From the Schrödinger Equation to a Unified Master Formula: Amplitude–Phase Dynamics with Dissipation and Decoherence

In this work, we derive a unified master formulation for quantum amplitude-phase dynamics, starting directly from the time-dependent Schrödinger equation (TDSE) . By applying the Madelung transformation and incorporating realistic dissipation and decoherence processes through non-Hermitian and Lindblad corrections, this formalism connects the evolution of pure states, the coherence dynamics of mixed states, and measurable phase shifts in precision experiments such as atomic clocks and interferometer systems. One important result is that we explicitly show why the amplitude term in the master formula appears as a linear combination of physically distinct contributions (baseline term, frequency (energy) shift, dissipation) rather than a single multiplicative scalar . This additive structure emerges naturally from the decomposition of the effective Hamiltonian under realistic environmental interactions. The ultimate master formula, It provides a concise and general representation of coherent quantum dynamics under decoherence, decay, and phase shift mechanisms. It includes simple numerical simulations and example code to demonstrate amplitude decay and phase changes. This research aims to provide a mathematically consistent bridge between fundamental quantum dynamics and effective models used in applied quantum technologies.