Proof of self-adjointness of negative Laplacian on an interval

Question

Consider the negative Laplacian $-\Delta:=-\frac{d^2}{dx^2}$ on $L^2(0,1)$ with domain $$D(-\Delta)=\{f\in H^2(0,1)\,:\,f(0)=f(1)=0\}.$$ How does one prove that $-\Delta:L^2(0,1)\supset D(-\Delta)\to L^2(0,1)$ is a self-adjoint operator? I know how to show symmetry, but I don't know which way to go after.