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

The quantum state of a particle is characterized by a wave function Ψ(r,t), which contains all the information about the system an observer can possibly obtain. 
The wave function Ψ(r,t) is interpreted as a probability amplitude of the particles presence.
|Ψ(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|Ψ(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.)

∫_{all space}P(r,t)d³r = 1,    C∫_{all space}|Ψ(r,t)|²d³r = 1,    ∫_{all space}|Ψ(r,t)|²d³r = finite.

A proper wave function must be square-integrable.

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 Ψ_a(r).
If Ψ(r,t₀) = Ψ_a(r) then a measurement of A at t = t₀ will yield the eigenvalue a.
Any Ψ(r,t₀) can be expanded in terms of eigenfunctions,

Ψ(r,t₀) = ∑_ac_aΨ_a(r).

The probability that a measurement at t = t₀ will yield the eigenvalue a' is

P_a' = |c_a'|²/∑_a|c_a|².

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

(iħ∂/∂t)Ψ(r,t) = (-ħ/(2m))∇²Ψ(r,t) + U(r,t) Ψ(r,t)

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