general form:
psi[x_,n_]=Sqrt[2/L] Sin[n Pi x/L]
Integrate[psi[x,1] psi[x,2],{x,0,L}]
Integrate[psi[x,2] psi[x,2],{x,0,L}]
Integrate[psi[x,1] x psi[x,2],{x,0,L}]

        -16 L
Out[4]= -----
            2
        9 Pi

dimensionless form; i.e., xnew=x/L, tnew=t/T to simplify
see: http://www.physics.csbsju.edu/QM/square.03.html
psi[x_,n_]=Sqrt[2] Sin[n Pi x]
w[n_]=n^2 Pi^2

ppsi[x]=(psi[x,1] Exp[-I w[1] t]+psi[x,2] Exp[-I w[2] t])/Sqrt[2]
Animate[Plot[Abs[ppsi[x, t]]^2,{x, 0, 1},PlotRange->{0, 3.3}],{t,0,2/(3 Pi)}]
Integrate[Abs[ppsi[x, 3]]^2, {x, 0, 1}] 
