This chapter stresses what may be called conventional or standard textbook quantum mechanics, that is, to the working hypothesis that quantum mechanics is complete. Some remarkable features of the general quantum formalism, often grouped under the general name "nonseparability," rather drastically limit the domain of validity of the idea. The two notions referred to in the title of this chapter are therefore, in a way, antagonistic. The state operator concept is of some use for investigating the properties of ensembles of subsystems of larger systems, sometimes uttered claim that its introduction makes it possible to remove "nondivisibility by thought" from quantum physics is definitely erroneous. When J. S. Bell's theorem appeared it was immediately clear that some expressions had to be comed in order to express in words its bearing and significance.
This chapter stresses what may be called conventional or standard textbook quantum mechanics, that is, to the working hypothesis that quantum mechanics is complete. Some remarkable features of the general quantum formalism, often grouped under the general name "nonseparability," rather drastically limit the domain of validity of the idea. The two notions referred to in the title of this chapter are therefore, in a way, antagonistic. The state operator concept is of some use for investigating the properties of ensembles of subsystems of larger systems, sometimes uttered claim that its introduction makes it possible to remove "nondivisibility by thought" from quantum physics is definitely erroneous. When J. S. Bell's theorem appeared it was immediately clear that some expressions had to be comed in order to express in words its bearing and significance.
Quantum measurement theories are interesting mainly because of the fundamental questions. Since quantum mechanics is primarily a theory of small, simple systems, it could a priori be expected that a full description of this formalism would be possible in a language that would refer only to the objective properties of such small systems. Unless quantum physics is objectively false in some of its predictions, limitations of such a kind are unavoidable. The chapter argues that such limitations extend to systems of arbitrary finite size and of arbitrary complexity. As a matter of fact, this is the first and perhaps the most significant conclusion in investigation of the conceptual foundations of quantum physics. An equally tenable standpoint is to assert that truly infinite systems have special features of their own that make them fall outside the realm of elementary quantum mechanics. Implicitly or explicitly, nonseparability is indeed present in all the various interpretations of the quantum rules which have been set forth.
This chapter temporarily departs from conventional "orthodoxy" in that the authors set aside the completeness hypothesis and its generalizations. In 1935 A. Einstein and his two younger collaborators, B. Podolsky and N. Rosen (EPR) issued a paper in which they showed that, assuming the validity of but very general and highly plausible assumptions, the incompleteness of quantum mechanics could be demonstrated. The opinion that quantum mechanics is a complete description of the atomic and subatomic phenomena was so much engrained in the mind of the physicists that this EPR result was soon referred to as a "paradox." The EPR argument was based on assumptions, or premises, that, prima facie look most plausible. On the theoretical side it has been shown that the D. Greenberger, M. Home, and A. Zeilinger argument can be developed without involving any spin, essentially by replacing the spin-component measurements by presence measurements performed by means of counters.
This chapter starts with the definition of postulates. Here, in quantum mechanics, the postulates concern the atomic and molecular properties, which are quite far removed from everyday experience. The postulates described here make no claim to those happy mathematical adjectives such as ‘rigorous’, ‘complete’, or irreducible. The ‘proof’ of the postulates is their ability to explain the adequate experimental observations. Here, we have incorporated six postulates. The first postulate opines that ‘the state of quantum mechanical objects is described by a wavefunction’, whereas the second postulate says that ‘every physical observable is expressed by a linear operator’. It also describes how a quantum mechanical operator is constructed. The third postulate gives an idea that the measurement of a physical observable will yield a result, which is one of the Eigen values of the corresponding operator for that observable. The fourth postulate gives how the average of a sequence of measurement is connected with an operator. The fifth postulate is called the time-dependent postulate. Here, it gives a relationship between the time-dependent wave function and an operator, whereas the sixth postulate says that the wave function must be antisymmetric (or symmetric) for the exchange of identical fermions. It must be noted that each and every postulate has been explained clearly. The references, solved problems, and questions on concepts have been given at the end of the chapter.
This chapter discusses some comments, and the rules of quantum mechanics. The quantum mechanics is universally valid for all the ensembles to which the more general formalism just alluded to is applicable. Any ensemble of systems the specification of which involves no reference to the results of future measurements that are to be performed on these systems obeys the quantum rules. One of the views most often expressed by the supporters of the "Copenhagen interpretation" is that quantum mechanics is a complete theory. The main motivation for the assumption is that the splitting of ensemble and subensembles cannot be made in any straightforward manner. However, this assumption implies the existence of particular, hitherto unknown forces, and it is therefore quite generally considered as unattractive. Thus the probabilities, or statistical distributions, that conventional quantum mechanics introduces into physics are inherently different from those found in classical mechanics, for example, in the kinetic theory of gases.
In most textbooks, measurement does not receive the full attention it deserves and sometimes is even not treated at all, apart a brief and cryptic mention of the “reduction of the wave packet.” However, in the last decades, the situation has profoundly changed and it is time to consider measurement a fundamental part of quantum mechanics, even, to a certain extent, an important generalization of the traditional theory (see also Chs. 14–15). This chapter consists of three major parts. In the first block (Secs. 9.1–9.4) we develop the main physical features of the measurement process: the heart of the argument is here represented by the aspects related to the interpretation. In the second part (Secs. 9.5–9.8) we discuss several special (and partly interdependent) topics of measurement: the heart here is represented by experimental aspects. In the third part (Secs. 9.9–9.12) we deal with the measurement process on a more formal plane, making use, in particular, of the generalization represented by the concepts of effect and positive operator valued measure (POVM). As we have said, the measurement problem is one of the most fundamental issues in the conceptual structure of quantum mechanics (as described in Sec. 9.1) and has a long history that will be examined in Sec. 9.2. In this context, the existence of apparently paradoxical quantum states comes about: the so-called Schrödinger cat states (see Sec. 9.3).
This book is not meant to contain a detailed analysis of the foundations of quantum mechanics. However, we felt an introduction to the general theory was required in order to make the presentation self-sufficient and clear the following chapters from possible ambiguities. We shall, wherever possible, avoid the intricacies of the mathematical description of infinite-dimensional spaces, as well as the conceptual subtleties related to the interpretation of the formalism (although the approach taken will necessarily betray aspects of the author's stand on these matters), and content ourselves with a ‘pragmatic’ introduction to the theory, as the chapter's title suggests. Throughout the chapter, we will even occasionally refer to a hypothetical ‘user’ of the theory as an artefact to elucidate relationships between the formalism and its operational interpretation. Notions related to quantum non-locality and the information content of quantum states, such as entropies and the characterisation of quantum correlations (also known as entanglement), will be the subject of the second half of the chapter, as they are very relevant to the material covered in the book.