The main focus of this book is many-particle systems such as electrons in a crystal. Such systems are studied within the framework of quantum mechanics, with which the reader is assumed to be familiar. Nevertheless, a brief review of this subject will provide an opportunity to establish notation and collect results that will be used later on.

The postulates

Quantummechanics is based on five postulates, listed below with some explanatory comments.

The quantum state

The quantum state of a particle, at time t, is described by a continuous, singlevalued, square-integrable wave function Ψ(r, t), where r is the position of the particle. In Dirac notation, the state is represented by a state vector, or ket, ∣Ψ(t)⟩, which is an element of a vector space V. We define a dual vector space V* whose elements, called bras, are in one-to-one correspondence with the elements of V: ket ∣α⟩ ∈ V ⟷ bra ⟨α∣ ∈ V*, as illustrated in Figure 1.1. The bra corresponding to ket c∣α⟩ is c* ⟨∣, where c*. is the complex conjugate of c. The inner product of kets ∣α⟩ and ∣β⟩ is denoted by ⟨β∣α⟩, and it is a complex number (c-number). Note that the inner product is obtained by combining a bra and a ket. By definition, ⟨β∣α⟩ = ⟨α∣β⟩*.