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Showing posts with the label Bohmian mechanics

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)}{\