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arXiv:2605.03324

Stationary Bohmian superposition under amplitude and phase modulation

Anand Aruna Kumar

quant-phSubmitted May 5, 2026• Announced May 6, 2026• 0 citations• Revised May 25, 2026 (v3)
v1 · May 5, 2026v2 · May 7, 2026v3 · May 25, 2026
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Abstract

In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering two near degenerate stationary branches, we derive a hierarchical reduction in which the mean amplitude satisfies an Ermakov Pinney equation, while the difference amplitude evolves through a forced Mathieu Hill type modulation induced by energy and stationary current differences. It is shown that energy coherence alone does not uniquely determine phase coherence, since independent stationary currents continue to enter both the modulation and phase difference equations. For weak amplitude modulation, a Wronskian based stationary branch obtained from an Ermakov Pinney solution admits a controlled amplitude phase construction, leading to an algebraic phase representation and a Jacobi Anger spectral expansion. As a result, a linear spectral structure emerges through Bessel weighted amplitude and phase modulation. Such a representation is naturally suited for modelling aperture geometries, as illustrated by rectangular and parabolic slit reductions exhibiting Fresnel type phase chirp and modulation driven sidebands. The present construction therefore provides an analytical route by which linear spectral superposition reemerges from nonlinear Bohmian amplitude phase dynamics. Keywords Bohmian mechanics; stationary superposition; amplitude and phase modulation; nonlinear superposition; Mathieu Hill equation; Fourier Bessel expansion.

Comments
Rev3. Twelve pages condensed and major revisions in title, organisation, with updated references. Resubmitting to a different journal

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In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering two near degenerate stationary branches, we derive a hierarchical reduction in which the mean amplitude satisfies an Ermakov Pinney equation, while the difference amplitude evolves through a forced Mathieu Hill type modulation induced by energy and stationary current differences. It is shown that energy coherence alone does not uniquely determine phase coherence, since independent stationary currents continue to enter both the modulation and phase difference equations. For weak amplitude modulation, a Wronskian based stationary branch obtained from an Ermakov Pinney solution admits a controlled amplitude phase construction, leading to an algebraic phase representation and a Jacobi Anger spectral expansion. As a result, a linear spectral structure emerges through Bessel weighted amplitude and phase modulation. Such a representation is naturally suited for modelling aperture geometries, as illustrated by rectangular and parabolic slit reductions exhibiting Fresnel type phase chirp and modulation driven sidebands. The present construction therefore provides an analytical route by which linear spectral superposition reemerges from nonlinear Bohmian amplitude phase dynamics.
Keywords Bohmian mechanics; stationary superposition; amplitude and phase modulation; nonlinear superposition; Mathieu Hill equation; Fourier Bessel expansion.
The Ermakov Pinney equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schrodinger 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.
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.
We investigate the stationary amplitude and current properties of a charged particle in a uniform magnetic field within the Bohm--Madelung formulation. The analysis is organised around two complementary questions: the effect of steady currents on amplitude evolution, and the influence of the singular radial behaviour on the energy spectrum of the Landau problem. For the zero-current system, corresponding to the standard Landau problem, the Hamilton--Jacobi equations lead to an Ermakov--Pinney reference structure for the separated amplitudes. The azimuthal sector additionally admits periodic branches in which the magnetic flux is constrained by the angular periodicity condition. Allowing nonvanishing component currents under global current conservation introduces nonlinear amplitude evolution, most prominently through the magnetic-vector-potential coupling in the azimuthal sector. Under the minimum choice of equal and opposite radial and azimuthal currents, the radial contribution remains analytically tractable, while the nonlinear angular momentum and action are bounded through two complementary solvable comparison equations. The Bohmian formulation also makes it possible to examine directly the singular structure of the radial energy equation. A Fisher-based regularisation associated with the current-branching construction is shown to closely parallel the role of Langer- and JWKB-type radial regularisation, while arising from the Bohm--Madelung amplitude--momentum description itself. The resulting modification of the radial index re-orders the Landau energy spectrum and lifts the corresponding azimuthal degeneracy. Keywords: Landau problem; Bohm--Madelung mechanics; nonlinear current dynamics; Ermakov--Pinney equation; radial regularisation; comparison bounds; spectral splitting.
Interference is commonly regarded as the most direct manifestation of the superposition principle. This association is natural for the linear Schrödinger equation, where coherent alternatives combine at the level of probability amplitudes. However, the situation becomes less transparent when nonlinear couplings are present, or when the field is only partially coherent. In this work, we argue that a more robust organizing principle is provided by the local flow generated by phase variations. In this sense, phase-induced flow acts as a unifying mechanism for interference-like dynamics in nonlinear and partially coherent Schrödinger systems. The discussion is developed from a hydrodynamic, or Bohmian, perspective, understood here as a practical probing tool rather than as an additional ontology. Three representative situations are considered: interfering Bose--Einstein condensates described by the Gross--Pitaevskii equation, nonlinear Schrödinger dynamics obtained by modifying the quantum-potential contribution, and partially coherent Airy beams described through their cross-spectral density. Although these systems differ in physical origin and mathematical implementation, they share a common dynamical structure: density-related observables are shaped by velocity fields determined by phase, or ensemble-phase, information. From this viewpoint, interference-like traits, localization, self-acceleration and coherence loss can be interpreted in terms of the preservation, deformation or breaking of the symmetries displayed by the underlying flow. This provides a compact way of connecting interference, nonlinear dynamics, classicality, coherence loss, and structured-light propagation within a single trajectory-based framework.
We develop a variational regularisation framework that enables analytical solutions of the stationary de~Broglie--Bohm wave equation. The formulation begins with a Fisher-information-augmented action functional for the probability density and phase fields, yielding the Madelung (Hamilton--Jacobi and continuity) equations and, upon complex recombination, a Schrödinger-type equation with a parametric information coupling μμμ. Beyond this density-based formulation, we introduce a variational regularisation scheme for the de~Broglie--Bohm equations that combines a global Fisher-information regularisation at the level of the action functional with a shell-level regularisation arising from stationary flux closure. This reduction isolates the regularisation mechanism in the spatial momentum flow and yields constrained Euler--Lagrange equations governing admissible amplitude configurations. The resulting first integral possesses an elliptic structure whose admissible asymptotic branch enforces a universal canonical relation p(x)x→μ/2p(x)x \to μ/2p(x)x→μ/2 near amplitude zeros. The framework yields closed-form analytical solutions for standard potentials and reveals a systematic inverse-square regularising term in the effective potential. The associated elliptic discriminant defines a geometric length scale that, for μ=ℏμ=\hbarμ=ℏ, naturally reduces to the reduced Compton wavelength. Canonical Bohmian regularisation therefore appears as a variational admissibility condition on density dynamics, producing structurally stable analytical branches and modified yet consistent energy spectra within stationary dBB mechanics.
The paraxial wave equation is formally equivalent to the Schrödinger equation, enabling quantum-mechanical concepts to be applied to classical optical fields. Within this framework, we investigate the Bohmian dynamics of superpositions of Laguerre-Gaussian beams. We derive the velocity field and quantum potential and analyze the resulting trajectory dynamics. Interference between modes with different orbital angular momentum generates vortex formation, trajectory exclusion from nodal regions, and spiral motion around phase singularities. Remarkably, despite the complexity of the Bohmian trajectories, the vortex cores follow nearly linear paths along propagation. The dynamics are governed by the amplitude ratio and OAM indices, while the relative phase only induces global rearrangements. These results establish a coherent trajectory-based description of structured light and highlight vortex cores as robust topological features.