This chapter discusses the mathematical formalism of the theory of quantum mechanics and present the differences between the theory of classical mechanics and quantum mechanics. The mathematics that is used to account for the indeterminacy of quantum mechanics renders the mathematical approach of quantum mechanics distinct from that of classical mechanics. In quantum mechanics, any dynamical variable that is an observable that can be measured experimentally is described as a linear Hermitian operator. In quantum mechanics, it is not only necessary to define an observable, but it is also necessary to define the states associated with it. When a system is in a linear superposition of states of an observable, the number of different values of an observable observed in an experiment is equal to the number of states that a quantum mechanical system can jump into upon measurement of an observable.