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--- cwf:bohmian_trajectory [2016/10/17 14:54] – created markus
+++ cwf:bohmian_trajectory [2016/10/24 15:54] (current) – markus
@@ Line 1: / Line 1: @@
 ====== Bohmian trajectories ======
-We define the usual probability density and current in Hartree atomic units ($e = \hbar = m_e = 1/(4\pi\varepsilon_0) = 1$).
+We define the usual probability density and current.
 \begin{align}
 \rho(t,x) &= |\Psi(t,x)|^2 \\
-j(t,x) &= \Im \{ \Psi(t,x) \nabla \Psi(t,x) \}
+j(t,x) &= \frac{\hbar}{m}\Im \{ \Psi(t,x) \nabla \Psi(t,x) \}
 \end{align}
@@ Line 14: / Line 14: @@
 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))} = \Im \left. \frac{\nabla \Psi(t,x)}{\Psi(t,x)} \right|_{x=X(t)}
+\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) = \Im \left. \frac{\nabla_1 \Psi(t,x)}{\Psi(t,x)} \right|_{x=X(t)} = \Im \left. \frac{\nabla_1 \psi_1(t,x_1)}{\psi_1(t,x_1)} \right|_{x_1=X_1(t)}
+\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 particles Bohmian trajectory.
+Thus knowledge of $\psi_1$ suffices for evaluation of the first particle's Bohmian trajectory.