Mage's Rhizome

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====== Potential network ======

The [[conditional evolution equation]] includes the wave function $\Psi$ on the full configuration space in the potentials terms $A$ and $B$, or more precisely in the ratios $\psi'/\psi$ and $\psi''/\psi$. We define

\begin{equation}
\phi_i(t,x) = \left. \frac{\nabla^i_y\Psi(t,x,y)}{\Psi(t,x,y)} \right|_{y=Y(t)}
\end{equation}

and thus have $\phi_1 = \psi'/\psi$, $\phi_2 = \psi''/\psi$ and so on. Note that this potential quantities with $i$ even are scalar valued while those with $i$ odd are 3d vector valued. To get a general evolution equation for the potentials $\phi_i$ we take the time derivative again obeying the chain rule for the time dependent [[Bohmian trajectory]] $Y(t)$ that substitutes the coordinate $y$.

\begin{equation}
\partial_t \phi_i = \left. \partial_t \frac{\nabla^i_y \Psi}{\Psi} \right|_{y=Y(t)} + \dot{Y}(t) \cdot\left. \nabla_y \frac{\nabla^i_y \Psi}{\Psi} \right|_{y=Y(t)}
\end{equation}

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}
\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{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 $i=2$. 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.