Fundamentals

Abstract

This chapter begins with the development of the formalism of quantum theory being sought. The first three sections deal with the early stages of the formalism, with setting up the language and the preliminary rules of computations. The procedure follows a method based on Schwinger’s elegant and incisive manner of direct analyses of selective measurements and extensions thereof, and has its roots in Dirac’s abstract presentation in terms of projection operators. Several examples of selective measurements will be given. The method developed provides tremendous insight into the physics behind the formalism and it leads naturally to the notion of probability associated with observations, to the generation of states, of wavefunctions in different descriptions and to various basic operations occurring in quantum mechanics, as well as to the emergence of Hermitian operators and inner-product spaces (§1.4). Preparation of pure ensembles of systems and mixtures is the subject matter of §1.5. In §1.6, the transmission of photons with given polarizations through polarizers is used to provide an illustration of rules developed earlier. The physical spaces in which computations are carried out are, in general, the Hilbert space, as an extension of the (finite) inner-product spaces encountered before, and the Rigged Hilbert space which are introduced in §1.7. Self-adjoint operators, representing observables, and their associated spectra are studied in §1.8. We will see that symmetry operations are implemented by either so-called unitary or anti-unitary operators and is the content of a famous theorem due E. P. Wigner (Wigner’s Theorem on symmetry transformations) which is proved in §1.9. This theorem is of central importance and a key one in quantum physics and deserves the special attention given here. The concept of probability and measurement with detailed illustrations are given in §1.10, emphasizing, in the process, the physical significance of a conditional probability associated with a measurement. This section also deals with non-ideal apparatuses that may disturb the physical system under consideration. Additional pertinent material related to this section will be given in §8.7–§8.9.