Main Question or Discussion Point
In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian operator as the Legendre transform of the Hamiltonian operator in quantum mechanics? Or a similar derivation for the Lagrangian density and Hamiltonian density in quantum field theory via the Legendre transform?