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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=dxdydzψ(x,y,z)ˆHψ(x,y,z), with the constraint dxdydzψ(x,y,z)ψ(x,y,z)=1, and a Hamiltonian ˆH=122+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.