We'll treat the problem only for a two-particle system.
Consider two identical particles which can be in the individual states ,which include the spin-dependence and are orthogonal.
The physical state of the system is:
Suppose we want to measure on each particle the physical quantity
What is the probability of finding
for one particle and
for the other?
The final state is:
The desired probability is given by
for bosons and
for fermions, that is:
is called direct term
, while the term
is called exchange term
This shows the appearance of a sort of interference phenomenon when we're dealing with identical particles.