Hydrodynamic Schrödinger Equation (HSE)

• HSE is a framework that reformulates quantum wavefunctions into density and velocity fields via the Madelung transformation.
• It encompasses nonlinear, many-body, and stochastic extensions, facilitating the modeling of quantum fluids, superfluids, and decoherence phenomena.
• Advanced numerical methods and analytical techniques using HSE allow simulation of fluid dynamics, self-similar blow-up, and quantum–classical transitions.

The Hydrodynamic Schrödinger Equation (HSE) designates the coupled evolution equations for density and velocity fields derived from the (generally nonlinear and/or stochastic) Schrödinger equation under the Madelung transformation. It provides an exact or approximate hydrodynamic representation of quantum or quantum-like systems, and offers a framework for describing quantum fluids, superfluids, nonlinear media, stochastic quantum dynamics, and quantum–classical transitions. Modern research extends HSEs well beyond linear quantum mechanics, including many-body, dissipative, and classical hydrodynamic limits, with applications in quantum fluid modeling, quantum plasmas, nonlinear optics, barotropic and rotational fluid flow, and numerical simulation frameworks.

1. Mathematical Formulation via Madelung Transformation

The HSE originates by rewriting the complex wavefunction in polar (Madelung) form: $$\psi(\mathbf r,t) = \phi(\mathbf r,t) \exp \left(\frac{i}{\hbar} S(\mathbf r,t)\right)$$ with $\rho(\mathbf r,t) = \phi^2(\mathbf r,t)$ the density and $v(\mathbf r,t) = \nabla S/m$ the velocity field. Substituting into the time-dependent Schrödinger equation,

$$i\hbar \partial_t \psi = -\frac{\hbar^2}{2m} \nabla^2 \psi + V(\mathbf r, t) \psi,$$

and separating real and imaginary parts yields:

• Continuity equation (mass conservation):

$\partial_t \rho + \nabla \cdot (\rho v) = 0$

• Quantum Euler (momentum/Hamilton–Jacobi) equation:

$$m \left(\partial_t v + v \cdot \nabla v \right) = -\nabla V(\mathbf r, t) - \nabla Q(\mathbf r, t)$$

where the quantum potential is

$$Q(\mathbf r, t) = -\frac{\hbar^2}{2m} \frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}}$$

These two coupled equations constitute the (linear) hydrodynamic Schrödinger system (Wong, 2014).

2. Generalizations: Nonlinear, Many-Body, and Stochastic Extensions

Nonlinear/Many-Body Case:

For the Gross–Pitaevskii (GP) or general nonlinear Schrödinger equation (NLSE),

$$i\hbar \partial_t \psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + g|\psi|^2\right)\psi$$

the hydrodynamic form becomes: $$\begin{aligned} &\partial_t \rho + \nabla\cdot(\rho v) = 0 \ &m (\partial_t v + v\cdot\nabla v) = -\nabla (g\rho) - \nabla Q(\rho) \end{aligned}$$ where the mean-field pressure $g\rho$ arises from contact interaction (Barna et al., 2018). For many-body Hamiltonians with two-body interactions,

$$i\hbar\,\partial_t\Psi(R,t) = \hat H\Psi(R,t),\ \hat H = \sum_i \left[ -\frac{\hbar^2}{2m}\nabla_i^2 + V_{\text{ext}} \right] + \frac{1}{2} \sum_{i\neq j}q^2/|r_i-r_j|$$

a closure at the mean-field/Hartree level leads to a nonlinear HSE with nonlocal potentials (Andreev et al., 2014).

Stochastic/Noisy HSE:

Including vacuum-induced mass-density noise leads to a stochastic extension: $$m (\partial_t v + v\cdot\nabla v) = -\nabla V - \nabla Q + \eta(x,t)$$ with associated stochastic Schrödinger equation: $$i\hbar \partial_t \psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V + i\hbar\Gamma(x,t)\right)\psi + \psi \cdot \eta(x,t)$$ where $\eta(x,t)$ is a colored noise with correlation length at the de Broglie scale (Chiarelli et al., 2020, Chiarelli, 2012). This construction enables modeling quantum–classical transitions and decoherence-driven emergence of classical hydrodynamics on large scales.

3. Physical Interpretation and Domain of Applicability

The HSE hydrodynamic variables admit the following interpretations:

• $\rho(\mathbf r, t)$: probability or mass density of the quantum fluid
• $v(\mathbf r, t)$: local velocity (from wavefunction phase)
• $-\nabla V$: classical force (external, mean-field, or self-consistent)
• $-\nabla Q$: purely quantum force responsible for wavepacket spreading, interference, and tunneling

Key assumptions and limitations include (Wong, 2014):

• Irrotationality: $v = \nabla S/m \implies \nabla \times v = 0$, no intrinsic vorticity except for multivalued phases or spinorial generalizations.
• Absence of viscosity and thermal pressure (ideal, isentropic flow); extensions require additional stress (thermal, viscous, or exchange-correlation) tensors.
• For strongly interacting or highly inhomogeneous systems, the quantum potential $Q$ can dominate, precluding a classical fluid limit.
• For dissipative, stochastic, or fluctuation-driven dynamics, generalizations must include random Langevin-type terms or energy-dissipating imaginary potentials.

