FIRST POSTULATE :
The physical states of a system at time ' t' is described by the wavefunction,
ψ ( x, t)
SECOND POSTULATE :
The wave function ψ ( x, t) and it's first and second derivatives is,
∂ψ (x,t ) / dx and d² ψ ( x, t) / ∂x² are continuous, finite and single - valued for all values of x also, the wave function ψ ( x, t) is normalized.
That is,
Where ψᵅ is the complex conjugate of ψ formed by replacing i with − i, wherever it's occurs in the function ψ.
[ i = √−1 ]
THIRD POSTULATE :
The physically observable quantity can be represented by a Hermitian operator.
An operator A is to be Hermitian if its satisfies the following conditions.
∫ ψᵢ* Â ψⱼ dx = ∫ ψⱼ ( Â ψᵢ )* dx .......(2)
Where ψᵢ and ψⱼ are the wave functions and representing the physical states of the quantum system,
i.e, a particle, an atom or molecule.
FOURTH POSTULATE :
The allowed values of an observable A are the eigen values, a i, in the operator equation
 ψᵢ = aᵢ ψᵢ......................... (3)
Equation (3) is known as eigenvalue equation.
Here,
 - is the operator for the observable physical quantity
ψᵢ is an eigenfunction of  with eigen value aᵢ.
In other words,
Measurement of the observable A yields the eigen value aᵢ.
FIFTH POSTULATE :
The average value, < A> , of an observable A, is obtained from the relation
............ (4)
Where, the function ψ is assumed to be normalized in accordance with equation (1)
Thus, the average value of, it say, the x- coordinate is given by
................. (5)
SIXTH POSTULATE :
The quantum mechanical operators corresponding to the observable are constructed by writing the classical expression in terms of the variables and converting the expressions to the operators.
SEVENTH POSTULATE :
The wave function ψ (x, t) is a solution of the time dependent equation.
Ĥ ψ (psi) (x, t) = i ħ ∂ψ ( x, t )
∂t
Where,
Ĥ is the Hamiltonian operator for this system.
