Modern atomic theory

Modern atomic theory:
Based on:

(Wave-Particle Dualism) the dual nature of the electron
Heisenberg Uncertainty Principle

Finding the mathematical expression that describes:

Wave motion of electron
Shape and energy of that wave motion

The dual nature of the electron: experimental data showed that the electron has a dual nature which means that it has:

A wave property
A particle property

This means that: {Every moving body such as electron – nucleus – molecules is associates with a wave motion or matter waves known as (MATTER WAVES) which have properties differ from electromagnetic waves}
The matter wave motion differs from the electromagnetic waves in
Matter waves:

They are not separating from moving body
Their speed is not = speed of light  

Electromagnetic waves:

They are separating from moving body
Their speed is = speed of light

Heisenberg Uncertainty Principle: Bohr's atomic model postulated that it is possible to determine both the position (location) and the velocity of the electron at the same time
BUT
By applying the principles of quantum mechanics Heisenberg concludes that:

{It is impossible to determine both the position and velocity of the electron exactly at the same time}. 

We can only say it is probably to a greater or loser extant to locate the electron in this or that place Because if we image that the electron moves with certain velocity in certain direction, we must use photon to measure its position, and then this photon will change the velocity of the electron
Schrödinger mathematical expression: He uses the ideas of Blank, Einstein, De Broglie and Heisenberg to show that:

It is possible to determine the allowed energy level (orbital) of the electron
It is possible to define the region of space around the nucleus where it is most probable to find the electron in each orbital
The motion of electron around the nucleus has wave properties, therefore it is prefer to use the term of electron cloud to describe orbital

Electron cloud: It is the region of space around the molecules where it is probable to find the electron in it
Difference between orbit and orbital:
Orbit

Used in Bohr's theory
Circular planer with particular radii

It has a constant value It has not a constant value

Orbital

Used in Schrödinger's wave mechanics theory
Its radius is not constant
It has not a constant value