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Physical Review A

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

O. V. Gritsenko

Published Sep 10, 2021• 0 citations
Publisher

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.

DOI
10.1103/physreva.104.032210

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In two celebrated experiments of quantum mechanics, the Stern-Gerlach (SG) and Heisenberg microscope (HM) experiments, the electron spin sz{s}_{z}sz​ and spatial \mathbitr\mathbit{r}\mathbitr coordinates are measured separately. In this paper, the combined SG+HM\mathrm{SG}+\mathrm{HM}SG+HM experiment is proposed to measure the full electron coordinate \mathbitx=\mathbitr,sz\mathbit{x}={\mathbit{r},{s}_{z}}\mathbitx=\mathbitr,sz​. 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 A\ensuremath-B\mathrm{A}\text{\ensuremath{-}}\mathrm{B}A\ensuremath-B. The theoretical description of a spin-collapsed and partially spatially collapsed state of the SG+HM\mathrm{SG}+\mathrm{HM}SG+HM experiment is given for the prototype ``perfectly'' entangled system, the dissociating H2{\mathrm{H}}_{2}H2​ molecule.
Most textbooks introduce the concept of spin by presenting the Stern-Gerlach experiment with the aid of Newtonian atomic trajectories. However, to understand how both spatial and spin degrees of freedom evolve over time and how the latter influence experimental outcomes, it is essential to employ a quantum approach. In this paper, we offer two simple methods, the Baker-Campbell-Hausdorff formula and the direct integration of the Schrödinger equation in an interaction picture, to determine the corresponding evolution operator. We not only provide an interpretation of the individual terms within this operator but also establish connections with semiclassical calculations, when feasible. Moreover, we compute the wave function and touch upon the concept of position-spin entanglement to illustrate how a full quantum description of the Stern-Gerlach experiment can open doors to topics like quantum measurement and nonlocality.
The Stem-Gerlach (SG) experiments considered as a quantum measurement process measuring the spin states of particles from their spatial distribution. The wave packet collapse or quantum decoherence can he described as a dynamical evolution governed by the interaction between the space degrees of freedom and the spin degrees of freedom. The analytic solution of this exactly solvable model shows that the decaying process of coherence with time in the SG experiment obeys no longer the linear exponential law.
The 1922 Stern–Gerlach experiment quickly proved fundamental to quantum physics. The benchmark experiment led to the quantization of angular momenta, discovery of electron spin, and study of the measurement problem and superposition. Immediately, Heisenberg and Einstein proposed multi-stage Stern–Gerlach experiments to explore deeper mysteries of directional quantization. In the classic multi-stage Stern–Gerlach experiment conducted by Frisch and Segrè, the Majorana (Landau–Zener) and Rabi formulae diverge afar from the experimental observation while the physical mechanism for electron-spin collapse remains unidentified. Here, introducing the physical co-quantum concept provides a plausible physical mechanism and predicts the experimental observation in absolute units without fitting (i.e., no parameters adjusted) with a p-value less than one per million, which is the probability that the co-quantum theory happens to match the experimental observation purely by chance. Further, the co-quantum concept is corroborated by statistically reproducing exactly the wave function, density operator, and uncertainty relation for electron spin in Stern–Gerlach experiments.
It is demonstrated that the collapse of the wavefunction in the quantum measurement process is reflected in the two-particle correlation function for the electron spin, which can in principle be studied by a triple Stern-Gerlach experiment. The change in momentum of a Stern-Gerlach magnet is itself sufficient to allow the determination of the electron spin, subject to the uncertainty principle, if the coherence of the two components of the spin is destroyed by the magnetic field. Similar considerations apply to the two-particle correlation function for the photon polarisation, and two-photon atomic cascades can be used in practice to investigate the measurement process and determine the extension of a photon wavepacket. The irreversibility of quantum mechanical measurements and its relation to the uncertainty principle is briefly considered.
We present an analytical solution to the decoherence time for the spin measurement and the diagonalization of the density matrix of spin variables in the Stern-Gerlach experiment.This solution requires the calculation of the Pauli spinor with a spatial extension, which is not found in quantum mechanics textbooks.With this full spinor and the measured position of the particle we demonstrate the three postulates of quantum measurement: quantization, spectral decomposition and wave function reduction.The transition from a quantum superposition to a statistical mixture is well explained in this way, but not the single result that always emerges from a particular experiment.The spinor spatial extension also allows the introduction of the de Broglie-Bohm trajectories which give a very simple explanation of the particles' impact.