ψ[x_,n_]=(Sqrt[(2/L)])Sin[((π n)/L)x]

ψ[y,m]

Integrate[ψ[x,n]ψ[x,m],{x,0,L}]

FullSimplify[Integrate[ψ[x,n]ψ[x,m],{x,0,L}]]

L=1

n=5
Plot[ψ[x,n],{x,0,L}]
Plot[((ψ[x,n])^2),{x,0,L}]