Why two individual sources can never be coherent? (Wave theory).

To understand why two separate light sources cannot be coherent, let us first consider the meaning of the term  coherent sources.

Two sources are said to be coherent if the phase difference (if any) between the waves of same frequency produced by the two sources remains constant with time. 

Let us consider the working of a simple light source like an incandescent bulb.

When an electric current passes through the filament of the bulb, the filament gets heated up as a result of its high resistance. This thermal energy excites the electrons of the filament material to higher excited states. When the electrons return to their ground states by de-excitation, energy is liberated in the form of light waves. 

Each excitation-de-excitation event produces a light wave. However, the precise occurrence of such an event is unpredictable and can occur at any time. All the available electrons do not get excited or de-excited simultaneously. Hence, two light sources cannot produce light waves of constant phase difference because the time lag between the generation of light waves by the two sources is unpredictable and random and hence, even the phase difference is continuously changing.

 

NOTE: The above explanation takes the simple example of an incandescent light bulb, but it is to be noted that the above explanation is valid even if other light sources like LEDs are used or if instead of light waves, sound waves are used because the basic principle is the same.