Abstract Assuming that the free energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm’s quantum potential and the Madelung equation are identically obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics admit a classical interpretation based on non-locality.
Abstract Assuming that the free energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm’s quantum potential and the Madelung equation are identically obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics admit a classical interpretation based on non-locality.
Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equations are obtained, showing explicitly that some of the hypotheses that led to the formulationb of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.
Assuming that the energy of a gas depends non-locally on the logarithm of its mass density, the body force in the resulting equation of motion consists of the sum of density gradient terms. Truncating this series after the second term, Bohm's quantum potential and the Madelung equation are obtained, showing explicitly that some of the hypotheses that led to the formulation of quantum mechanics do admit a classical interpretation based on non-locality. Here, we generalize this approach imposing a finite speed of propagation of any perturbation, thus determining a covariant formulation of the Madelung equation.
A recent attempt by Lohmiller and Slotine to derive the Schrödinger wave function exactly from classical action and classical density was shown by Vattay to silently drop the Madelung–Bohm quantum potential Q = −(ℏ²/2M) ∇²√ρ/√ρ, producing the standard semiclassical (WKB) approximation rather than an exact result. Vattay's mathematical correction is unambiguous, but it leaves the quantum potential unexplained: Q is treated as an irreducible mathematical fact about quantum mechanics with no underlying mechanism. We show that within Quantum Wavespace Theory (QWST) — a framework based on a finite propagation speed C and a finite saturation density P₀ — the quantum potential acquires an explicit physical origin. Starting from the QWST Lagrangian and applying the Madelung decomposition, we show how Q emerges from the gradient-energy term (C²/2)|∇Ψ|² of the wavespace pattern. In the small-amplitude limit, QWST is mathematically equivalent to standard quantum mechanics, with Q identified as the gradient energy of the wavespace pattern written in polar coordinates. When the saturation constraint is reinstated, the Madelung–Bohm equation acquires a calculable nonlinear modification governed by V_sat(|∇Ψ|; P₀) in the QWST Lagrangian. The correction is parametrically negligible in atomic physics (in agreement with all current measurements) and becomes order-unity within nucleon cores, where standard linear quantum mechanics is known to be insufficient. The saturation scale P₀ entering this correction is consistent with the order-of-magnitude proton internal pressure reported by Burkert et al., and the framework places this scale near ~10³⁵ Pa through a cosine-envelope integration of the proton mass.
Abstract In single-particle Madelung mechanics, the single-particle quantum state Ψ ( x → , t ) = R ( x → , t ) e i S ( x → , t ) / ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R ( x → , t ) 2 and local momentum ∇ → S ( x → , t ) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q ( x → , t ) = − ℏ 2 2 m ∇ → R ( x → , t ) R ( x → , t ) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q ( x → , t ) < 0 . For energy superposition states with band-limit
In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation [Formula: see text] outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.
Bohmian mechanics is a deterministic theory of quantum mechanics that is based on a set of n velocity functions for n particles, where these functions depend on the wavefunction from the n-body time-dependent Schroedinger equation. It is well know that Bohmian mechanics is not applicable to stationary states, since the velocity field for stationary states is the zero function. Recently, an alternative to Bohmian mechanics has been formulated, based on a conservation of energy equation, where the velocity fields are not the zero function, but this formalism is only applicable to stationary states with real valued wavefunctions. In this paper, Bohmian mechanics is merged with the alternative to Bohmian mechanics. This is accomplished by introducing an interpretation of the Bohm quantum potential. The final formalism gives dynamic particles for all states, including stationary states. The final main working equation contains two kinetic energy terms and a term that contains a factor that can be interpreted as a pressure. The derivation is a simple n-body generalization of the recent generalization, or refinement, of the Madelung equations.