4. HSE in Numerical and Computational Frameworks

HSEs have been employed as alternative numerical frameworks for fluid and quantum-fluid simulation:

Context	HSE Model Basis	Key Features
Shallow water equations	Madelung-mapped NLS equation with drift	Dispersive regularization, robust vacuum handling, Strang–splitting + SEM solver (Fu et al., 5 Jan 2026)
Burgers' equation	Nonlinear, density-dependent quantum potential	Finite-difference stable, spectral unstable for high $N$ (Kerni et al., 4 Feb 2026)
Metallic nanostructures (QHT)	Effective (single-orbital) nonlinear SE	Treats nonlocality, spill-out, strong-field quantum plasmonics with FDTD coupling (Takeuchi et al., 2021)
Smoothed-particle hydrodynamics	SPH discretization of Madelung equations	Lagrangian adaptivity, conservative, handles collapse/singularity (Mocz et al., 2015)
Quantum computing of fluids	Two-component or quaternionic HSE	Capable of encoding vorticity and turbulence for quantum simulation (Meng et al., 2023)

In all cases, the central hydrodynamic variables $(\rho, v)$, or their equivalents, are either evolved directly or reconstructed from the wavefunction.

5. Advanced Generalizations and Classical Limit Constructions

Recent approaches reformulate the HSE to reconcile quantum and classical hydrodynamics, particularly for barotropic or rotational flows. Two main methodologies have emerged:

• Correction/cancellation approach: Introduction of a counter-term designed to cancel the quantum potential in the HSE, yielding equations that recover classical Bernoulli and Euler structure exactly without reliance on the formal limit $\hbar\to0$ (Ciou, 2 Jan 2025, Ciou, 25 May 2025). This class of variant HSEs implements Lagrangian densities and Hamiltonians with mass-independent coefficients, and leverages Clebsch parameterization or multicomponent wavefunctions to encode vorticity.
• Euler–Schrödinger transformations: Allomorphism between incompressible Euler equations and a wave-equation form in which the quantum potential maps to fluid surface tension, allowing direct identification of genuine classical analogues for the Bohm term (Zareei, 2021).

These advanced constructs clarify the structural discrepancies between canonical quantum hydrodynamics and classical fluid dynamics, especially regarding rotational flows, conservation laws, and the physical significance of the "quantum pressure" term.

6. Self-Similar, Blow-Up, and Singular Dynamics

The hydrodynamic representation exposes the structure of self-similar (intermediate-asymptotic) and blow-up dynamics in nonlinear quantum fluids. For focusing NLS in mass-supercritical regimes, self-similar HSE reductions produce coupled ODEs for scaled density and velocity profiles which serve as formal blow-up candidates. The existence and qualitative properties of these solutions have been established, with explicit connections to the hydrodynamic equations' ill-posedness in Sobolev spaces and their role in blow-up phenomena (Cao-Labora et al., 21 Mar 2025, Barna et al., 2018).

7. Quantum–Classical Transition, Fluctuations, and Measurement

HSE frameworks incorporating colored mass-density noise or stochastic pressure terms address the quantum-to-classical transition. On scales large compared to the de Broglie length, the quantum force becomes negligible and emergent classical hydrodynamics is recovered. This stochastic HSE formalism predicts that quantum nonlocality is confined to short scales (set by de Broglie wavelength) and that quantum measurement must satisfy uncertainty relations determined by vacuum fluctuation correlations and relativistic causality constraints (Chiarelli et al., 2020, Chiarelli, 2012). The ensemble interpretation, including limitations of tracking back to linear Schrödinger dynamics and the breakdown caused by noise-induced phase singularities, is carefully delineated.

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The Hydrodynamic Schrödinger Equation thus constitutes both a foundational and versatile formalism for connecting wave-mechanical, fluid-mechanical, and statistical representations of quantum and classical systems. Its modern deployment encompasses nonlinear, stochastic, and computational generalizations, positioning it as a core analytical and numerical tool in quantum hydrodynamics, fluid dynamics, and quantum-inspired simulation research (Wong, 2014, Barna et al., 2018, Andreev et al., 2014, Chiarelli, 2012, Ciou, 2 Jan 2025, Ciou, 25 May 2025, Kerni et al., 4 Feb 2026, Fu et al., 5 Jan 2026, Takeuchi et al., 2021, Mocz et al., 2015, Meng et al., 2023).