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--- cwf:potential_network [2016/10/27 15:02] – markus
+++ cwf:potential_network [2016/10/27 23:09] (current) – markus
@@ Line 15: / Line 15: @@
 A calculation using the Schrödinger equation with external potential $V$ and which includes the general product rule at one point yields the following lengthy expression.
 \begin{equation}
-\begin{aligned}
 \mathrm{i}\hbar\partial_t \phi_i = -\frac{\hbar^2}{2m} \left( \Delta_x\phi_i + 2\nabla_x\phi_i \cdot \frac{\nabla_x\psi}{\psi} + \phi_{i+2}-\phi_i \phi_2\right) + \sum_{k=1}^i {i \choose k} \left.\left( \nabla_y^k V \right)\right|_{y=Y(t)} \phi_{i-k} + \mathrm{i}\hbar \dot{Y}(t) (\phi_{i+1}-\phi_i\phi_1)
-\end{aligned}
 \end{equation}
-Note that we define $\phi_0=1$. The sum index really starts at 1 (instead of 0) because the 0th term has been cancelled. This results fits to the one given in [(:cite:cwf:norsen2015)] (35-36) for $i=1$ and $i=2$. A possible self-consistent iteration scheme for each time-step could start with $\phi_i=0$ for $i\geq 2,3,\ldots$ and then solve for $\phi_i$ in the order 1,2,1,2,3,1,2,3,4 etc. The original wave function $\Psi$ is indeed completely absent from this scheme.
+Note that we define $\phi_0=1$. The sum index really starts at 1 (instead of 0) because the 0th term has been cancelled. This results fits to the one given in [(:cite:cwf:norsen2015)] (35-36) for $i=1$ and $i=2$. The original wave function $\Psi$ is indeed completely absent from this scheme. We see that for small $|\psi|$ the method might be unstable because of the occurence of this quantity in the denominator, a problem that does not arise in the related [[coupled system]] of $\psi,\psi',\psi''$ etc.
 Substituing the formula for the [[Bohmian trajectory]] $\dot{Y}(t) = \hbar/m \Im \phi_1(t,x)|_{x=X(t)}$ shows that also the last term has a factor $\hbar^2$ in front, thus higher orders $\phi_{i+1}$ and $\phi_{i+2}$ of the potential network are generally weighted less which hints towards a possible quick convergence of the iteration scheme. It is also interesting to note that the high frequency components of the external potential $V$ get connected to the lower orders of $\phi_i$ and vice versa with $\nabla_y^i V$ appearing by itself in the evolution equation. Of course any application must cut the infinite network of potentials at one point and then considers only the remaining orders as an [[approximations]] to the problem with full complexity.