Stationary Bohmian Amplitudes

Updated 8 February 2026

Stationary Bohmian amplitudes are the time-independent spatial moduli of quantum wavefunctions that guide particle trajectories via nontrivial phase gradients.
They integrate techniques from quantum hydrodynamics, chaos theory, and invariant variational methods to reveal detailed kinetic and osmotic flows.
The framework underpins applications in quantum gravity, atomic physics, and photonics by linking amplitude geometry with observable dynamics.

A stationary Bohmian amplitude refers to the spatial modulus $R(\mathbf{x})$ of a quantum wavefunction $\psi(\mathbf{x},t) = R(\mathbf{x})e^{i S(\mathbf{x},t)/\hbar}$ in a stationary (typically energy eigenstate) quantum system, within the Bohmian or Bohm–Madelung formulation. While the probability density $|\psi(\mathbf{x},t)|^2$ is time-independent (stationary), the gradients of the amplitude and phase play a central dynamical role in the guidance of Bohmian trajectories, the emergence of kinetic and “osmotic” flows, and the structure of the quantum potential. The study of stationary Bohmian amplitudes integrates quantum hydrodynamics, the theory of quantum chaos, variational and invariant approaches, and nontrivial ontologies of quantum mechanics, including quantum gravity.

1. Formalism: Decomposition and Stationary States

In the Bohmian framework, any pure state is written as

$\psi(\mathbf{x},t) = R(\mathbf{x},t) \exp \left( \frac{i S(\mathbf{x},t)}{\hbar} \right)$

where $R = |\psi|$ (the amplitude) and $S = \hbar \arg \psi$ (the phase). For a stationary (energy eigen) state,

$\psi(\mathbf{x},t) = e^{-iEt/\hbar} \phi(\mathbf{x}),\qquad \phi(\mathbf{x}) = R(\mathbf{x}) e^{i S_0(\mathbf{x})/\hbar}$

the amplitude and phase are time-independent (except for the global oscillation of phase).

The Bohmian guidance equation for a system of $n$ particles specifies the velocity field as

$\frac{d\mathbf{x}_j}{dt} = \frac{1}{m_j} \nabla_j S(\mathbf{x}_1, ..., \mathbf{x}_n, t)$

For stationary eigenstates with real amplitudes (i.e., $S_0$ constant), all Bohmian velocities vanish, while for eigenstates with nontrivial $S_0(\mathbf{x})$ the velocity is proportional to the stationary phase gradient, typically yielding nonzero local motion (Finley, 2021).

In the Madelung (hydrodynamic) picture, the quantum potential is

$Q(\mathbf{x}) = -\frac{\hbar^2}{2m} \frac{\nabla^2 R(\mathbf{x})}{R(\mathbf{x})}$

which contributes, together with the classical kinetic and external potential terms, to the quantum Hamilton–Jacobi equation.

2. Dynamical Content of Stationary Bohmian Amplitudes

Even in stationary states, Bohmian motion is possible if $S_0(\mathbf{x})$ has a nontrivial spatial structure. This arises, for example, in entangled multipartite states, superpositions, and in relativistic contexts.

Phase complexity and Bohmian chaos: While earlier studies linked quantum-chaotic Bohmian trajectories to moving nodes, it was established that static (stationary) nodes can also generate exponential sensitivity and chaos. The crucial ingredient is the spatial complexity of $S_0(\mathbf{x})$ , whose gradients define a nontrivial, often intricate velocity field. This field can exhibit vortices, X-points, and other singular structures, producing sensitive dependence on initial conditions (Cesa et al., 2016).
Quantifying chaos: The average (maximal) Lyapunov exponent $\bar{h}$ measures the rate of separation of nearby Bohmian trajectories. It is computed numerically by rescaling separations and averaging the local exponential growth rates over many trajectories (Cesa et al., 2016).
Entanglement and superposition structure: Quantities like the participation ratio (PR), Meyer–Wallach entanglement $Q(\psi)$ , geometric entanglement $E_G$ , and the three-tangle $\tau_3$ sometimes correlate with the degree of chaos as quantified by $\bar{h}$ , but are independent of the basis and the phases of wavefunction components. Chaos in Bohmian dynamics depends critically on both the spatial form and phases of the basis elements, which cannot be captured by these scalar quantum information measures alone.

