Variational Calculus and Lagrangian Multipliers, simple example

Let us specialize to the expectation value of the energy for one particle in three-dimensions. This expectation value reads $$ E=\int dxdydz \psi^*(x,y,z) \hat{H} \psi(x,y,z), $$ with the constraint $$ \int dxdydz \psi^*(x,y,z) \psi(x,y,z)=1, $$ and a Hamiltonian $$ \hat{H}=-\frac{1}{2}\nabla^2+V(x,y,z). $$ We will, for the sake of notational convenience, skip the variables \( x,y,z \) below, and write for example \( V(x,y,z)=V \).