Superposition of quasi coherent Bohm Madelung waves

Abstract: We examine the problem of superposition of stationary quantum states in the Bohm Madelung formulation, where amplitude and phase obey coupled nonlinear equations and linear superposition is not generally valid. In the quasi-coherent regime of near degenerate stationary branches, the dynamics separates into a hierarchical structure: the mean amplitude evolves according to an Ermakov-Pinney equation governed by a Wronskian invariant, while the difference amplitude obeys a parametrically driven Hill Mathieu equation determined by the energy splitting. Despite this intrinsic nonlinearity, a linear spectral structure re-emerges through a Jacobi Anger expansion, yielding a Fourier Bessel representation with square-summable coefficients and translation covariant weights. Applications to aperture geometries and spatially separated sources demonstrate how amplitude modulation and phase induced sidebands organise interference patterns within a nonlinear amplitude phase framework. Keywords Bohm Madelung, Ermakov Pinney, nonlinear superposition, quasi coherent states, Mathieu Hill, Fourier Bessel, quantum interference.

• The paper develops a perturbative superposition framework in which the mean amplitude follows an Ermakov–Pinney equation while near-degenerate differences obey a forced Hill/Mathieu equation.
• Quasi-coherent states produce an exact Jacobi–Anger Fourier–Bessel expansion with square-summable sidebands, whose norm agrees with the averaged amplitude envelope through first order in modulation.
• The framework models apertures and translated sources, but remains limited to first-order, stationary, largely separable regimes and does not yet demonstrate experimentally distinct predictions from standard quantum superposition.

The paper "Superposition of quasi-coherent Bohm–Madelung waves" (2605.03324) addresses a structural question in the polar (amplitude–phase) representation of quantum mechanics: since the Bohm–Madelung equations for $R(x)$ and $S(x)$ are coupled and nonlinear, linear superposition of wavefunctions does not translate into any primitive operation on amplitudes and phases. The author constructs a controlled regime — near-degenerate, quasi-coherent stationary branches of a free particle or constant-potential system — in which a consistent superposition structure nevertheless emerges. The central result is hierarchical: the mean amplitude obeys a closed Ermakov–Pinney equation fixed by a Wronskian invariant, the difference amplitude obeys a parametrically driven Hill/Mathieu equation set by the energy splitting $\Delta E$, and despite this nonlinearity a linear spectral structure reappears through a Jacobi–Anger expansion into Fourier–Bessel sidebands.

Background: superposition in the polar representation

Writing stationary states as $\psi_j = R_j e^{iS_j/\hbar}$, each branch independently satisfies a quantum Hamilton–Jacobi equation,

$$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$

together with the continuity constraint $(R_j^2 S_j')' = 0$. Because these equations are nonlinear in $R$ and $S$, adding two solutions at the amplitude level is not equivalent to adding wavefunctions. The paper positions itself against prior treatments of Bohmian superposition via trajectory dynamics and open systems (2605.03324), arguing instead that superposition should be understood as an emergent structure valid only within controlled dynamical regimes. A useful preliminary observation is that when two branches are strongly dissimilar in amplitude and phase, decoherence is strong enough that they may be treated as effectively independent; the interesting case is therefore the near-degenerate one.

Hierarchical decomposition of near-degenerate branches

For two branches with nearly equal energies, the author introduces the splitting $R_{1,2} = R_0 \pm \rho$ and $S_{1,2} = S_0 \pm \Delta S/2$, subject to $S(x)$0, $S(x)$1. Adding and subtracting the Hamilton–Jacobi equations yields coupled equations for the mean and difference sectors. In the leading order, the mean branch reduces to the standard Ermakov–Pinney form

$S(x)$2

where $S(x)$3 is the stationary flux invariant and $S(x)$4. The difference amplitude satisfies

$S(x)$5

i.e., a linear oscillation evolving over the nonlinear Ermakov background. This hierarchy — nonlinear mean dynamics carrying a linearised perturbation — is explicitly analogous to small orbital oscillations about a closed nonlinear trajectory, where stability is governed by Hill- or Mathieu-type equations.

A key reduction follows when the branches are equi-energy ($S(x)$6): the difference equation becomes homogeneous,

$S(x)$7

a Mathieu–Hill equation whose periodic coefficient is inherited from the spatial modulation of the quasi-coherent background. The author notes plainly that this reduction is not automatic: an amplitude-level Ermakov superposition requires both branches to share the same leading phase, a constraint absent from naive amplitude-addition pictures.

Wronskian-controlled phase and the quasi-coherent branch

The phase in the stationary sector is nonlocal, determined by the current invariant through $S(x)$8. Choosing Pinney solutions built from $S(x)$9, $\Delta E$0 with Wronskian $\Delta E$1, and imposing the constraint $\Delta E$2 with $\Delta E$3, the quasi-coherent amplitude takes the form

$\Delta E$4

An adiabatic expansion of $\Delta E$5 away from zeros of $\Delta E$6 then integrates to

$\Delta E$7

so that to leading order the phase is algebraic in $\Delta E$8 with only $\Delta E$9 oscillatory corrections. The distinction drawn here between "near-degeneracy" (two weakly differing branches) and "quasi-coherence" (a single Ermakov amplitude from two linearly independent solutions linked by their constant Wronskian) is important: quasi-coherence is not conventional phase coherence but a Wronskian-controlled compatibility with a stationary current.

