Volume 12, Issue 2, pages 1-22
The Ermakov–Pinney equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schr\”{o}dinger equation is expressed in Bohm–Madelung form and the Hamiltonian is diagonal and separable. Under these conditions, the stationary continuity constraint induces a nonlinear amplitude equation of Ermakov–Pinney type in each degree of freedom, revealing a hidden invariant structure that is independent of whether the evolution parameter is time or space. By reformulating the separated stationary equations in Sturm–Liouville form and applying Liouville normalization, we demonstrate that the quantum potential is encoded as a curvature contribution of the self-adjoint operator rather than appearing as an additional dynamical term. This correspondence preserves the standard probabilistic predictions of quantum mechanics while yielding exact stationary Bohmian amplitudes and their associated invariants. The resulting invariant-based formulation provides stationary guiding fields and clarifies the ontological status of Bohmian amplitudes as geometrically encoded structures rather than auxiliary dynamical additions. The results further show that stationary constrained Bohm–Madelung systems naturally admit variational formulations whose extremals preserve the Ermakov–Lewis invariant.
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