Ghose’s Interpolating Equation in Quantum-Classical Dynamics

• Ghose’s interpolating equation is a nonlinear modification where a continuous parameter λ attenuates the quantum potential, enabling a smooth transition from quantum to classical dynamics.
• It bridges stochastic quantum mechanics, Madelung hydrodynamic formulations, and Koopman–von Neumann classical mechanics while preserving the underlying Schrödinger structure.
• The polar decomposition into amplitude and phase variables clarifies how controlled suppression of the quantum potential influences interference effects and classical behavior.

Ghose’s interpolating equation is a nonlinear modification of Schrödinger dynamics in which a continuous parameter $\lambda$ suppresses the quantum potential $Q[\psi]$ and thereby yields a one-parameter transition between quantum and classical dynamics. In the formulation presented in “Quantum, Stochastic, and Classical Dynamics Within A Single Geometric Framework,” the equation is

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$

with $\psi=\sqrt{\rho}\,e^{iS/\hbar}$ and quantum potential

$$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$

In that source, the equation is identified as one “first proposed by Ghose (2002)” and is used as the central bridge between Nelson-type stochastic quantum mechanics, hydrodynamic/Madelung dynamics, and Koopman–von Neumann classical mechanics (Ghose, 31 Oct 2025).

1. Defining form and basic variables

Ghose’s interpolating equation is written in configuration space for a wavefunction $\psi(x,t)$, particle mass $m$, and classical potential $V(x,t)$ as

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi.$$

The notation $Q[\psi]$ is essential: the quantum potential depends on $Q[\psi]$0 through its amplitude $Q[\psi]$1, where $Q[\psi]$2. Because the term $Q[\psi]$3 depends nonlinearly on the wave amplitude, the equation is nonlinear in $Q[\psi]$4 (Ghose, 31 Oct 2025).

The standard polar decomposition

$Q[\psi]$5

splits the dynamics into amplitude and phase variables. In this representation, $Q[\psi]$6 plays the role of the Hamilton–Jacobi phase and $Q[\psi]$7 the role of a transported density. The quantum potential is the standard Madelung expression

$Q[\psi]$8

which is the term selectively attenuated by $Q[\psi]$9.

The defining feature of the construction is not a deformation of the kinetic or classical potential terms, but a deformation of the quantum-potential sector alone. That is the sense in which the equation “interpolates”: it preserves the overall Schrödinger-type structure while continuously reducing the contribution that distinguishes quantum Hamilton–Jacobi dynamics from classical Hamilton–Jacobi dynamics.

2. Interpolation parameter and limiting regimes

The interpolation parameter satisfies

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$0

Its role is to suppress the quantum potential continuously. At $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$1, Ghose’s interpolating equation reduces exactly to the ordinary Schrödinger equation,

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$2

At $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$3, the equation becomes

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$4

which the source identifies as the “classical Schrödinger” or Rosen equation (Ghose, 31 Oct 2025).

The limiting interpretations are explicit. The case $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$5 is standard quantum mechanics. The case $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$6 removes the quantum potential from the Hamilton–Jacobi sector, leaving the classical Hamilton–Jacobi equation coupled to continuity transport. Intermediate values $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$7 retain a residual quantum correction proportional to $$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$8.

This structure is best seen in the hydrodynamic form. The phase equation becomes

$$i\hbar \frac{\partial \psi}{\partial t} = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V - \lambda Q[\psi] \right]\psi, \qquad 0\le \lambda \le 1,$$9

Thus $\psi=\sqrt{\rho}\,e^{iS/\hbar}$0 does not alter the continuity equation or the velocity field definition; it changes only the strength of the quantum-potential contribution in the phase dynamics. The interpolation is therefore continuous at the level of Hamilton–Jacobi theory rather than being merely a formal replacement of one linear wave equation by another.

The same source states that $\psi=\sqrt{\rho}\,e^{iS/\hbar}$1 is introduced phenomenologically. The parameter is not derived there from a microscopic theory; instead, it is inserted to encode a controlled transition from stochastic/quantum to classical dynamics. A plausible implication is that the equation is best understood as an effective deformation scheme rather than as a completed fundamental theory.

3. Madelung structure and stochastic origin

In polar variables, the equation decomposes into two coupled real equations. The continuity equation remains unchanged:

$\psi=\sqrt{\rho}\,e^{iS/\hbar}$2

The phase equation is the modified Hamilton–Jacobi equation

$\psi=\sqrt{\rho}\,e^{iS/\hbar}$3

This pair makes the interpolation especially transparent: density transport remains of continuity type for all $\psi=\sqrt{\rho}\,e^{iS/\hbar}$4, while the phase dynamics shifts from quantum to classical as the coefficient of $\psi=\sqrt{\rho}\,e^{iS/\hbar}$5 is reduced (Ghose, 31 Oct 2025).

The stochastic background used in the same paper is Nelson’s stochastic mechanics. There the Schrödinger equation is written as

$\psi=\sqrt{\rho}\,e^{iS/\hbar}$6

with the identification

$\psi=\sqrt{\rho}\,e^{iS/\hbar}$7

Here $\psi=\sqrt{\rho}\,e^{iS/\hbar}$8 is the diffusion coefficient associated with forward–backward Brownian motion. Under that identification, the stochastic evolution reproduces standard Schrödinger dynamics.

