Postulate of Operators and State Functions

The fundamental connection between the classical and quantum mechanics is achieved through the operators concept. They symbolize the processes, the operations, through which a (measurable) quantity/property or a function turns in another structural property or in another function. For example, the square root V when operates to the function /turns it in a different function There is clear therefore how the quantum formal- 

The first postulate of the quantum mechanics refers thus to the correspondence principle by which the functions and the classical quantities (as coordinate, momentum, energy, or orbital momentmn) become operators, having as the connection element (of correspondence) the Planck constant h or its reduced form h = h/(27r). The list of these correspondences is shown in Table 3.1. 

In Table 3.1, the S5mibol means the object under focus upon which the quantum operation/operator is applied. 

One also notes how only the space dependency function retains the function quality and not becomes a quantum operator (i.e., multiplies with the identity operator which leaves invariant any other function upon which is applied). 

TABLE 3.1 The Classical (left) to Quantum (right) Correspondences for the Main Structural Properties of Matter (Putz, 2006) 

---