QUANTUM MECHANICAL POSTULATES

The formulation of wave mechanics or quantum mechanics, by wave mechanical treatment of the, structure of the atom and rest upon a few postulates which, for a system moving in one - dimensional, and it's say, the x- coordinates are given below.

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,

    

.......... (1) 

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  Â 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.