Ermakov Invariants in Stationary Quantum Mechanics: A Bohm-Madelung and Hamilton-Jacobi Perspective

The Ermakov-Pinney (EP) equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schr\"{o}dinger equation is written in Bohm-Madelung (BM) form and the Hamiltonian is diagonal and separable. Under these conditions, the quantum continuity constraint induces a nonlinear amplitude equation of EP type for each degree of freedom, revealing a hidden invariant structure independent of whether the evolution parameter is time or space. This framework is illustrated using the one-dimensional harmonic oscillator, clarifying the role of the second independent solution, which is typically suppressed in standard quantum mechanics. The results establish Ermakov invariants as an intrinsic amplitude-space structure underlying separable stationary problems and motivate further investigation using extended variational formulations of Bohmian dynamics.