Postulates of Quantum Mechanics

Quantum mechanics deals with the wave nature of electron in which complete description of an electron is obtained as a mathematical function known as wave function. It has following main postulates-

Postulate I The state of a quantum-mechanical system is completely specified by a function Ψ(r, t) that depends on the coordinates of the particle and on time. This function, called the wave function or the state function, has the important property that the product of Ψ*(r, t) Ψ(r, t) dxdydz is the probability that the particle lies in the volume dxdydz located at r(x, y,z) at time t.

Note that completely specified means that the wave function contains all the information that can be obtained about the system using quantum mechanics provides tremendous motivation to solve the Schrödinger equation and find the explicit wave function.

Corollary In order for the wave function to be used in the Schrödinger equation, it must have several mathematical properties:
1. It must be finite.
2. It must be continuous and single valued.
3. It must be defined for at least first and second derivatives.

Postulate II To every observable laboratory measurement in classical mechanics there corresponds an operator in quantum mechanics.

Corollary Cartesian coordinates (x, y, z), spherical polar coordinates (r, θ, φ, or in general any set of coordinates, q, merely become multiplicative operators, while the corresponding momentum operators, P_q, become differential operators such as (ћ δ/i δ_q).

Postulate III In any measurement of the observable associated with the operator A, the only exact values that will ever be observed are the eigenvalues a_j which satisfy the eigenvalue equation
AΨ_j = a_jΨ_j

Corollary If the state function is not an eigenfunction of the operator A, then only an average value can be obtained as from many measurements; see Postulate IV.

Postulate IV If a system is in a state described by a normalized wave function Ψ, then the average value of the observable corresponding to the operator A is given by

Corollary If the wave function is not normalized, then the average value of the observable corresponding to the operator A is given by

Postulate V The wave function or state function of a system evolves in time according to the time dependent Schrödinger equation
HΨ(r,t) = i ћ(δΨ/δt)

Source: Essentials of Physical Chemistry by Don Shillady