# Derivation of the Local-Mean Stochastic Quantum Force

## TL;DR

This paper models the Schrödinger equation as an ensemble average of a particle influenced by a stochastic quantum force, revealing its relation to vacuum fluctuations and providing a new interpretation of the Bohm potential.

## Contribution

It derives the local mean stochastic quantum force from the Schrödinger equation and interprets the Bohm potential as energy from vacuum fluctuations.

## Key findings

- Quantum force's local mean is proportional to the third derivative of probability density.
- Quantum pressure relates to the second derivative of probability density.
- The stochastic force has zero mean and is uncorrelated with particle position.

## Abstract

We regard the non-relativistic Schrodinger equation as an ensemble mean representation of the stochastic motion of a single particle in a vacuum, subject to an undefined stochastic quantum force. The local mean of the quantum force is found to be proportional to the third spatial derivative of the probability density function, while its associated pressure is proportional to the second spatial derivative. The latter arises from the single particle diluted gas pressure, and this observation allows to interpret 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 an uncertainty relation. We analyze the dynamic Gaussian solution to the Schrodinger equation as a simple example for exploring the mean properties of this quantum force. We conclude with a few possible interpretations as to the origins of quantum stochasticity.