Emergent spectral linearity via Jacobi–Anger expansion

Substituting the quasi-coherent amplitude and phase into the parent wavefunction and applying the Jacobi–Anger expansion produces

$\psi_j = R_j e^{iS_j/\hbar}$0

with coefficients that simplify, via the Bessel recurrence relation, to the compact principal-index form

$\psi_j = R_j e^{iS_j/\hbar}$1

This is the paper's strongest analytical result: although the underlying amplitude–phase dynamics is nonlinear, the state admits an exact plane-wave decomposition with square-summable coefficients. Using standard Bessel identities, the coefficient norm evaluates to

$\psi_j = R_j e^{iS_j/\hbar}$2

which matches the spatial average of the amplitude envelope, $\psi_j = R_j e^{iS_j/\hbar}$3, to first order in $\psi_j = R_j e^{iS_j/\hbar}$4. Absolute convergence follows from the factorial decay $\psi_j = R_j e^{iS_j/\hbar}$5, which dominates the polynomial prefactor. The physical interpretation is a discrete momentum distribution centered at $\psi_j = R_j e^{iS_j/\hbar}$6 with Bessel-weighted sidebands generated by nonlinear phase modulation — structurally identical to spectral broadening in nonlinear fibre optics. Notably, the local intensity $\psi_j = R_j e^{iS_j/\hbar}$7 is governed entirely by the envelope; the sideband structure manifests only in momentum space and in interference patterns, not in the density.

Two-dimensional apertures and the parabolic slit reduction

Extension to two dimensions proceeds under Cartesian separability with the local flux-closure condition $\psi_j = R_j e^{iS_j/\hbar}$8, which eliminates nonlocal current redistribution and preserves the Ermakov structure per direction. The rectangular aperture yields a product Fourier–Bessel lattice,

$\psi_j = R_j e^{iS_j/\hbar}$9

with box quantisation $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$0, $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$1.

For a narrow slit, the transverse direction is taken coherent ($$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$2) while modulation is retained along propagation. A parabolic reduction $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$3 maps the effective wavefunction onto a Fresnel-type form: the amplitude acquires an oscillatory envelope $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$4, while the phase factorises into Bessel-weighted quadratic chirp modes $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$5. The consequence is that the parabolic reduction converts the Bohm–Madelung aperture directly into a nonlinear optical slit in which both amplitude and phase contribute dynamically to diffraction — in contrast to classical formulations where only the phase carries the Fresnel structure.

Spatially separated sources and translation covariance

Two translated copies of the parent state at $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$6 generate symmetric and antisymmetric combinations. For small separation, Taylor expansion gives $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$7 and $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$8, so the symmetric branch isolates even-order interference while the antisymmetric branch captures the gradient response. In the spectral domain the construction is exact: translation acts as a phase rotation, weighting each odd harmonic by $$\frac{(S_j')^2}{2m} + V - \frac{\hbar^2}{2m}\frac{R_j''}{R_j} = E_j,$$9 (symmetric) or $(R_j^2 S_j')' = 0$0 (antisymmetric), preserving orthogonality and normalisation. Since observables depend only on derivatives of $(R_j^2 S_j')' = 0$1, the transformation $(R_j^2 S_j')' = 0$2 leaves all measurable quantities invariant, combining translation covariance with a gauge-like phase freedom.

Limitations and open questions

Several restrictions bound the validity of the results. All derivations hold to first order in the modulation parameter $(R_j^2 S_j')' = 0$3 and rely on the strict hierarchy $(R_j^2 S_j')' = 0$4, $(R_j^2 S_j')' = 0$5; the regime outside this perturbative window, where the Mathieu equation may enter parametric instability bands, is not analysed. The equi-energy reduction ($(R_j^2 S_j')' = 0$6) required for the homogeneous difference equation means genuinely energy-split superpositions are treated only at the level of the forced equation, without solving it. The flux-closure condition $(R_j^2 S_j')' = 0$7 in two dimensions excludes circulating-current configurations, limiting applicability beyond separable geometries. The connection to double-slit interference is asserted as "applicable" rather than demonstrated through explicit fringe calculations or comparison with experiment. Finally, the claim that the framework may illuminate coherence in light, matter waves, and macroscopic quantum systems remains qualitative; no quantitative prediction distinguishing this description from standard Schrödinger-picture treatment is offered, which raises the open question of whether the emergent spectral structure yields observable signatures beyond those of linear superposition.

Conclusion

The paper establishes that in the Bohm–Madelung formulation, superposition is not a primitive operation but an emergent spectral feature of nonlinear amplitude–phase dynamics. Within the quasi-coherent regime, the mean branch closes on an Ermakov–Pinney equation governed by a Wronskian invariant, the difference branch obeys a Mathieu–Hill equation over that background, and the Jacobi–Anger expansion restores a linear, square-summable Fourier–Bessel representation whose norm matches the spatially averaged envelope. Applications to rectangular apertures, parabolic slit reduction, and translated source pairs show how amplitude modulation and phase-induced sidebands organise interference within a nonlinear framework consistent with known phenomena in nonlinear optics. The analysis is confined to first-order perturbation theory in stationary, separable settings, and its empirical content relative to the standard linear picture remains to be established.