The Hilbert Space Formulation of Quantum Physics

Abstract

In this chapter, we introduce the basic concepts of quantum theory. In Section 1.1, the state-space of a quantum physical system, the Hilbert space, is presented in axiomatic form and the concept of a closed linear manifold (subspace) is defined. In Section 1.2, we investigate the algebra of the subspaces of a Hilbert space and show that these subspaces form an orthocomplemented quasimodular lattice, which, moreover, has some additional properties. Closed linear manifolds are very closely related to projection operators, which are introduced in Section 1.3. On the other hand, projection operators represent observable quantities of the physical system. A physical system is characterized by its state and by its properties. These concepts will be defined in Section 1.4, and their relations to the elements and the subspaces of Hilbert space will be established.