$Assumptions = Element[{l,n},Integers] 


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

psi1=u[n,x1] u[l,x2]
psi2=(u[n,x1] u[l,x2]+u[n,x2] u[l,x1])/Sqrt[2]
psi3=(u[n,x1] u[l,x2]-u[n,x2] u[l,x1])/Sqrt[2]

Integrate[Integrate[ psi1^2,{x1,0,a}],{x2,0,a}]
Integrate[Integrate[ psi2^2,{x1,0,a}],{x2,0,a}]
Integrate[Integrate[ psi3^2,{x1,0,a}],{x2,0,a}]

Integrate[Integrate[(x1-x2)^2 psi1^2,{x1,0,a}],{x2,0,a}]
N[%] /. {n->2,l->3}

Integrate[Integrate[(x1-x2)^2 psi2^2,{x1,0,a}],{x2,0,a}]
N[%] /. {n->2,l->3}

Integrate[Integrate[(x1-x2)^2 psi3^2,{x1,0,a}],{x2,0,a}]
N[%] /. {n->2,l->3}

                   2
Out[31]= 0.148373 a

		     2
Out[33]= 0.0726835  a

		    2
Out[35]= 0.224062  a



p1=u[n,x1] u[l,x2]  /. {a->1,n->2,l->3}
ContourPlot[p1^2,{x1,0,1},{x2,0,1}]

p2=psi2  /. {a->1,n->2,l->3}
ContourPlot[p2^2,{x1,0,1},{x2,0,1}]

p3=psi3  /. {a->1,n->2,l->3}
ContourPlot[p3^2,{x1,0,1},{x2,0,1}]


