This note develops the final form of Lagrangian Quantum Mechanics and shows that it produces the correct wavefunctions of the hydrogen atom. It produces the same result as the usual hamiltonian quantization of the H atom given Eq. (15), a new fundamental equation of the H atom. The great advantage of this method is that it can be extended to all atoms and molecules given the general potential (23) between the electrons and protons of an atom or molecule. It therefore produces a new subject area of computational quantum chemistry. It also shows that a three dimensional orbit of a mass m about M, using the inverse square law, quantizes to the same type of wavefunctions as those of the H atom. This method can also be extended to relativistic quantum mechanics. I will write up UFT371 now, Sections 1 and 2 as usual.