Beyond the Wave Function: A Classical Path to Quantum Systems

Author: Denis Avetisyan

New research explores a Bohmian mechanics-based approach to bridge the gap between classical statistical descriptions and quantum phenomena.

The study of a potential well with depth $V_0 = -0.5$ demonstrates how solutions to the Bohm and Schrödinger equations reveal a ground-state energy of $E = -0.36$ alongside a scattering solution characterized by a wave number $k = 1.5$ (corresponding to $E = 1.125$), illuminating the dynamics described by the underlying nonlinear Schrödinger equation detailed in Eq. (4).

This review details how the Vlasov equation and quantum potential can be used to quantize classical statistical ensembles, offering an alternative foundation for quantum mechanics.

Despite the century-long success of quantum mechanics, alternative interpretations and foundational approaches continue to offer novel perspectives on microscopic phenomena. This work, ‘A Vlasov-Bohm approach to Quantum Mechanics for statistical systems’, explores a quantization strategy rooted in Bohmian mechanics, leveraging the Vlasov equation to describe the collective behavior of quantum particles. By incorporating the Bohm quantum potential within a mean-field framework, we demonstrate a pathway to consistently recover quantum mechanical results-specifically within the Random Phase Approximation-from a classically-inspired starting point. Could this approach offer new insights into the interplay between classical and quantum statistical mechanics, and potentially illuminate the foundations of quantum field theory?

The Illusion of Randomness: Questioning Quantum Foundations

The Copenhagen Interpretation, a cornerstone of quantum mechanics, posits that the fundamental nature of reality is inherently probabilistic; a particle’s properties are not defined until measured, and exist as a superposition of states described by a wave function. While remarkably successful in predicting experimental outcomes, this interpretation deliberately avoids addressing the question of what, if anything, determines the outcome of a measurement. It essentially accepts probability as a fundamental aspect of existence, rather than a reflection of incomplete knowledge of underlying deterministic processes. This pragmatic approach, while allowing for accurate predictions, leaves open the possibility that a deeper, more complete theory could reveal hidden variables or mechanisms governing particle behavior, restoring a sense of predictability to the quantum realm. The interpretation, therefore, functions as a powerful predictive tool, but simultaneously obscures the potential for a fully deterministic understanding of physical reality, prompting ongoing research into alternative interpretations and foundational theories.

The initial frameworks of quantum mechanics, namely Matrix Mechanics and Wave Mechanics, though revolutionary in their predictive power, left a critical question unanswered regarding particle trajectories. While these formulations successfully described how to calculate probabilities of finding a particle in a certain state, they did not delineate what a particle is actually doing between measurements. Heisenberg’s Matrix Mechanics, focused on observable quantities, deliberately eschewed explicit descriptions of particle paths, while Schrödinger’s Wave Mechanics, though providing an equation governing wave-like behavior, treated the wave function as a mathematical tool rather than a direct representation of a physical entity tracing a path. This meant that, despite the equations’ success, the fundamental nature of how a particle moves – its trajectory through space and time – remained largely undefined, creating a conceptual gap in the understanding of quantum phenomena and prompting further investigation into the underlying reality beyond probabilistic descriptions.

Quantum mechanics describes reality through the wave function, a mathematical object representing the probability of finding a particle in a specific state. However, a fundamental difficulty arises when attempting to connect this probabilistic description to the concrete, definite positions and momenta observed in experiments. The wave function evolves smoothly, offering no inherent mechanism for the “collapse” into a single, localized state upon measurement. This disconnect suggests that the standard formulation may be incomplete, failing to fully account for how a particle transitions from a superposition of possibilities-existing as a spread of probabilities-to a definite, measurable value. Resolving this challenge requires either a refinement of the wave function’s interpretation or the introduction of new physical principles that bridge the gap between the quantum realm of probabilities and the classical world of definite properties, potentially hinting at underlying deterministic mechanisms beyond current understanding.

Reintroducing Order: The Bohmian Path

Bohmian mechanics describes particle motion through definite trajectories determined by the wave function and a related potential, termed the Quantum Potential. Unlike standard quantum mechanics where particles are described by probability distributions, Bohmian mechanics assigns each particle a precise position at all times. This trajectory is not the classical path, but one guided by the wave function’s phase, resulting in velocities determined by the equation $v = \frac{1}{m} \nabla S$, where $S$ is the phase of the guiding wave. The Quantum Potential, derived from the wave function, accounts for the non-classical behavior observed in quantum systems and introduces a fundamentally deterministic element to particle dynamics.

Bohmian mechanics diverges from the Copenhagen interpretation by explicitly incorporating hidden variables to account for particle positions. The Copenhagen interpretation, while accurately predicting experimental outcomes, describes particle location only through probability distributions defined by the wave function. In contrast, Bohmian mechanics asserts that particles possess definite positions at all times, even when not undergoing measurement. These positions are not merely probabilistic; they are determined by the initial conditions of the particle and the quantum potential, derived from the wave function itself. Consequently, while the wave function still governs the system’s evolution, it acts as a “guiding wave” influencing the trajectory of these definite particle positions, effectively restoring a deterministic element to quantum mechanics.

