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Acta Physica Polonica A

Nonexistence of the Classical Trajectories in the Stern-Gerlach Experiment

M. Arsenijević, M. Dugić

Published May 1, 2010• 1 citation
Publisher

Abstract

The Stern-Gerlach experiment is a paradigm of the quantum measurement of spin.Its physical interpretation is in intimate relation with the physical basis of the current research in the atomic (molecular) nanofabrication procedures.Nevertheless, interpretation of the experiment is an open issue yet.Here, we give the arguments for the physical nonexistence of the so-called classical trajectories of the atoms (molecules) in front of the screen.Some nanotech-related consequences are distinguished.

DOI
10.12693/aphyspola.117.760

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The Stern-Gerlach experiment is a paradigm of the quantum measurement of spin.Its physical interpretation is in intimate relation with the physical basis of the current research in the atomic (molecular) nanofabrication procedures.Nevertheless, interpretation of the experiment is an open issue yet.Here, we give the arguments for the physical nonexistence of the so-called classical trajectories of the atoms (molecules) in front of the screen.Some nanotech-related consequences are distinguished.
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The Stern-Gerlach experiment holds key role for students, particularly those studying quantum mechanics, modern physics, atomic physics, and physics education.It is one of the simplest experiments that demonstrates that the microscopic world does not behave accordingly withdynamics of classical physics, thusthe Stern-Gerlach experiment allows students to follow a logical progression: Classical expectation → Experimental observation → Failure of classical explanation → Quantum hypothesis → Spin states → Mathematical formulation → Quantum measurement.
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 claim of Kocsis et al. to have experimentally determined "photon trajectories" calls for a re-examination of the meaning of "quantum trajectories". We will review the arguments that have been assumed to have established that a trajectory has no meaning in the context of quantum mechanics. We show that the conclusion that the Bohm trajectories should be called "surreal" because they are at "variance with the actual observed track" of a particle is wrong as it is based on a false argument. We also present the results of a numerical investigation of a double Stern-Gerlach experiment which shows clearly the role of the spin within the Bohm formalism and discuss situations where the appearance of the quantum potential is open to direct experimental exploration.
The standard interpretation of the Stern-Gerlach experiment assumes that the atomic center-of-mass plays the role of "quantum apparatus" for the atomic spin. Following a recent, decoherence-based, model fitting with this interpretation, we investigate whether or not such model can be constructed. Our conclusions are somewhat surprising: only if the screen capturing the atoms in the experiment brings the information about the atomic-nucleus center-of-mass, one may construct the model desired. The nucleus CMCMCM system is monitored by the nucleus "relative system (RRR)". There appear the effective (the electrons-mediated) interaction between CMCMCM and RRR that is possibly responsible for decoherence. For larger atoms, the interaction scales as Z2Z^2Z2 (ZZZ is the "atomic number"), being totally independent on the atomic mass. The interaction selects the CMCMCM-wave-packet states as the approximate pointer basis. Interestingly enough, the model stems nonoccurrence of decoherence due to the internal environment for the larger systems (such as the macromolecules and the macroscopic systems). Certainly, disproving this model (e.g. in an experiment) stems the active role of the screen, which becomes responsible for the "CMCMCM + spin"-state "reduction" ("collapse"), i.e. for the irreversible retrieval of the classical information from the quantum world.