ABSTRACT The probability continuity equation is solved within the Madelung–Bohm framework, assuming a separable phase expressed as . Using operator methods, the amplitude of the wave function is reformulated in a form that is more practical for application to any given initial condition. This results in the function , which characterizes the amplitude of the wave function, with being the transformation of the position operator via a squeeze‐like operator. To demonstrate how this result can be applied, two scenarios are examined: one where the external potential is zero, causing the dynamics to originate only from the Bohm potential, and another where the form of mirrors the characteristics of wave propagation within optical waveguide arrays. In the second scenario, the amplitude, phase, and potentials might become complex, yet still comply with the Schrödinger equation.
ABSTRACT The probability continuity equation is solved within the Madelung–Bohm framework, assuming a separable phase expressed as . Using operator methods, the amplitude of the wave function is reformulated in a form that is more practical for application to any given initial condition. This results in the function , which characterizes the amplitude of the wave function, with being the transformation of the position operator via a squeeze‐like operator. To demonstrate how this result can be applied, two scenarios are examined: one where the external potential is zero, causing the dynamics to originate only from the Bohm potential, and another where the form of mirrors the characteristics of wave propagation within optical waveguide arrays. In the second scenario, the amplitude, phase, and potentials might become complex, yet still comply with the Schrödinger equation.
In this work, we show that the Madelung–Bohm trajectories determined by any solution, Ψ( ξ , ζ ), of the paraxial wave equation in an arbitrary two-dimensional optical medium, where ζ is the propagation coordinate and ξ the transversal one, are given by the contours of the function G ( ξ , ζ )≡∫|Ψ( ξ , ζ )| 2 d ξ . Furthermore, we remark that an analogous result follows directly for the one-dimensional Schrödinger equation for a particle evolving in an arbitrary classical potential, where ξ gives the position of the particle at time ζ . We apply this result to several examples in free and quadratic optical media. They include the Airy beam, Hermite–Gaussian beams, and some of their superpositions. Additionally, this work provides an alternative derivation of Hermite–Gaussian (HG) beams in free space by employing the quantum potential equations and the framework of quantum trajectories. By analyzing known trajectories, we demonstrate that the fundamental parameters characterizing these beams—the Gouy phase, beam radius, and wavefront curvature—emerge naturally within the quantum potential formalism.
The Schrödinger equation and the Born rule constitute two of the most successful structures in quantum mechanics. The former governs the continuous evolution of the wave function, while the latter connects its squared amplitude to the statistical distribution of experimental outcomes. Yet these two structures operate at different descriptive levels: the Schrödinger state is continuous, spatially extended, and phase-sensitive, whereas individual experiments terminate in localized physical events. This paper develops a theoretical framework for reconstructing the physical relation between these two levels. Rather than proceeding only in the conventional direction from Schrödinger dynamics to Born probabilities, we introduce an inverse explanatory strategy: from Born back to Schrödinger. Born statistics are treated as an empirical constraint from which the physical role of the wave function is reconsidered. We propose that, for the single-particle regime examined here, the wave function may be interpreted functionally as a dynamical propagation structure rather than as the localized particle itself. Its amplitude, phase, conserved current, gauge-covariant propagation momentum, and associated velocity field jointly define a structured family of possible motions. This interpretation leads to three separations: propagation from localization, propagation from physical population, and arrival from detector registration. A physical preparation generates a source measure (\mu_0^{\rm phys}); Schrödinger dynamics generates a propagation map (\Phi_\Psi); and microscopic detector dynamics is represented by a response map (\mathcal R_D). The Born problem can then be formulated as the closure condition RD[(ΦΨ)∗μ0phys]=?μBorn,D. This relation is not claimed here as a derived law, but as the central dynamical problem that a physical reconstruction of quantum probability must solve without inserting the Born measure at either boundary. The paper synthesizes six preceding studies on wave-packet localization, Born's scattering interpretation, the emergence of probability, the physical meaning of the wave function, source-to-detector transport, and unified propagation dynamics. It places the resulting framework in dialogue with Madelung and Hamilton–Jacobi formulations, Bohmian mechanics, decoherence, Quantum Darwinism, structural and symmetry-based derivations of the Born rule, Everettian approaches, objective-collapse theories, and detector models. The resulting proposal is best understood as a propagation-first theoretical reconstruction program: Schrödinger dynamics organizes possible propagation, while Born statistics constrain how that propagation becomes physically realized. Keywords: Schrödinger equation; Born rule; wave function; quantum foundations; propagation structure; quantum probability; quantum trajectories; source-to-detector transport; detector response; wave–particle duality; Bohmian mechanics; quantum measurement
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 μ.
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/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=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.
Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position-momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.
Comment on arXiv:2605.02621 [23] and arXiv:2606.05197 [24]:
In contrast to his earlier comment [23] on rspa.2025.0413 [12] and answered in arXiv:2605.20443 [11], the same author of [24] does not seem to dispute any longer that the use in [12] of tools from Feynman's [5] namely
- the kernel approach, which computes a wave starting at a given initial position or momentum, and is a key element in [12] to ensure that the Bohm-like quantum potential vanishes. This is directly verified in all examples of [12] as well as in the Pauli, Dirac and Maxwell derivations from classical action. This approach is very different from the standard Bohm-Madelung potential on the overall wave, as illustrated in detail in [11] for the two-slit experiment.
- for the harmonic oscillator, the kernel computation directly from a Taylor series expansion, which does not involve any circularity.
His second post [24] now
- criticizes the use in [12], [11] of Duru and Kleinert's time-independent eigenwave equivalence of the original and the time-scaled Schrödinger equations [4], confusing this standard result with a multivariable chain rule in the time-dependent kernel computation from action in [12].
- claims that "The mathematical framework presented by the authors remains identical to the well-established semiclassical Van Vleck propagator [sic]" which requires that the "Hamiltonian is at most quadratic in position and momentum [sic]". This confuses a quadratic action in the vector case with a harmonic propagated density based on the Laplacian of the action (with harmonicity always fulfilled e.g., for holomorphic complex functions [10]. In addition, it ignores the extension of a real position to a complex or quaternion position or spin, the key result on multi-valued action branches (which enables a classical derivation of quantum randomness) and the quantization of classical action in [12].
We show that two-mode squeezed vacuum-like states may be engineered in the Bohm–Madelung formalism by adequately choosing the phase of the wave function. The difference between our wave function and the one of the squeezed vacuum states is given precisely by the phase we selected. We would like to stress that the engineering of two-mode vacuum states is possible due to the existence of the Bohm potential, and it is relevant because of its potential use in the propagation of optical fields, where it may render a variety of applications in optics. The approach to generate non-classical states, namely, two-mode squeezed states of a quantum mechanical system is one of the first applications of the Madelung–Bohm formalism.