3. Stationary Bohmian Amplitudes in Alternative and Generalized Models

Osmotic and symmetric velocities: In addition to the phase-gradient velocity $v_B = (1/m) \nabla S$ , the amplitude-gradient velocity $v_s = -(\hbar/m) \nabla R / R$ emerges from the real part of the quantum momentum density. Notably, $v_B = 0$ for all real stationary eigenstates, but $v_s$ can be nonzero, as shown for one-dimensional potential steps and in the context of evanescent wave regions. The inclusion of $v_s$ forms the basis of the generalized Madelung–fluid models and Nelson's stochastic mechanics (Waegell, 21 Nov 2025).
Energy-balance and merged Bohmian-Madelung approaches: Recent work formulates a merged dynamical law incorporating both $v_B$ and $v_s$ -like contributions, yielding nontrivial kinetic terms even for real stationary states. The quantum potential is decomposed into a part quadratic in amplitude-gradient velocities and a pressure-like term. This realizes genuinely dynamic trajectories even in stationary states (Finley, 2021).
Phase-space kinetic theories: The kinetic Bohmian equation—a nonlinear Vlasov-type equation on phase-space densities—admits stationary solutions whose configuration-space marginals reproduce standard stationary quantum mechanics. The monokinetic (Lagrangian) ansatz $f(x,p) = \rho(x) \delta(p - \nabla S(x))$ recovers the stationary quantum Hamilton–Jacobi system and clarifies that Bohmian amplitudes parameterize the unique minimizer of a Fisher-information-based variational principle (Gangbo et al., 2018).

4. Structure, Invariants, and Variational Principles

Ermakov–Lewis invariants and amplitude equations: For separable Hamiltonians, the stationary continuity equation forces the amplitude sector $R_i(q_i)$ in each coordinate $q_i$ to obey an Ermakov–Pinney (EP) type nonlinear ODE. The EP equation automatically admits a conserved Ermakov–Lewis invariant

$I_i = \frac{1}{2} \left[ (\rho_i y_i' - \rho_i' y_i)^2 + k_i \frac{y_i^2}{\rho_i^2} \right]$

where $y_i$ is a solution of the linearized sector Schrödinger equation, $\rho_i$ is the amplitude factor, and $k_i$ is a separation constant linked to the quantum flux (Kumar, 31 Jan 2026).

Sturm–Liouville and geometric encoding: The stationary amplitude equations are recast in Sturm–Liouville form, with explicit normalization (Liouville transform) isolating the purely geometric (curvature) contributions of the quantum potential. Thus, stationary Bohmian amplitudes are not physically extraneous fields but are embedded in the geometry of the self-adjoint kinetic operator (Kumar, 31 Jan 2026).
Variational formulations: The stationary amplitude equations result from extremizing an action akin to that for the EP system, guaranteeing conservation of the Ermakov–Lewis invariant and giving a geometric, invariant-based foundation for the construction of stationary Bohmian amplitudes (Kumar, 31 Jan 2026, Gangbo et al., 2018).

5. Physical Realizations and Applications

Single-photon polarization states: The Bohmian construction for stationary states describes the phase-space of polarization as an effective 2D isotropic oscillator, with stationary distributions determined by amplitude topology and circulating Bohmian trajectories. The stationary ensemble average reproduces the classical concept of a random superposition of polarization ellipses, but derives from a rigorous, time-independent quantum amplitude and associated phase field (Luis et al., 2013).
Hydrogen atom ground state: In the non-relativistic (Schrödinger) Bohm model, the ground-state electron is at rest due to a real and positive wavefunction. In the Dirac–Bohm approach (relativistic), the ground-state spinor's nontrivial phase yields a persistent azimuthal current and nonzero Bohmian velocity—consistent with observed muon time-dilation in muonium atoms (Hiley, 2014).
Quantum gravity and stationary universal amplitudes: In the context of loop quantum gravity, the universal stationary wavefunction $\Psi_\Gamma$ of the Wheeler–DeWitt equation defines a probability density over space of graphs, with the Bohmian guiding law ordering configurations (spin networks) so that the emergent macroscopic limit recovers classical spacetime. The phase of the stationary amplitude provides the nonlocal law of evolution for the “atoms of space” (Vassallo et al., 2013).

6. Limitations, Correlations, and Ontological Significance

Correlation between amplitude complexity and chaos: Participation ratio and certain entanglement measures can track the complexity of the stationary amplitude structure, but are insensitive to the spatial and phase organization critical to chaos. No simple scalar measure suffices; direct analysis of $v(\mathbf{x}) = \nabla S/m$ and Lyapunov exponents is necessary for a complete account (Cesa et al., 2016).
Physical meaning of stationary amplitudes: In Bohmian mechanics, stationary amplitudes carry not just the probability density but also encode possible kinetic, osmotic, and geometric flows that are inaccessible to projective quantum measurements. Recognition of this dynamical content is crucial in extended ontological and stochastic models, and informs debates on the completeness of Bohmian mechanics (Waegell, 21 Nov 2025, Finley, 2021).
Applications beyond standard quantum mechanics: The explicit construction and analysis of stationary Bohmian amplitudes via invariant-based theory, variational principles, and generalized kinetic modeling provide a basis for further study in fields including optimal transport, quantum hydrodynamics, and non-perturbative quantum gravity (Kumar, 31 Jan 2026, Gangbo et al., 2018, Vassallo et al., 2013).