The deterministic dynamics of a single-particle quantum ensemble is equivalent to the stochastic one due to the indistinguishability of quantum particles

TL;DR

The paper demonstrates that the deterministic evolution of a single-particle quantum ensemble can be viewed as equivalent to a stochastic process due to particle indistinguishability, linking quantum and classical Brownian motion concepts.

Contribution

It reveals that quantum ensemble dynamics are fundamentally equivalent to classical stochastic processes, bridging deterministic quantum evolution with stochastic models.

Findings

Quantum ensemble dynamics impose momentum restrictions at each point.

Indistinguishability leads to equivalence with classical Brownian collision.

Deterministic quantum evolution can be modeled as a stochastic process.

Abstract

It is shown that the wave function describing the pure state of a single-particle quantum ensemble, in addition to statistical restrictions, imposes restrictions on the particle momentum at points in the configuration space $\mathbb{R}^3$: at time $t$, each point $\mathbf{r}$ is a ``meeting'' point of two (non-interacting) particles of the ensemble with momenta $\mathbf{p}_1(\mathbf{r},t)$ and $\mathbf{p}_2(\mathbf{r},t)$. Their peculiarity is that the velocities $\mathbf{p}_1(\mathbf{r},t)/m$ and $\mathbf{p}_2(\mathbf{r},t)/m$ coincide with the velocities $\mathbf{b}(\mathbf{r},t)$ and $\mathbf{b}_*(\mathbf{r},t)$, which are introduced in Nelson's stochastic approach as key characteristics of frictionless Brownian particle motion. This means that the instantaneous dynamics of a pair of non-interacting quantum particles of an ensemble at point $\mathbf{r}$ at time $t$, due to their fundamental indistinguishability, is equivalent to the collision of two classical Brownian particles. And since this is true at all times for all points in $\mathbb{R}^3$, the unitary deterministic dynamics of a single-particle quantum ensemble is equivalent to a stochastic process.