Noninterfering and simultaneous Stern-Gerlach and Heisenberg microscope experiments to measure the full electron coordinate in an entangled state

Abstract

In two celebrated experiments of quantum mechanics, the Stern-Gerlach (SG) and Heisenberg microscope (HM) experiments, the electron spin ${s}_{z}$ and spatial $\mathbit{r}$ coordinates are measured separately. In this paper, the combined $\mathrm{SG}+\mathrm{HM}$ experiment is proposed to measure the full electron coordinate $\mathbit{x}={\mathbit{r},{s}_{z}}$. To this end, noninterfering and (virtually) simultaneous SG and HM experiments are proposed to apply to the individual fragments A and B of an entangled dissociating system $\mathrm{A}\text{\ensuremath{-}}\mathrm{B}$. The theoretical description of a spin-collapsed and partially spatially collapsed state of the $\mathrm{SG}+\mathrm{HM}$ experiment is given for the prototype ``perfectly'' entangled system, the dissociating ${\mathrm{H}}_{2}$ molecule.