Kinetic Energy

The kinetic energy operator is given as:

Its expectation value is given by the following equation:

In answering the questions that were posed at the beginning of this section, the form and purpose of the wave function has been introduced.
So far, a mathematical structure in the formalism describing a quantum mechanical system has been constructed. The diagram that follows illustrates such a structure (Figure 1). Why do we calculate averages of observables in quantum mechanics?
As previously discussed, quantum mechanics has indeterminacy associated with the measurements of observables. Therefore, the only way to gain valuable information from the wave function is to evaluate statistically correct values of the observables.
However, how is a wave function for any system determined? The answer is by solving the Schrodinger equation.

FIGURE 1: This schematic describes the structure in the formalism of quantum mechanics. It shows that as a first step, observables must be defined, followed by the state vector and so on.

The time dependence of a state vector is excluded so far. To describe the time-dependence of a state vector of a quantum mechanical system or simply dynamics of a quantum mechanical system, the solution to the Schrodinger’s equation is required. This equation will be introduced in the section that follows.