Derivation of the Schr¨odinger Equation from Feynman's Path Integral: Explicit Scale Cut-off and Gaussian Integration

this paper explicitly reconstructs the derivation of the time-dependent Schr¨odinger equationfrom Richard Feynman’s path integral formulation (1948). By rigorously applying the scalecut-off condition (η ∼ √ϵ) and evaluating the Gaussian probability amplitudes term by term,we demonstrate the bottom-up emergence of the Hamiltonian operator without relying ona priori top-down classical assumptions.