https://mage.uber.space/dokuwiki/ 2026-05-03T14:52:20+00:00 https://mage.uber.space/dokuwiki/ https://mage.uber.space/dokuwiki/lib/tpl/dokuwiki/images/favicon.ico text/html 2016-10-31T10:59:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/approximations?rev=1477907996&do=diff 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 … text/html 2016-10-24T15:54:39+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/bohmian_trajectory?rev=1477317279&do=diff Bohmian trajectories We define the usual probability density and current. \begin{align} \rho(t,x) &= |\Psi(t,x)|^2 \\ j(t,x) &= \frac{\hbar}{m}\Im \{ \Psi(t,x) \nabla \Psi(t,x) \} \end{align} The well-known continuity equation can then be derived directly from the Schrödinger equation. \begin{equation} \partial_t \rho + \nabla \cdot j = 0 \end{equation} Note that the symbol $\nabla$ includes partial derivatives with respect to all particle coordinates, thus $j$$3N$$j = \rho v$$x = X(t)$$3N$\b… text/html 2017-01-19T16:12:39+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/conditional_evolution_equation?rev=1484838759&do=diff Conditional evolution equation We follow the derivation in Can the wave function in configuration space be replaced by single-particle wave functions in physical space? (§3) wich seems easier to understand than the equivalent treatment in Quantum-trajectory approach to time-dependent transport in mesoscopic systems with electron-electron interactions$\psi(t,x) = \Psi(t,x,Y(t))$$V(t,x,y)$$Y(t)$$\psi' = \nabla_y \Psi|_{y=Y(t)}$$\psi'' = \Delta_y\Psi|_{y=Y(t)}$\begin{equation} \begin{aligned} \mat… text/html 2016-10-30T17:33:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/conditional_wave_function?rev=1477845203&do=diff The conditional wave function The concept was apparently introduced in Quantum equilibrium and the origin of absolute uncertainty 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$$y$$y$$Y(t)$\begin{equation} \psi(t,x) = \Psi(t,x,Y(t)). \end{equation}$\Psi(t,x_1,x_2) = \alpha(t,x_1)\otimes\b… text/html 2017-01-24T14:27:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/coupled_system?rev=1485264478&do=diff 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 evolution equation), $\psi',\psi''$ etc. This was first noted in The theory of (exclusively) local beables, a numerical realization can be found in \begin{equation} \psi^{(i)}(t,x) = \nabla_y^i\Psi|_{y=Y(t)} \end{equation}\begin{equation} \mathrm{i}\hbar\partial_t \psi^{(i)} = -\frac{\hbar^2}{2m} \left( \Delta_x\p… text/html 2016-10-27T23:09:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/potential_network?rev=1477602556&do=diff 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… text/html 2016-11-07T17:15:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/start?rev=1478535355&do=diff Conditional wave functions and trajectory methods in quantum dynamics References for this namespace are collected here. The formulation of quantum mechanics on 3N-dimensional configuration space has been considered unfavourably since the early advent of the theory and is the main source of computational difficulties due to the exponential growth of data when the particle number $N$ text/html 2016-10-24T09:59:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://mage.uber.space/dokuwiki/cwf/wave_function_collapse?rev=1477295982&do=diff Wave function collapse In the language of Bohmian mechanics or when simply looking at conditional wave functions the collapse of the wave function during measurement processes arises naturally. Measurement here means a coupling of two quantum systems that evntually leads to strong entanglement. As a very basic example take $\mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2$$\mathcal{H}_i = \mathbb{C}^2$$(\sigma_+,\sigma_-)$$(\pi_+,\pi_-)$$\Psi_0 = (c_+\sigma_+ + c_-\sigma_-)\otimes \pi_-$\[ \Ps…