Introduction to Quantum Chemistry - Postulates of Quantum Theory

Quantum Mechanics

-         It’s is mechanics of small particles; electron, protons, photons, …etc. that describe their properties.

-         In quantum mechanics we talk about system which may be a group of particles (ex: group of electrons)

Postulates of quantum mechanics: 

Postulate (1):

For every state of system consist of (n) particles, they can be described with wave function, which is function of coordinates and time.

This function has some conditions to act as wave function, these conditions are: -

1)    Single value

2)    Continuous

3)    Finite through space

Interpretation of wave function:-

-         Y itself has no physical meaning

-         The square of the function is proportional to the probability at any point.

Postulate (2):

For every experimentally observable variable (i.e. mechanical property), such as momentum, position, or energy, there is a corresponding linear mathematical operator.

 

-     Any operator is designed with a “ “above the quantity and capital letter.

-         Operator means a mathematical operation or group of operations that carried out on a function to obtain new function such as multiplication by any number or square root …. Etc.

-         Example of mechanical properties and their corresponding operators are given in the following table:- 

Postulate (3):

Any mechanical observable is calculated using the Eigen function equation

Where:

-         Y is Eigen function

-     A is operator

-         a is Eigen value (constant)

Postulate (4):

The expected (mean) value of any observable is calculated using:

There are some characters are sharp, i.e. have only one value. But there are other characters have more than one value this character undergoes average value. 

Born interpretation of wave function

Born stated that Y corresponds to square root the probability density (it is the probability of finding particle per unit volume). In other words, the square of the function Y is proportional to the probability of finding the particle at this point.

Hamiltonian operator:

The Hamiltonian operator 

Where,  T is the kinetic energy operator, V is the potential operator

When Hamiltonian operator applied to the wave function of the system it yields the energy of this system (E) as Eigen value multiplied by the wave function of the system as the Eigen function.

In other words, the wave function of the system must be Eigen function of the Hamiltonian operator and its Eigen function is the energy of this system (E)