A=-6.104263*10^-39;B=I*6.625*10^-34;
out=NDSolve[{A*D[\[CapitalPsi][x,t],x,x]+Sin[2\[Pi]*t]*\[CapitalPsi][x,t]+B*D[\[CapitalPsi][x,t],t]==0,\[CapitalPsi][0,t]==0,\[CapitalPsi][0.005,t]==0},\[CapitalPsi],{x,0,0.005},{t,0,1}]
sol=\[CapitalPsi]/.First[out]
Plot3D[Im[sol[x,t]],{x,0,0.005},{t,0,1},WorkingPrecision->100]
As you can see the solution is kinda boring; it is zero everywhere. That's because your boundary conditions are 0...