## 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 $$V(x,y,z)=V$$.