Bohmian Mechanics

Bohmian Mechanics is a theory about $N$-particles moving along trajectories with position

\[\boldsymbol{q}_{k}(t): \R \to \R^{3},\]

where $k$ indicates the $k$-th particle.

For the sake of convenience, we denote the set of all particle positions as \(\boldsymbol{Q}(t) := \{ \boldsymbol{q}_{1}(t), ..., \boldsymbol{q}_{N}(t) \} \subset \mathbb{R}^{3N}\) .

Those particle positions are guided by a (universal) wavefunction

\[\Psi(\boldsymbol{Q}, t): \R^{3N} \times \R \mapsto \mathbb{C}, \Psi(\boldsymbol{Q},t) = R(\boldsymbol{Q}, t) \mathrm{e}^{\frac{\mathrm{i}}{\hbar} S(\boldsymbol{Q},t)},\]

which is defined on configuration space.

The wavefunction evolves according to the Schrödinger equation, which is a linear differential equation

\[\mathrm{i} \hbar \frac{\partial}{\partial t} \Psi(\boldsymbol{Q}, t) = - \sum_{k = 1}^{N}\frac{\hbar^{2}}{2m_{k}} \boldsymbol{\nabla}_{k}^{2} \Psi(\boldsymbol{Q}, t) + V(\boldsymbol{Q}) \Psi(\boldsymbol{Q}, t),\]

where $\boldsymbol{\nabla}_{k}$ acts on the $k$-th particle position.

The particle positions are guided by the wavefunction according to the guiding equation

\[\frac{\mathrm{d}}{\mathrm{d} t} \boldsymbol{q}_{k}(t) = \frac{\hbar}{m_{k}} \text{Im} \biggl[\frac{\Psi^{*}(\boldsymbol{Q},t) \boldsymbol{\nabla}_{k} \Psi(\boldsymbol{Q},t)}{\Psi^{*}(\boldsymbol{Q},t) \Psi(\boldsymbol{Q}, t)} \biggr].\]

The guiding equation can be rewritten in the terms of the phase of the wavefunction $S(\boldsymbol{Q}, t)$, such that

\[\frac{\mathrm{d}}{\mathrm{d} t} \boldsymbol{q}_{k}(t) = \frac{1}{m_{k}} \boldsymbol{\nabla}_{k} S(\boldsymbol{Q},t).\]