ψ[x_,y_,nx_,ny_]=2(Sqrt[(1/(Lx Ly))])Sin[((π nx)/Lx)x]Sin[((π ny)/Ly)y]

D[ψ[x,y,nx,ny],{x,2}]

FullSimplify[Integrate[ψ[x,y,10,3]ψ[x,y,1,2],{x,0,Lx},{y,0,Ly}],Assumptions->{nx∈Integers,ny∈Integers,mx∈Integers,my∈Integers}]

Lx=1
Ly=1

Plot3D[ψ[x,y,2,3],{x,0,Lx},{y,0,Ly},PlotPoints->50]
Plot3D[((ψ[x,y,2,3])^2),{x,0,Lx},{y,0,Ly},PlotPoints->50]

nx=2;
ny=2;
DensityPlot[ψ[x,y,nx,ny],{x,0,Lx},{y,0,Ly},PlotPoints->50,Mesh->False,AspectRatio->1]
DensityPlot[((ψ[x,y,nx,ny])^2),{x,0,Lx},{y,0,Ly},PlotPoints->50,Mesh->False,AspectRatio->1]