Quantifying some properties of the stochastic quantum force

TL;DR

This paper models the quantum force in the Schrödinger equation as a stochastic force, analyzing its properties and implications for quantum pressure and uncertainty relations, with a focus on the Bohm potential and vacuum fluctuations.

Contribution

It introduces a novel interpretation of the quantum force as a stochastic force with specific statistical properties and links it to the Bohm potential and vacuum fluctuations.

Findings

Quantum force's local mean proportional to third derivative of probability density

Imaginary part of quantum momentum as root mean square fluctuation

Quantum Bohm potential interpreted as energy to place a particle in fluctuating vacuum

Abstract

We consider for clarity the simple case of the one dimensional non-relativistic Schr\"{o}dinger equation and regard it as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. By analyzing the Bohm potential it is found that the imaginary part of the quantum momentum is the root mean square fluctuation of the particle around its mean velocity, where the latter is the real part of the quantum momentum. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, and its associated pressure to the second spatial derivative. The latter is decomposed from the single particle diluted gas pressure, and this pressure partition allows the interpretation of the quantum Bohm potential as the energy required to put a particle in a bath of fluctuating vacuum at constant entropy and volume. The stochastic force expectation value is zero and is uncorrelated with the particle location, thus does not perform work on average. Nonetheless it is anti-correlated with volume and this anti-correlation leads to a new type of an Heisenberg like relation. We imply the dynamic Gaussian solution to the Schr\"{o}dinger equation as a simple example for exploring the mean properties of this quantum force. Still an interesting remained open task is the identification of the stochastic law that leads to these obtained mean properties.