Postulates of Quantum Mehanics

The foundations of Quantum Mechanics are a setof postulates. 
We know that we have no proofs for any postulate. A postulate 
is a statement taken for true in order to process a reasonning, 
do computations and make interpretations. 

Quantum theory has no pure reality. It is only probabilistic 
as it is set in the so-called "Copenhagen interpretation". 

Postulate 1 
 
The state of a quantum system is
described by a function, ψ(x, y, z,t) or ψ(r,t) ,
This function is the state function or the wavefunction. It contains 
all the information that can be determined about the
system. This function is continuous, differentiable, and integrable.

All linear combination of wave functions is also a wave function.
This is the Principle of Linear Superposition of States.

We define a scalar product on a related set of wave functions 
as:
<ψ(r,t)|φ(r,t)> = ∫ ψ*(r,t)|φ(r,t)dr

The scalar product, <ψ(r,t)|φ(r,t)> is a real or
complex number.

Postulate 2

To every physical observable there
corresponds a linear Hermitian operator.
 
Suppose that A is an operator. Then, the
equation:
A ψ  = a ψ
is called an eigenvalue equation if , i.e., a constant. The
function ψ is referred to as an eigenfunction or
eigenvector of A with eigenvalue  a. "a" is either a
real or complex number.

The result of a measurement must be a real number. Then the 
related operator is Hermitian or self-adjoint.

Postulate 3 
 
The only possible values that can
arise from measurements of a physical observable
A are the eigenvalues ai of the eigenvalue
equation
A ψi = ai ψi


Postulate 4 

If A is a Hermitian operator
which represents a physical observable, A, then
the eigenfunctions ψi of the eigenvalue equation
A ψi = ai ψi
form a complete set.


Postulate 5 
 
If ψ(r,t) is the normalized state
function of a system at time t, then the average
value or expectation value of a physical
observable A at time t is
<A> = ∫ dr ψ*(r,t)A ψ(r,t),
where ψ* is the complex conjugate of ψ


Postulate 6 

The time development of the state of an undisturbed system 
is given by the time-dependent Schrödinger equation:
Hψ(r,t) = iℏ ∂ ψ(r,t)/∂t 

Where H is the Hamiltonian (energy operator).