal point of view it appears to contain some ambiguities. Such a transition is always possible if polynomials in p have constant coefficients.
Quantum mechanics is one of the most fundamental yet difficult subjects in physics. Nonrelativistic quantum theory is presented here in a clear and systematic fashion, integrating Born's probabilistic interpretation with Schrödinger dynamics. Basic quantum principles are illustrated with simple examples requiring no mathematics beyond linear algebra and elementary probability theory. The quantum measurement process is consistently analyzed using fundamental quantum principles without referring to measurement. These same principles are used to resolve several of the paradoxes that have long perplexed physicists, including the double slit and Schrödinger's cat. The consistent histories formalism used here was first introduced by the author, and extended by M. Gell-Mann, J. Hartle and R. Omnès. Essential for researchers yet accessible to advanced undergraduate students in physics, chemistry, mathematics, and computer science, this book is supplementary to standard textbooks. It will also be of interest to physicists and philosophers working on the foundations of quantum mechanics.