A Vlasov-Bohm approach to Quantum Mechanics for statistical systems

Quantum mechanics is the most successful theory to describe microscopic phenomena. It was derived in different ways over the past 100 years by Heisenberg, Schrödinger, and Feynman. At the same time, other interpretations have been suggested, including the Bohm-De Broglie interpretation and the so-called Bohmian mechanics. Here, we show that Bohmian mechanics, which utilizes the concept of the Bohm quantum potential, can also serve as a starting point for quantizing classical non-relativistic systems. By incorporating the Bohm quantum potential into the Vlasov framework, we obtain a mean-field theory that captures the corpuscular nature of matter, in agreement with quantum mechanics within the Random Phase Approximation (RPA).

💡 Research Summary

In this paper the authors propose a novel route to quantize non‑relativistic statistical systems by embedding the Bohm quantum potential into the classical Vlasov framework. The work begins with a concise historical overview of quantum mechanics, emphasizing the multiple formulations (matrix, wave‑function, path‑integral) and the hydrodynamic picture introduced by Madelung, which later inspired Bohm’s deterministic interpretation. Rather than treating Bohmian mechanics as a mere reinterpretation of the Schrödinger equation, the authors invert the logical order: they start from the Bohm quantum potential and ask whether it can serve as a primary building block for quantization.

In Section 2 the Bohm quantum potential (Q(\mathbf r)=-(\hbar^{2}/2m),\nabla^{2}A/A) is introduced, where (A) is the amplitude of the wave function. For one‑dimensional stationary states the authors exploit the classical relation that the probability density is inversely proportional to the particle velocity, (\rho(x)\propto 1/v(x)). By expressing the total potential as (V_T(x)=V(x)+Q(x)) and eliminating (Q) they obtain a nonlinear Schrödinger‑type equation for the amplitude: \