A regularisation method to obtain analytical solutions to de Broglie-Bohm wave equation

TL;DR

The paper develops an information-theoretic regularization framework to derive analytical solutions of the de Broglie-Bohm equations by coupling Hamilton-Jacobi, continuity, and Fisher information with a parametric coupling mu.A generalized Euler-Lagrange formalism is employed to obtain closed-form, regularized wave functions for selected 1D and 2D stationary potentials, avoiding delta-function artifacts typical of standard approaches.The approach yields Schrödinger-like dynamics without presupposing hbar a priori and reproduces standard energy spectra (with minor regularization-induced shifts) for free, harmonic-oscillator, and Coulomb-like systems.Overall, the method offers a systematic route to analytic dBB-based wave functions and energy states, with potential extensions to higher dimensions, open systems, and numerical comparisons.

Abstract

We present an analytical method to solve de Broglie Bohm equation for the wave function by combining concepts from the Hamilton Jacobi equations of mechanics, continuity equations, and information theory. From a statistical point of view, the probabilistic description of particle motion provides a middle ground for transitioning from classical to quantum mechanics. An action functional obtained by coupling a Fisher information term produces a parametrically equivalent form of the de Broglie Bohm dBB equations. Next, we show that generalized Euler Lagrange equations can provide analytical solutions of dBB equations for certain simple stationary systems. One- and two-dimensional examples are provided to illustrate the similarities and differences between regular QM and the wave functions obtained from our generalization. To retain the generality of the method, we do not use hbar apriori but instead introduce a parametric information-error coupling term, mu. Our emphasis is strictly limited to the regularization method and its implications for energy states. We defer formal interpretations of the foundational aspects of this subject to a separate communication.