1. The Postulates of Quantum Mechanics

2. Postulate I: the Board and the Game Tools
The system state can be described by a wavefunction defined in the domain of all possible states of the system.
For a single particle the possible state space is the set of the spatial coordinates in which the particle can reside*
For a system composed of numerous particles the state space includes all the states of all the particles. The aspiration is to move to a reduced description of the system (for example: moving to the center of mass coordinates )
* In the future an additional coordinate will be considered. For example the spin coordinate.

3. Postulate II: the Rules of the Game
The change of the system state in time can be described by the evolution operator:
The evolution operator is a linear operator, which fulfills properties of a group (Unitary Operator)
The generator of the evolution group is the energy operator (The Hamiltonian)
The eigenfunctions of the Hamiltonian operator are also the eigenfunctions of the evolution operator (stationary states)

4. Postulate III: the Trial Interface
A measuring action performed on the system can be described as an action of filtering the wavefunction into its components.
Each measurable quantity has a corresponding Hermitian Operator, whose eigenfunctions constitute a basis set for filtering. Each eigenfunction has a corresponding real eigenvalue which constitutes a possible result of the measurement:
The state of the system can be written as a superposition of the basis states:

5. Postulate III: the Trial Interface
The result of the measurement is the realization of the system in only one of its basis states. The measuring process causes a reduction of the superposition (wave-like) to a single component (particle-like)
The probability for a system in a state | to be realized in a basis state|n is:
The measurement produces for each particle an experimental single value, equal to the eigenvalue n of the realized eigenstate |n

6. The Normalization Requirement
The following must be fulfilled for the square of the wavefunction to have a probability density significance:
On the other hand:
And therefore:

7. Average and Expected Values in Statistics
When a fair dice whose faces are marked by the digits 1,2,2,4,4,4 is tossed once, the domain of possible events is:
If the dice is tossed N times, the average value obtained is:
After numerous tosses the value ni/N converges to the probability for event Pi and the expected value is:

8. Average and Expectation Values in Quantum Mechanics
For each single measurement an eigenvalue of the measuring operator is obtained:
The domain of possible events is {n}
An experimental measurement is composed of a huge number of single measuring actions, and therefore the average equals the expectation value. The experimental result is called “the expectation value of the observation”: