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Ermakov Invariants in Stationary Quantum Mechanics: A Bohm-Madelung and Hamilton-Jacobi Perspective

Anand Aruna Kumar

Published Mar 20, 2026• 0 citations
Publisher

Abstract

The Ermakov-Pinney (EP) equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schr\"{o}dinger equation is written in Bohm-Madelung (BM) form and the Hamiltonian is diagonal and separable. Under these conditions, the quantum continuity constraint induces a nonlinear amplitude equation of EP type for each degree of freedom, revealing a hidden invariant structure independent of whether the evolution parameter is time or space. This framework is illustrated using the one-dimensional harmonic oscillator, clarifying the role of the second independent solution, which is typically suppressed in standard quantum mechanics. The results establish Ermakov invariants as an intrinsic amplitude-space structure underlying separable stationary problems and motivate further investigation using extended variational formulations of Bohmian dynamics.

DOI
10.20944/preprints202601.0205.v2

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The Ermakov-Pinney (EP) equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schr"{o}dinger equation is written in Bohm-Madelung (BM) form and the Hamiltonian is diagonal and separable. Under these conditions, the quantum continuity constraint induces a nonlinear amplitude equation of EP type for each degree of freedom, revealing a hidden invariant structure independent of whether the evolution parameter is time or space. This framework is illustrated using the one-dimensional harmonic oscillator, clarifying the role of the second independent solution, which is typically suppressed in standard quantum mechanics. The results establish Ermakov invariants as an intrinsic amplitude-space structure underlying separable stationary problems and motivate further investigation using extended variational formulations of Bohmian dynamics.
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.
In this work, we examine the problem of stationary superposition in the Bohmian amplitude phase formulation, where amplitude and phase obey coupled nonlinear equations and direct linear superposition is not generally preserved. Considering two near degenerate stationary branches, we derive a hierarchical reduction in which the mean amplitude satisfies an Ermakov Pinney equation, while the difference amplitude evolves through a forced Mathieu Hill type modulation induced by energy and stationary current differences. It is shown that energy coherence alone does not uniquely determine phase coherence, since independent stationary currents continue to enter both the modulation and phase difference equations. For weak amplitude modulation, a Wronskian based stationary branch obtained from an Ermakov Pinney solution admits a controlled amplitude phase construction, leading to an algebraic phase representation and a Jacobi Anger spectral expansion. As a result, a linear spectral structure emerges through Bessel weighted amplitude and phase modulation. Such a representation is naturally suited for modelling aperture geometries, as illustrated by rectangular and parabolic slit reductions exhibiting Fresnel type phase chirp and modulation driven sidebands. The present construction therefore provides an analytical route by which linear spectral superposition reemerges from nonlinear Bohmian amplitude phase dynamics. Keywords Bohmian mechanics; stationary superposition; amplitude and phase modulation; nonlinear superposition; Mathieu Hill equation; Fourier Bessel expansion.
The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schrödinger equation, which are determined by any two independent solutions to the classical equation of motion. Ermakov and Pinney have shown that a general solution to the time-dependent oscillator with an inverse cubic term can be expressed in terms of two independent solutions to the time-dependent oscillator. We explore the connection between linear quantum invariants and the Ermakov-Pinney solution for the time-dependent harmonic oscillator. We advance a novel method to construct Ermakov-Pinney solutions to a class of time-dependent oscillators and the wave functions for the time-dependent Schrödinger equation. We further show that the first and the second Pöschl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.
The time-evolution of the maximum and the width of exact analytic wave packet (WP) solutions of the time-dependent Schrödinger equation (SE) represents the particle and wave aspects, respectively, of the quantum system.The dynamics of the maximum, located at the mean value of position, is governed by the Newtonian equation of the corresponding classical problem.The width, which is directly proportional to the position uncertainty, obeys a complex nonlinear Riccati equation which can be transformed into a real nonlinear Ermakov equation.The coupled pair of these equations yields a dynamical invariant which plays a key role in our investigation.It can be expressed in terms of a complex variable that linearizes the Riccati equation.This variable also provides the time-dependent parameters that characterize the Green's function, or Feynman kernel, of the corresponding problem.From there, also the relation between the classical and quantum dynamics of the systems can be obtained.Furthermore, the close connection between the Ermakov invariant and the Wigner function will be shown.Factorization of the dynamical invariant allows for comparison with creation/annihilation operators and supersymmetry where the partner potentials fulfil (real) Riccati equations.This provides the link to a nonlinear formulation of timeindependent quantum mechanics in terms of an Ermakov equation for the amplitude of the stationary state wave functions combined with a conservation law.Comparison with SUSY and the time-dependent problems concludes our analysis.
We investigate the stationary amplitude and current properties of a charged particle in a uniform magnetic field within the Bohm--Madelung formulation. The analysis is organised around two complementary questions: the effect of steady currents on amplitude evolution, and the influence of the singular radial behaviour on the energy spectrum of the Landau problem. For the zero-current system, corresponding to the standard Landau problem, the Hamilton--Jacobi equations lead to an Ermakov--Pinney reference structure for the separated amplitudes. The azimuthal sector additionally admits periodic branches in which the magnetic flux is constrained by the angular periodicity condition. Allowing nonvanishing component currents under global current conservation introduces nonlinear amplitude evolution, most prominently through the magnetic-vector-potential coupling in the azimuthal sector. Under the minimum choice of equal and opposite radial and azimuthal currents, the radial contribution remains analytically tractable, while the nonlinear angular momentum and action are bounded through two complementary solvable comparison equations. The Bohmian formulation also makes it possible to examine directly the singular structure of the radial energy equation. A Fisher-based regularisation associated with the current-branching construction is shown to closely parallel the role of Langer- and JWKB-type radial regularisation, while arising from the Bohm--Madelung amplitude--momentum description itself. The resulting modification of the radial index re-orders the Landau energy spectrum and lifts the corresponding azimuthal degeneracy. Keywords: Landau problem; Bohm--Madelung mechanics; nonlinear current dynamics; Ermakov--Pinney equation; radial regularisation; comparison bounds; spectral splitting.
The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parametrization is at the heart of the recently formulated equivalence principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the quantum stationary HJ equation (QSHJE). This is a third order nonlinear differential equation which provides a trajectory representation of quantum mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p-q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second order linear differential equations. This reflects in an intrinsic ψ D -ψ duality between linearly independent solutions of the Schrödinger equation. Unlike Bohm's theory, there is a nontrivial action even for bound states and no pilot waveguide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave function is made. For example, tunneling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of particle's self-energy. Furthermore, the QSHJE is defined only if the ratio ψ D /ψ is a local homeomorphism of the extended real line into itself. This is an important feature as the L 2 (ℝ) condition, which in the Copenhagen formulation is a consequence of the axiomatic interpretation of the wave function, directly follows as a basic theorem which only uses the geometrical gluing conditions of ψ D /ψ at q=±∞ as implied by the EP. As a result, the EP itself implies a dynamical equation that does not require any further assumption and reproduces both tunneling and energy quantization. Several features of the formulation show how the Copenhagen interpretation hides the underlying nature of QM. Finally, the nonstationary higher dimensional quantum HJ equation and the relativistic extension are derived.