Mage's Rhizome

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====== Approximations ======

If one wants to solve the [[conditional wave function]] to get a guiding field for the [[Bohmian trajectory]] one needs an approximated expression for the effective potential originating from the [[potential network]] $\phi_i$.

In the case of an unentangled state that factorizes as $\Psi(t,x,y) = \alpha(t,x)\beta(t,y)$ we see that $\phi_i(t) = \nabla_y^i \beta(t,y)/\beta(t,y)|_{y=Y(t)}$ with no $x$-dependence left. Thus we cannot expect any real influence from these auxiliary potentials on the [[conditional wave function]] $\psi(t,x)$. This leads us directly to a very rough approximation where all such entanglement effects are ignored and $\phi_i=0$ for all $i \geq 1$ and thus $A=B=0$. Another reasoning would not set them to zero but take the zero-order Taylor expansion in $x$ at $X(t)$ thus simply setting $x=X(t)$. This means the auxiliary potentials are time-dependent but constant in space and only contribute a global phase to the conditional wave function.[(Note that this phase makes a difference as soon as the wave function is defined as a Slater determinant and each component is treated individually as explained here and then summed up. This procedure is employed in Oriols, 2007 (second algorithm).)]

This kind of approximation has been used in [(:cite:cwf:oriols2007)] and [(:cite:cwf:norsen2015)] (where it is called "small entanglement approximation"). Note that even though the effective potential is reduced to the pure external potential this quantity enters the [[conditional evolution equation]] as $V(t,x,Y(t))$ and thus depends on the path $Y(t)$ of the remaining particles.

A higher order approximation could take $\phi_1$ and $\phi_2$ into account and ignore higher orders in the respective evolution equations. The potentials are then evolved by a time-stepping procedure alongside the conditional wave functions. Note however that the effective potential is different for every [[conditional wave function]]. This is in contrast to other effective potential techniques like the Kohn-Sham approach in density functional theory, where the effective potential acts like a usual external potential on non-interacting particles.