Axiomatic Construction of First Quantization and Deduction of the Schrödinger Equation - A theory for analytical Hamiltonians

This work develops non-relativistic quantum mechanics from four fundamental postulates, treated axiomatically, using the Hilbert space formalism of kets and operators. It shows that the canonical commutation relations, the correspondence with classical mechanics, and the identification of the Hamiltonian as the time-evolution generator arise naturally from this framework. The analysis highlights that the consistency of Schrödinger’s theory depends on the analyticity of the Hamiltonian, which ensures valid operator expansions and the quantum–classical correspondence. The generalized Schrödinger equation is then derived, unifying the abstract operator approach with the physical wave description under a single analytical foundation