The Quantum Bridge: Emergence of Quantum Mechanics from an Amplitude–Phase State Function

We present a mathematical derivation showing how standard quantum mechanical structuresemerge from a complex state function written in amplitude–phase form. Using the state function℘ = Aeiθ , we demonstrate that the Born density, probability current, continuity equation, andHamilton–Jacobi structure arise without postulation. Probability is shown to be an emergentquantity resulting from coarse-graining of phase degrees of freedom. This construction providesa precise “quantum bridge” between deterministic amplitude–phase dynamics and conventionalquantum mechanics.