The De Broglie Postulate, foundational to Bohmian Mechanics, asserts that every particle possesses an associated pilot wave which governs its motion. This postulate extends the wave-particle duality by assigning a physical wave to every particle, relating the particle’s momentum $p$ to the wave vector $k$ through $p = \hbar k$, and its energy $E$ to the wave frequency $\omega$ via $E = \hbar \omega$. Crucially, the wave function, traditionally interpreted as a probability amplitude, is in this framework a real, physically guiding entity. The particle’s trajectory is not determined by probabilistic collapse, but by the gradient of the phase of this guiding wave, thus establishing a deterministic connection between the wave and particle behavior.

Mathematical Underpinnings: The Nonlinear Schrödinger Equation

The Nonlinear Schrödinger Equation (NLSE) within Bohmian Mechanics governs the time evolution of both the guiding wave, represented by the wave function $\Psi$, and the positions of the particles it directs. Specifically, the NLSE comprises a Schrödinger equation for $\Psi$ coupled with a guidance equation determining particle trajectories. The equation for $\Psi$ includes a potential term dependent on the particle density, reflecting the self-interaction of the guiding wave. The guidance equation dictates that particle velocities are proportional to the gradient of the phase of $\Psi$, ensuring that particles follow paths consistent with the wave’s behavior. This formulation contrasts with the standard Schrödinger Equation, which only describes the evolution of the wave function and does not provide explicit particle trajectories.

The standard Schrödinger equation, while accurately predicting the probability distribution of a particle, does not explicitly describe the forces acting on the particle itself. The Nonlinear Schrödinger Equation (NLSE) differentiates itself by incorporating the quantum potential, $Q$, directly into the equation of motion. This potential, arising from the wavefunction’s curvature, effectively acts as a force influencing the particle’s trajectory. The NLSE thus modifies the momentum equation to include a term proportional to the gradient of the quantum potential, $\nabla Q$, allowing for a deterministic calculation of particle movement guided by both the conventional potential and this quantum mechanical force. This contrasts with the standard formulation where the wavefunction only dictates probabilities, not direct particle dynamics.

The formulation of the Nonlinear Schrödinger Equation enables the direct computation of particle trajectories, yielding deterministic predictions of quantum phenomena-a departure from the probabilistic nature of standard quantum mechanics. This is achieved by explicitly solving for particle positions as functions of time, guided by the quantum potential. Furthermore, the equation allows for the reconstruction of the quantum potential itself from classical statistical ensembles; the resulting formalism is mathematically equivalent to the standard quantum formalism, demonstrating that Bohmian mechanics and conventional quantum mechanics are empirically indistinguishable despite their differing ontological interpretations.

From Microscopic Rules to Macroscopic Order

Bohmian Mechanics offers a compelling pathway to reconcile the deterministic foundations of classical physics with the probabilistic nature of quantum systems, particularly when considering many-particle systems. Unlike traditional quantum approaches which often rely on wave function evolution and probabilistic interpretations, this framework posits the existence of definite particle trajectories guided by a quantum potential. This allows for the application of classical statistical methods – those traditionally used to describe systems with known positions and velocities – to systems governed by the Schrödinger equation. Consequently, statistical properties like ensemble averages and distributions can be derived directly from these trajectories, offering a novel and deterministic basis for understanding complex phenomena in areas ranging from condensed matter physics to quantum fluids. The framework doesn’t merely mimic statistical mechanics; it genuinely extends its reach by providing a concrete, albeit non-classical, foundation for its principles at the quantum level.

Bohmian Mechanics offers a unique pathway to understanding statistical properties by tracing the evolution of individual particle trajectories, rather than relying solely on wave function descriptions. This approach reveals that seemingly probabilistic behavior emerges from the deterministic unfolding of these trajectories, influenced by the quantum potential. Consequently, concepts central to statistical mechanics – such as entropy, free energy, and transport coefficients – can be derived directly from analyzing the collective behavior of these particles over time. This perspective isn’t limited to systems at equilibrium; it also provides a novel framework for investigating non-equilibrium phenomena, offering insights into relaxation processes and the emergence of macroscopic order from microscopic dynamics. By connecting the quantum and classical realms through particle trajectories, the framework allows for a reinterpretation of statistical mechanics, presenting a deterministic underpinning to traditionally probabilistic descriptions of matter.

The successful reproduction of established statistical mechanics results within the Bohmian mechanics framework offers a compelling validation of its approach. Specifically, calculations of the $Dielectric Function$ – a crucial property describing a material’s response to electromagnetic fields – yield outcomes demonstrably equivalent to those derived from standard quantum mechanical treatments. This isn’t merely a formal equivalence; the Bohmian derivation provides a trajectory-based understanding of how collective material properties emerge from the behavior of individual particles. Consequently, Bohmian mechanics doesn’t simply predict the same phenomena as traditional quantum mechanics, but offers an alternative, deterministic pathway to arrive at those same predictions, reinforcing its potential as a complementary tool for exploring complex systems and bridging the gap between microscopic and macroscopic descriptions of matter.

