Observables and states in the Hilbert space formalism of quantum mechanics

Abstract

In this chapter the main elements of the Hilbert space formalism of quantum mechanics are summed up briefly. Usually this formalism is considered the formalism of quantum mechnics but it is not the only and not even the most “natural” one. We shall see in Chapter 6 that the so-called “algebraic” quantum mechanics is equally important. The Hilbert space formalism has a long history that goes back to the twenties and it was worked out rigorously by von Neumann in [168]. In Chapter 7 we shall have to say a few words about von Neumann’s contribution to establishing this formalism systematically and about his subsequent critical attitude towards the Hilbert sapce formalism. In the present chapter only the two key concepts, the observables and states and their mathematical representatives are discussed. Summarizing the formalism also serves the purpose of fixing some of the notations that will be used subsequently. As far as possible, the formalism is presented here without any interpretation, that is, without addressing and discussing those controversial interpretative issues that have been with us since the birth of this theory. One of these problems is whether what is called “quantum logic” can be viewed as the logic of the quantum system. This issue will be discussed in Section 5.3.