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--- cwf:coupled_system [2016/10/27 23:01] – created markus
+++ cwf:coupled_system [2017/01/24 14:27] (current) – markus
@@ Line 1: / Line 1: @@
 ====== Coupled system ======
-Instead of deriving further evolution equations in the form of a [[potential network]] one can directly write down a system of coupled equations for $\psi$ (the [[conditional wave equation]]), $\psi',\psi''$ etc. This was first noted in [(:cite:cwf:norsen2010)], a numerical realization can be found in [(:cite:cwf:marian_phd)] (Appendix A). We write
+Instead of deriving further evolution equations in the form of a [[potential network]] one can directly write down a system of coupled equations for $\psi$ (the [[conditional evolution equation]]), $\psi',\psi''$ etc. This was first noted in [(:cite:cwf:norsen2010)], a numerical realization can be found in [(:cite:cwf:marian_phd)] (Appendix A). We write
 \begin{equation}
 \psi^{(i)}(t,x) = \nabla_y^i\Psi|_{y=Y(t)}
@@ Line 9: / Line 9: @@
 \mathrm{i}\hbar\partial_t \psi^{(i)} = -\frac{\hbar^2}{2m} \left( \Delta_x\psi^{(i)} + \psi^{(i+2)}\right) + \sum_{k=0}^i {i \choose k} \left.\left( \nabla_y^k V \right)\right|_{y=Y(t)} \psi^{(i-k)} + \mathrm{i}\hbar \dot{Y}(t) \psi^{(i+1)}.
 \end{equation}
+Written down in matrix form we have the following.
+\begin{equation}
+\mathrm{i}\hbar\partial_t
+\left(\begin{array}{c}
+\psi^{(0)} \\ \psi^{(1)} \\ \psi^{(2)} \\ \psi^{(3)} \\ \vdots
+\end{array}\right) =
+\left(\begin{array}{cccccc}
+H_x & \mathrm{i}\hbar\dot{Y} & -\frac{\hbar^2}{2m} & 0 & 0 & \cdots \\
+\nabla_y V & H_x & \mathrm{i}\hbar\dot{Y} & -\frac{\hbar^2}{2m} & 0 & \cdots \\
+\nabla_y^2 V & \nabla_y V & H_x & \mathrm{i}\hbar\dot{Y} & -\frac{\hbar^2}{2m} & \cdots \\
+\nabla_y^3 V & \nabla_y^2 V & \nabla_y V & H_x & \mathrm{i}\hbar\dot{Y} & \cdots \\
+\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
+\end{array}\right)
+\cdot
+\left(\begin{array}{c}
+\psi^{(0)} \\ \psi^{(1)} \\ \psi^{(2)} \\ \psi^{(3)} \\ \vdots
+\end{array}\right)
+\end{equation}
+Here $H_x = -\frac{\hbar^2}{2m}\Delta_x+V$ is the usual Hamiltonian with only the $x$-coordinates present and all terms with the potential $V$ have set $y=Y(t)$ after differentiation. Now we also notice that the given scheme is just the Taylor expansion of $\Psi(t,x,y)$ at $y=Y(t)$ (assumed to exist) and all orders together would give the full wave function $\Psi$ again. For this to hold it is imperative to check under which conditions a [[:cwf:space-analyticity of wave function:|wave function is space-analytic]] and, even more restrictive, allows for a convergent Taylor series with infinite radius of convergence at the location of the Bohmian trajectory $Y(t)$.