  The purpose of the paper is to study the foundations of the main axioms of
Quantum Mechanics. From a general study of the mathematical properties of the
models used in Physics to represent systems, we prove that the states of a
system can be represented in a Hilbert space, that a self-adjoint operator is
associated to any observable, that the result of a measure must be the eigen
value of the operator and appear with the usual probability. Furthermore an
equivalent of the Wigner's theorem holds, which leads to the Schr{\"o}dinger
equation. These results are based on well known mathematics, and do not involve
any specific hypothesis in Physics. They validate and explain the methods
currently used, which are made simpler and safer, and open new developments. In
the third edition of this paper developments have been added about the
estimation of physical anomalies.
