Superposition of quasi coherent Bohm Madelung waves

TL;DR

This paper investigates the superposition of stationary quantum states in the Bohm Madelung framework, revealing a hierarchical nonlinear structure that can be analyzed using Fourier Bessel expansions and applied to interference phenomena.

Contribution

It introduces a hierarchical nonlinear model for superposition in Bohm Madelung quantum states and demonstrates a linear spectral structure through a Fourier Bessel approach.

Findings

Amplitude evolves via Ermakov Pinney equation governed by Wronskian invariant.

Difference amplitude follows a parametrically driven Hill Mathieu equation.

Fourier Bessel representation captures interference patterns in nonlinear regimes.

Abstract

The problem of superposition of stationary quantum states in the Bohm Madelung formulation is examined in a regime 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 one dimensional shifted 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.