ABSTRACT
Dirac introduced a very useful notation of state vectors Ψi in terms of "bra" vectors. This notation allows to make the formal expressions of quantum physics more transparent and easier to manipulate. The bra-ket notation extends to action of operators on state vectors. The projection operator and the completeness relation are very important quantities in quantum physics. They are very useful in calculations and are often employed to represent an arbitrary operator and a wave function (state) in terms of orthonormal states. To illustrate the procedure of representations, this chapter considers an arbitrary operator  and shows how to represent the operator in terms of projection operators of known orthonormal states. The example in the chapter illustrates the concept of orthonormal states. It presents the application of the completeness relation. Using the completeness relation, that matrix elements of the product operator in the same orthonormal basis can be found from the multiplication of the matrix elements.
