Formation of quantum mechanics or wave mechanics for the wave mechanical treatment of the structure of an atom rests upon a few postulates for a system moving in one dimension along the x- coordinate as,
First Postulate: A system’s physical state at a time is described by the wave function Ψ (x, t).
Second Postulate: The wave function Ψ (x, t) and its first and second derivatives ꝺ Ψ (x, t)/ ꝺx and ꝺ2 Ψ (x, t)/ ꝺx2 are continuous, finite, and single-valued for all values of x. The wave function Ψ (x, t) is normalized, i.e.
Here, Ψ* = complex conjugate of Ψ formed by replacing I with -i it occurs in the function Ψ (i =√-1 )
Third Postulate: A Hermition operator is a physically observable quantity. An operator  is a Hermition if it satisfies the following condition;
Here, Ψi and Ψj is the wave function representing the physical state of a quantum system, like a particle, an ato or a molecule.
Fourth Postulate: Allowed values of observable A are the eigenvalues, ai, in the operator equation,
Equation (3) is called the eigenvalue equation.
 is an operator for the observable (physical quantity), and Ψi is an eigenvalue ai or the measurement of the appreciable A yields the eigenvalue ai.
Fifth Postulate: The expectation value, <A>, of an observable A, corresponding to the operator Â, from the relation,
Function Ψ is assumed to be normalized by equation (1). So, the average value of the x-coordinate is,
Sixth Postulate: Quantum mechanical operators related to the observable are constructed by writing the classical expression in terms of variables and converting the expression to the operators, as shown below,
Seventh Postulate: The wave function Ψ (x, t) is the solution of the time-dependent Schrödinger equation,
Here, Ĥ = Hamiltonian operator of the system
