The concept of Hilbert space is at the heart of the body of quantum mechanics. In this chapter, we introduce the concept of Hilbert space and develop the algebra and calculus of operators in it. It is preceded by introduction of the notions of ‘linear vector space’ and ‘scalar product space’ on which the concept of Hilbert space is based.

Vector Space

A set V of elements {|u, |vi, |w,…g, called vectors, is said to constitute a vector space V over the field F if it is closed under the operations of addition, and multiplication by scalars in F governed by following axioms:

The symbol ji, introduced by Dirac to denote a vector, is called a ket. In the study of the representation of the vector space, we will see that a vector can be represented in different ways. A representation of interest is the one in which it is represented by a column vector. We denote column vectors by a tilde under a letter. Thus, for example,ṵ ewould represent a column vector.

Because of its properties being similar to those of the scalar zero, the null vector j0i may be denoted by the scalar 0. Whether 0 in an expression involving vectors refers to the null vector j0i or to the scalar 0 is to be inferred from the context.

The vector spaces of interest to us are:

1. If the field F in the axioms defining the vector space V is the field of real numbers, then it is called the vector space over the field of real numbers or simply a real vector space.