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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 Schrödinger 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.