BOSONS VS. FERMIONS
For a two-particle system, the wave function is 
.
The system is a function of the coordinates of both particle one and particle
two. There are two cases to consider. First, the case when the particles
are distinguishable gives the solution for the wave function as 
.
For the case when the particles are indistinguishable, the solution is
. Classically,
we can view all particles as distinguishable because we can label them
to keep track of which particle is which, but in quantum mechanics the
situation is much more complicated. There is no way to observe particles
without changing their state. When dealing with two electrons, they are
identical for all practical purposes. This is why quantum systems are
considered to consist of indistinguishable particles. The solution for
the wave function has two possible values which correspond to the + and
- cases: .
The + case is for bosons and the - case is for fermions. Two fermions
cannot occupy the same state, but two bosons can occupy the same state.
To show the difference between fermions and bosons, I will animate the
probability density for both bosons and fermions. 
The Interference term is simplified to 
.
Now integrate over the second particle which results in 
.
My animations show the differences in the probability densities of bosons
and fermions.
The bottom part of the animations represents the gaussian wave functions of the two particles with one particle at rest in the center. The top part is the probability density of both the fermion and the boson case. The one with the dip in the middle is for fermions. The reason for this is because there is destructive interference since no two fermions can occupy the same state whereas for bosons there is constructive interference since two bosons can occupy the same state. The first graph is with a small intial displacement between the particles.
Click on the picture below for the animated gif -or-
click here for an mpg file
This second graph shows the two particles with a wide intial displacement. In this case, it is almost impossible to tell the difference between the fermion and boson cases. The reason for this is because when the initial displacement is large, the momentum is very large and consequently the difference in momentum of the two particles is large and the interference term is almost negligible.
Click on the picture below for the animated gif -or-
click here for an mpg file