The paper describes a “$\psi=\sqrt{\rho}\,e^{iS/\hbar}$9–$$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$0 hierarchy” in which $$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$1 labels the stochastic scale responsible for quantum behavior and $$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$2 independently controls suppression of the quantum potential. In that construction, the origin of $$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$3 is stochastic, while the interpolation parameter is an added deformation. This yields a hybrid picture: stochastic mechanics motivates the quantum-potential term, and the $$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$4-deformation organizes the passage toward classical behavior.

This also clarifies why the equation is nonlinear. The nonlinearity is not an arbitrary extra potential, but the result of feeding back the amplitude-dependent functional $$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$5 into the wave equation itself. In hydrodynamic variables, the same effect appears as a controlled attenuation of the quantum correction in the Hamilton–Jacobi equation.

4. Classical endpoint and Koopman–von Neumann embedding

At $$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$6, the quantum-potential term disappears from the phase equation, leaving

$$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$7

together with

$$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$8

These are the classical Hamilton–Jacobi and continuity equations. In the source, this limit is not treated as an isolated endpoint, but as the entry point to Koopman–von Neumann (KvN) classical mechanics (Ghose, 31 Oct 2025).

The KvN evolution equation is given in Hilbert-space form as

$$Q[\psi] = -\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt{\rho}}{\sqrt{\rho}}.$$9

This is the Liouville-operator representation of classical mechanics in phase space. Writing

$\psi(x,t)$0

one obtains Liouville’s equation for the phase-space density $\psi(x,t)$1 and a convective equation for the phase $\psi(x,t)$2.

The bridge from Ghose’s interpolating equation to KvN mechanics is established by restricting the phase-space density to a single Lagrangian manifold,

$\psi(x,t)$3

Under this monokinetic, single-sheet ansatz, the Liouville dynamics reproduces exactly the classical Hamilton–Jacobi plus continuity system obtained from the $\psi(x,t)$4 limit. The source therefore concludes that the classical endpoint corresponds to KvN mechanics restricted to a single Lagrangian sheet.

The same discussion emphasizes that this restriction is not the full classical theory. General classical ensembles require phase-space mixtures,

$\psi(x,t)$5

Accordingly, the $\psi(x,t)$6 limit yields exact classical configuration-space dynamics, but only for a special class of Liouville solutions. Full Liouville/KvN classical mechanics reappears only after allowing multi-sheet or mixed phase-space distributions.

5. Geometric interpretation and phase superselection

A major interpretive layer in the same source is geometric. Quantum pure states are associated with the complex projective Hilbert manifold $\psi(x,t)$7, while the classical endpoint is associated with a quotient structure described as $\psi(x,t)$8, with some notational inconsistency also noted in the source text (Ghose, 31 Oct 2025).

Within that framework, $\psi(x,t)$9 is interpreted as a “continuous projection flow from configuration-space Hilbert dynamics (quantum) to phase-space Hilbert dynamics (classical).” It is also described as a contraction of the quantum symplectic contribution. The proposed interpolating symplectic structure is written schematically as

$m$0

As $m$1, the quantum contribution vanishes, interference structure is removed, and the dynamics contracts toward the classical phase-space description.

Phase superselection is central to this interpretation. In KvN mechanics, the phase of $m$2 is convected but unobservable because classical observables act by multiplication on phase-space functions. The source therefore treats identification of states differing only by $m$3 phase as part of the classicalization mechanism. In this picture, classicality is not only suppression of $m$4, but also elimination of physically meaningful interference phases.

The paper also stresses that the $m$5 “classical Schrödinger equation” still retains a single-wave description and therefore a vestige of nonclassical structure. It explicitly notes that this representation can still exhibit non-crossing trajectories. This indicates that suppression of the quantum potential alone does not automatically produce the most general or fully dephased classical theory; phase superselection and phase-space ensemble structure remain necessary.

6. Scope, limitations, and nomenclatural ambiguities

Several limitations are explicit. Ghose’s interpolating equation is nonlinear because $m$6 depends on $m$7 through $m$8. The interpolation parameter $m$9 is introduced phenomenologically rather than derived microscopically. The $V(x,t)$0 limit reproduces classical Hamilton–Jacobi plus continuity equations exactly, but only on a single Lagrangian sheet. Full classical Liouville behavior requires either KvN phase-space dynamics or incoherent mixtures of sheets (Ghose, 31 Oct 2025).

The arXiv record also shows that the label “interpolating equation” is easily conflated with unrelated constructions. In “Interpolating between random walk and rotor walk,” the relevant implicit equation is

$V(x,t)$1

which defines a doubly perturbed Brownian motion arising as the scaling limit of the $V(x,t)$2-rotor walk; that paper explicitly states that it does not use the phrase “Ghose’s Interpolating Equation” (Huss et al., 2016). In “Interpolating between torsional rigidity and principal frequency,” the interpolating object is the semilinear Dirichlet problem

$V(x,t)$3

derived from a variational quotient interpolating between torsional rigidity and the first Dirichlet eigenvalue; Ghose is not mentioned there either (Carroll et al., 2010). A further, distinct usage appears in “Polynomial parametrisation of the canonical iterates to the solution of $V(x,t)$4,” where the iterative differential equation

$V(x,t)$5

is explicitly connected to “Ghose’s interpolating equation” in a normalized inverse-iterate setting (Miyamoto, 2024).

These multiple usages imply that the name is not uniform across the literature represented here. In the quantum–classical transition literature, the most specific and direct meaning is the nonlinear wave equation with the $V(x,t)$6 term (Ghose, 31 Oct 2025). In other areas, structurally different interpolation equations may be described in Ghose-related or Ghose-type terms. The safest technical usage is therefore to specify the equation explicitly rather than rely on the name alone.