u[n_,x_]=Sqrt[2/a] Sin[Pi n x/a]

psid=u[n,x1] u[l,x2]
psif=(u[n,x1] u[l,x2]-u[n,x2] u[l,x1])/Sqrt[2]
psib=(u[n,x1] u[l,x2]+u[n,x2] u[l,x1])/Sqrt[2]

Integrate[Integrate[(x1-x2)^2 psid^2,{x1,0,a}],{x2,0,a}]
Simplify[%,Element[{l,n},Integers]]
N[%] /. {n->1,l->2}

         2      2    2        2   2
        a  (-3 n  + l  (-3 + n  Pi ))
Out[6]= -----------------------------
                    2  2   2
                 6 l  n  Pi

                  2
Out[7]= 0.103341 a

Integrate[Integrate[(x1-x2)^2 psif^2,{x1,0,a}],{x2,0,a}]
Simplify[%,Element[{l,n},Integers]]
N[%] /. {n->1,l->2}

          2      10   2    10   2        2   2       6  4   2        2   2
Out[9]= (a  (-3 n   Pi  + l   Pi  (-3 + n  Pi ) + 6 l  n  Pi  (-1 + n  Pi ) + 
 
          2  8   2       2   2     8     2   2      4   4
>        l  n  Pi  (9 + n  Pi ) + l  (9 n  Pi  - 4 n  Pi ) - 
 
            4  4                l + n       2   2      4   4
>        2 l  n  (192 (-1 + (-1)     ) + 3 n  Pi  + 2 n  Pi ))) / 
 
         2        4  2        4   4
>    (6 l  (l - n)  n  (l + n)  Pi )

                   2
Out[10]= 0.168232 a

see the term (-1+(-1)^M)...that is zero if M=even, i.e., if l & n have same parity

           2      10   2    10   2        2   2
Out[12]= (a  (-3 n   Pi  + l   Pi  (-3 + n  Pi ) + 
 
            6  4   2        2   2     2  8   2       2   2
>        6 l  n  Pi  (-1 + n  Pi ) + l  n  Pi  (9 + n  Pi ) + 
 
          8     2   2      4   4
>        l  (9 n  Pi  - 4 n  Pi ) - 
 
            4  4                 l + n       2   2      4   4
>        2 l  n  (-192 (-1 + (-1)     ) + 3 n  Pi  + 2 n  Pi ))) / 
 
         2        4  2        4   4
>    (6 l  (l - n)  n  (l + n)  Pi )

                    2
Out[13]= 0.0384498 a

<(x1-x2)^2> is the average separation between the particles squared.  Clearly
this is smallest for boson (i.e., bosons go together) and largest for fermions
(i.e., fermions stay apart)




