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 d^{3}r situated at **r** is

dP(**r**,t) = C|Ψ(**r**,t)|^{2}d^{3}r.

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)d^{3}r = 1,
C∫_{all space}|Ψ(**r**,t)|^{2}d^{3}r = 1,
∫_{all space}|Ψ(**r**,t)|^{2}d^{3}r = 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**,t_{0}) = Ψ_{a}(**r**) then a
measurement of A at t = t_{0 }will yield the eigenvalue a.

Any Ψ(**r**,t_{0}) can be expanded in terms of eigenfunctions,

Ψ(**r**,t_{0}) = ∑_{a}c_{a}Ψ_{a}(**r**).

The probability that a measurement at t = t_{0} 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))**∇**^{2}Ψ(**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).