Fundamental Assumptions

Quantum Mechanics divides the world into two parts, commonly called the system and the observer.  Except at specified times the system and the observer do not interact.  An interaction at those specified times is called a measurement.  Quantum Mechanics predicts all the information that the observer can possibly obtain about the system.  This information can be represented in different ways.  It is often represented in terms of a wave function.  A measurement changes the information an observer has about the system and therefore changes the wave function of the system.

Fundamental assumptions of Quantum Mechanics

In a particular representation and applied to a system consisting of a single, structureless particle the fundamental assumptions of Quantum Mechanics are:

The quantum state of a particle is characterized by a wave function y(r,t), which contains all the information about the system an observer can possibly obtain.
The wave function y(r,t)is interpreted as a probability amplitude of the particles presence.  |y(r,t)|2 is the probability density.  The probability that a particle is at time t in a volume element d3r situated at r is  dP(r,t)=C|y(r,t)|2d3r.  For a single particle the total probability of finding it anywhere in space at time t is equal to 1.  (In non-relativistic Quantum Mechanics material particles, unlike photons, are neither created nor destroyed.)

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A proper wave function must be square-integrable.

Link:
Probability Illustrator

The principle of spectral decomposition applies to the measurement of an arbitrary physical quantity A.
The result of a measurement belongs to a set of eigenvalues {a}.
Each eigenvalue is associated with an eigenfunction ya(r).
If y(r,t0)=ya(r) then a measurement of A at t=t0 will yield the eigenvalue a.
Any y(r,t0) can be expanded in terms of eigenfunctions, 
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The probability that a measurement at t=t0 will yield the eigenvalue a’ is

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If a measurement of A yields a, then the wave function immediately after the measurement is ya(r).
The Schroedinger equation describes the evolution of y(r,t);

 

is the Schroedinger equation for a particle of mass m whose potential energy is given by V(r,t).