**Postulate III.-** Each physical observable in Quantum Mechanics corresponds to a linear and Hermitian operator. To find such an operator, we write the mechanoclassical expression of the observable in terms of the Cartesian coordinates and the corresponding linear moments. Next, we replace each x-coordinate with the operator $\hat{x}$ (multiply by x) and each momentum \(p_x\) with the operator \(-i\hbar\frac{\partial}{\partial x }\).

Let's see how this postulate works in the construction of the most important operators of Quantum Mechanics.

**Position operator of a particle**

**Linear momentum operator of a particle.**

**Operator kinetic energy of a particle**

In Classical Mechanics, kinetic energy is given by the following expression:

We write this equation in terms of linear momentum \(p_x = mv_x\).

Substituting the amount of movement by:

The quantum mechanical operator kinetic energy is obtained

**hamiltonian operator**