psi[x_,n_]=Sqrt[2] Sin[n Pi x]
w[n_]=n^2 Pi^2

ppsi[x_,t_]=(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}]