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Ermakov-Lewis Invariants in Stationary Bohm-Madelung Quantum Mechanics

Anand Aruna Kumar

TL;DR

This work shows that Ermakov–Pinney (EP) equations and the Ermakov–Lewis (EL) invariant emerge naturally in stationary Bohm–Madelung form for diagonal, separable Hamiltonians. After expressing the Schrödinger equation in a self-adjoint Sturm–Liouville framework and applying Liouville normalization, each separable coordinate yields a nonlinear EP equation for the amplitude with a coordinate-independent EL invariant, while the quantum potential is encoded as curvature of the Liouville-normalized operator. The authors demonstrate the construction in canonical 1D systems (free particle, harmonic oscillator, Coulomb) using EP amplitudes based on Weber and Whittaker/Macdonald-type solutions, and discuss well-posedness, Hamilton–Jacobi connections, and gauge implications. The results offer a unified, invariant-based interpretation of stationary Bohmian guiding fields and suggest a constrained variational perspective for stationary quantum dynamics, extendable to other separable geometries.

Abstract

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.

Ermakov-Lewis Invariants in Stationary Bohm-Madelung Quantum Mechanics

TL;DR

This work shows that Ermakov–Pinney (EP) equations and the Ermakov–Lewis (EL) invariant emerge naturally in stationary Bohm–Madelung form for diagonal, separable Hamiltonians. After expressing the Schrödinger equation in a self-adjoint Sturm–Liouville framework and applying Liouville normalization, each separable coordinate yields a nonlinear EP equation for the amplitude with a coordinate-independent EL invariant, while the quantum potential is encoded as curvature of the Liouville-normalized operator. The authors demonstrate the construction in canonical 1D systems (free particle, harmonic oscillator, Coulomb) using EP amplitudes based on Weber and Whittaker/Macdonald-type solutions, and discuss well-posedness, Hamilton–Jacobi connections, and gauge implications. The results offer a unified, invariant-based interpretation of stationary Bohmian guiding fields and suggest a constrained variational perspective for stationary quantum dynamics, extendable to other separable geometries.

Abstract

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.
Paper Structure (22 sections, 84 equations, 1 figure, 2 tables)

This paper contains 22 sections, 84 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Stationary BM Formulation and EP Structure
  3. Continuity Equation and Constraint
  4. EP Structure and the EL Invariant
  5. Sturm--Liouville encoding of the quantum potential
  6. Implication for trajectories (stationary fields).
  7. Canonical One-Dimensional Examples
  8. Free particle: Bohm continuity -> Ermakov amplitude and invariant
  9. Harmonic oscillator: Weber basis and Bohmian Ermakov amplitude
  10. Why the Weber Basis?
  11. Coulomb potential
  12. Well-Posedness of the Stationary Bohm–Ermakov Problem
  13. Relation to Hamilton--Jacobi Separability
  14. Implications
  15. Well-Posedness Criteria and Invariant Constraints
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Geometry of the symmetric two--center Coulomb problem in two dimensions. The test charge is influenced by two identical charges $+Ze$ located at the foci $x=\pm a$ of an ellipse. The distances $r_1$ and $r_2$ enter naturally in confocal elliptic coordinates.