Mage's Rhizome

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====== Bohmian trajectories ======

We define the usual probability density and current.
\begin{align}
\rho(t,x) &= |\Psi(t,x)|^2 \\
j(t,x) &= \frac{\hbar}{m}\Im \{ \Psi(t,x) \nabla \Psi(t,x) \}
\end{align}

The well-known continuity equation can then be derived directly from the Schrödinger equation.
\begin{equation}
\partial_t \rho + \nabla \cdot j = 0
\end{equation}

Note that the symbol $\nabla$ includes partial derivatives with respect to all particle coordinates, thus $j$ is a $3N$-component vector. By analogy to fluid dynamics it is natural to introduce a probability velocity vector field by setting $j = \rho v$ that defines the flow of any given fluid element at position $x = X(t)$ in $3N$-dimensional configuration space.
\begin{equation}
\dot{X}(t) = v(t,X(t)) = \frac{j(t,X(t))}{\rho(t,X(t))} = \frac{\hbar}{m}\Im \left. \frac{\nabla \Psi(t,x)}{\Psi(t,x)} \right|_{x=X(t)}
\end{equation}

Problems with this definition clearly arise for small densities $\rho \approx 0$.
Note that if one is only interested in the propagation of the first particle coordinates $x_1$ we are naturally led to the concept of the [[conditional wave function]] $\psi_1(t,x_1) = \Psi(t,x_1,X_2(t),X_3(t),\ldots)$.
\begin{equation}
\dot{X}_1(t) = \frac{\hbar}{m}\Im \left. \frac{\nabla_1 \Psi(t,x)}{\Psi(t,x)} \right|_{x=X(t)} = \frac{\hbar}{m}\Im \left. \frac{\nabla_1 \psi_1(t,x_1)}{\psi_1(t,x_1)} \right|_{x_1=X_1(t)}
\end{equation}

Thus knowledge of $\psi_1$ suffices for evaluation of the first particle's Bohmian trajectory.