Mage's Rhizome

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====== The conditional wave function ======

The concept was apparently introduced in [(:cite:cwf:DGZ1992)] and is given by the full wave function $\Psi$ of a system with some coordinates fixed at the position of the Bohmian trajectories for a given initial position. Thus it is possible to define the notion of the wave function of a subsystem with coordinates $x$, with the rest of the system described in coordinates $y$. If we assume the coordinates $y$ following the [[Bohmian trajectory]] $Y(t)$ in time then the conditional wave function of the subsystem is

\begin{equation}
\psi(t,x) = \Psi(t,x,Y(t)).
\end{equation}

Of course we cannot expect this quantity to follow a linear evolution equation like the full wave function. Also it will not stay normalized with the passage of time and even includes the [[wave function collapse]] naturally. Here is a derivation of the [[conditional evolution equation]].

If the subsystems under consideration are non-entangled and also no assumed interaction leads to entanglement afterwards then it is shown that the conditional wave functions just obey seperate Schrödinger equations. Take $\Psi(t,x_1,x_2) = \alpha(t,x_1)\otimes\beta(t,x_2)$ and a potential $V_1(t,x_1)+V_2(t,x_2)$ (no interaction term). Then putting this into Schrödinger's equation
\begin{equation}
\mathrm{i}\hbar \partial_t \Psi = \left(-\frac{\hbar^2}{2m}(\Delta_1+\Delta_2) + V_1 + V_2\right)\Psi
\end{equation}
yields
\begin{equation}
\mathrm{i}\hbar \partial_t\alpha \otimes \beta + \alpha \otimes \mathrm{i}\hbar\partial_t\beta = \left( -\frac{\hbar^2}{2m}\Delta_1\alpha + V_1\alpha\right)\otimes \beta+ \alpha\otimes\left( -\frac{\hbar^2}{2m}\Delta_2\beta + V_2\beta\right).
\end{equation}

Now the equation has to be fulfilled for the $\ldots\otimes\beta$ and the $\alpha\otimes\ldots$ components of the tensor product separately and thus we get two decoupled Schrödinger equations. In switching to the conditional wave function $\psi(t,x_1) = \Psi(t,x_1,X_2(t)) = \alpha(t,x_1)\beta(t,X_2(t))$ we notice that this is just the $\alpha$ with some time-varying phase factor. Thus in the non-entangled setting the conditional wave function modulo a phase will follow the same single-particle Schrödinger equation.