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Probability evolution in Bohmian mechanics

According to the pilot wave theory of de Broglie and Bohm, a single electron is both a wave and a point-like particle. The wave is described by a position-space wave function $$\Psi(\boldsymbol{r}, t)$$, whose propagation is governed by the SchrÃ¶dinger equation $i \hbar \frac{\partial \Psi(\boldsymbol{r}, t)}{\partial t} = - \frac{\hbar^2}{2 m} \boldsymbol{\nabla}^2 \Psi(\boldsymbol{r}, t) + U(\boldsymbol{r}) \Psi(\boldsymbol{r}, t) \,,$ with $$\hbar$$, $$m$$ and $$U$$ denoting the Planck constant, electron mass, and external potential, respectively. The particle is described by its (well-defined!) position $$\boldsymbol{x}(t)$$ and velocity $$\boldsymbol{v}(t)$$, that evolve in time according to $\frac{d \boldsymbol{x}(t)}{d t} = \boldsymbol{v}(t) \,, \qquad \boldsymbol{v}(t) = \boldsymbol{u}\big( \boldsymbol{x}(t), t \big) \,,$ where $\boldsymbol{u}(\boldsymbol{r}, t) = \frac{\hbar}{m} \, \operatorname{Im} \left( \frac{\boldsymbol{\nabla} \Psi(\boldsymbol{r}, t)}{\Psi(\boldsymbol{r}, t)} \right)$ is a velocity field associated with the wave function $$\Psi$$, and $$\operatorname{Im}(\cdot)$$ denotes the imaginary part.

Problem

Suppose the initial position of the electron, $$\boldsymbol{x}(0)$$, is not known with certainty. It is known however that $$\boldsymbol{x}(0)$$ is distributed in accordance with some initial probability density $$P(\boldsymbol{r},0)$$. Generally, the probability density will change in the course of time. Let $$P(\boldsymbol{r},t)$$ denote the corresponding probability density at a later time $$t$$.

• Show that the time-evolution of $$P(\boldsymbol{r},t)$$ is governed by $\frac{\partial P(\boldsymbol{r},t)}{\partial t} = -\boldsymbol{\nabla} \cdot \Big( P(\boldsymbol{r},t) \boldsymbol{u}(\boldsymbol{r},t) \Big) \,.$

• Consider a special case such that $$P(\boldsymbol{r},0) = |\Psi(\boldsymbol{r},0)|^2$$. Find $$P(\boldsymbol{r},t)$$, and discuss the physical meaning of the result.

Solution

The solution is provided in Issue #5 - Probability evolution in Bohmian mechanics of Quantum Newsletter.