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

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.