The continuous spectrum of a quantum mechanical (QM) system contains important information on the system. However, the normalization of wave functions of the continuous spectrum is often difficult and therefore is often omitted from the books on QM. We discuss this problem and show how we can overcome this problem technically, with some examples.
The continuous spectrum of a quantum mechanical (QM) system contains important information on the system. However, the normalization of wave functions of the continuous spectrum is often difficult and therefore is often omitted from the books on QM. We discuss this problem and show how we can overcome this problem technically, with some examples.
Although• quantum mechanics has achieved great success in the explanation of physical phenomena, the mathematics used in quantum mechanics is not rigorous.Unrigorous treatments occur in the case of continuous spectrum.In a Hilbert • space, let A be a self-adjoint operator.When we define the eigenvalue and the eigenelement by the ' ' relation Af=J.f, (1) ' ' h d -. -f(q)=).f(q)2m dq has solutions 2,ri f(q)=ce"Aq for any ;.., But these are not quadratically integrable.E. Hellinger and H. Weyi< 0 have already investigated the problem of continuous spectrum and introduced the " eigendifferential."But they are concerned with a special problem, i. e. bounded quadratic forms and solutions of differential equation.In respect of generality, they do not satisfy.< 2 > E. Fues< 3 > and some other writers<" have used this conception " eigendifferential " in quantum mechanics.But the treatments are either complicated or not rigorous.J. v. Neumann< 5 > has proved that in Hilbert space, for any selfadjoint operator A, there exists a resolution of identity E(;..) such that Af= J:and he has applied this theory to quantum mechanics.Although his theory gave a rigorous mathematical foundation to quantum mechanics, it did not give any interpretation to the representation theory of Dirac's form.Recently I investigated the "theory of vector-valued set functions."This theory removes the difficulties in the treatement of continuous spectrum, and gives a rigorous interpretation to the representation theory.< 5 > The vector-valued set function q(U) is a function of set U, and its functional value is an element (vector) in the Hilbert ,space, (1) E. Hellinger [l], 231-271.H. Weyl [1], 238-251.(2) T. Carlemann used this conception "eigendifferential" in the treatise of singular integral equations.Cf.T. Carlemann [1].(3) E. Fues [1], 295-303.(4) For example, L. de Broglie [2], 140-159.(5) J. v. Neumann [1], 53-121.(6) F. Maeda [2], [5], [6].A summary of the theory of vector-valued set functions is given in the first part Qf the present paper.(1) G. Temple has formulated a principle of quantum theory in which projective operators are given priority over self-adjoint operators.(C(.G. Temple [1)).But. the .whole discussion differs from that of my present paper.
Von Neumann's theory of measurement in Quantum Mechanics was spelled out in the last chapter of his book, which was published in 1932. This book was highly mathematical for its time, and in 1939 London and Bauer provided a simplified account of the measurement theory part of it (London and Bauer, 1939). Von Neumann died in 1957. Thirty years after the publication of von Neumann's book, Wigner published a review containing his views on the shortcomings of von Neumann's theory, but omitting any discussion of its mathematical core, namely von Neumann's analysis of composite systems (Wigner, 1963). He also published a set of lecture notes entitled Interpretation of Quantum Mechanics (Wigner, 1983) in which some of his concerns were spelled out in greater detail, and an article addressed ‘to an audience of non-physicists’ (Wigner, 1964). Wigner's own contributions to measurement theory were discussed by Shimony in a talk at the Wigner centennial conference (Shimony, 2002). The English translation of von Neumann's book was published in 1955.1 The account that follows is based on these sources. We shall assume that the reader is acquainted with notions such as wave function collapse and the Heisenberg cut, but we shall not assume familiarity with the technicalities of von Neumann's theory. This chapter is organized accordingly. Section 8.1 explains what we mean by the term von Neumann's measurement theory and gives an overview of the subject. It is followed by Sections 8.2 and 8.3, in which the theory is spelled out in detail. Section 8.4 recounts Wigner's reservations.
Scope of this book Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner. The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.
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