Beyond Prediction: Charting New Territories

A compelling alternative to the probabilistic foundations of kinetic theory emerges through the application of Bohmian Mechanics to systems traditionally described by the Vlasov Equation. This approach postulates a deterministic underpinning for plasma and other collective behaviors, suggesting particles possess definite trajectories guided by a quantum potential derived from the overall particle distribution. Importantly, recent work demonstrates that solutions obtained within this Bohmian framework are not merely analogous to those from the Vlasov equation – they are demonstrably consistent with it, recovering established results for particle distributions and collective effects. This derivation offers a pathway to explore plasma dynamics with a fully deterministic model, potentially resolving ambiguities inherent in statistical descriptions and providing a richer understanding of phenomena like wave-particle interactions and turbulence, while simultaneously offering a novel avenue for testing the boundaries of quantum mechanics in complex systems.

Investigations leveraging the Random Phase Approximation (RPA) within the Bohmian mechanics framework hold the potential to illuminate the underlying mechanisms governing collective behaviors in complex systems. This approach seeks to move beyond traditional perturbative methods by explicitly accounting for particle trajectories and correlations. By applying RPA, researchers aim to model the intricate interplay between numerous particles, predicting emergent phenomena such as plasma oscillations, superconductivity, and collective excitations in condensed matter. The deterministic nature of Bohmian mechanics, combined with the RPA’s ability to capture screening effects and collective modes, could offer a more complete and intuitive understanding of these phenomena, potentially resolving discrepancies between kinetic theory and observed behavior, and opening new avenues for controlling and manipulating collective systems at the quantum level. This method may provide a pathway toward modeling complex systems where traditional kinetic approaches falter due to inherent non-equilibrium conditions and many-body interactions.

Investigations into the Heisenberg Uncertainty Principle within the framework of Bohmian Mechanics offer a unique pathway to refine the understanding of quantum limits, challenging conventional interpretations. While traditionally viewed as a fundamental constraint on measurement precision, Bohmian Mechanics, with its deterministic trajectories, suggests the principle might arise from the observer’s lack of complete knowledge of particle positions. This perspective doesn’t negate the principle’s validity but recontextualizes it as an epistemological limitation rather than an ontological one. Current research focuses on how the ‘quantum potential’-a key element of Bohmian Mechanics-influences the spread of particle trajectories and contributes to the observed uncertainty. By meticulously examining the relationship between particle positions, the quantum potential, and the limits of predictability, scientists hope to establish a more nuanced comprehension of the $ \Delta x \Delta p \geq \hbar/2 $ relationship and its implications for the very fabric of quantum reality.

The pursuit of a unified framework bridging classical and quantum realms, as demonstrated in this work through the Vlasov-Bohm approach, echoes a fundamental tenet of emergent phenomena. It suggests that the macroscopic behaviors of statistical systems aren’t simply extrapolations of individual quantum states, but rather arise from the collective interplay of underlying dynamics. As Werner Heisenberg observed, “Not only does God play dice with the universe, but He throws them where we cannot see.” This sentiment aptly captures the inherent uncertainty and holistic nature of complex systems. The quantization of classical ensembles, central to this paper, isn’t about imposing order, but about recognizing the patterns that self-organize from local interactions-a principle wherein the effect of the whole is not always evident from the parts.

Where Do We Go From Here?

The presented work, by framing quantum mechanics through a Vlasov-Bohm lens, doesn’t so much solve the measurement problem as relocate it. The focus shifts from the collapse of a wavefunction to the evolution of a statistical ensemble, guided by a quantum potential. This is not a victory over indeterminacy, but a re-description of it. Global regularities emerge from simple rules, and the quantization process itself appears as a consequence of the system’s inherent dynamics, rather than imposed from without. The enduring challenge remains: how to rigorously delineate the boundary between quantum and classical regimes, and whether such a boundary is even fundamental.

Further investigation must address the limitations of the dielectric function approximation and its impact on the validity of the Vlasov approach for strongly correlated systems. Any attempt at directive management often disrupts this process, suggesting a need for methods that allow for emergent behavior rather than pre-determined outcomes. A fruitful avenue lies in exploring the connection to other kinetic equations used in plasma physics and condensed matter theory – perhaps revealing a deeper, underlying structure governing the behavior of complex systems.

Ultimately, the value of this approach isn’t in offering a new ‘interpretation’ of quantum mechanics, but in providing a computational framework for studying statistical systems where quantum effects are significant. The future likely resides not in seeking a ‘true’ quantum gravity, but in developing increasingly sophisticated tools for navigating the complex interplay between order and chaos that defines the universe, and accepting that control is an illusion.

Original article: https://arxiv.org/pdf/2512.11772.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/