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.
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.
A dynamical model for the collapse of the wave function in a quantum measurement process is proposed by considering the interaction of a quantum system (spin-1/2) with a macroscopic quantum apparatus interacting with an environment in a dissipative manner. The dissipative interaction leads to decoherence in the superposition states of the apparatus, making its behaviour classical in the sense that the density matrix becomes diagonal with time. Since the apparatus is also interacting with the system, the probabilities of the diagonal density matrix are determined by the state vector of the system. We consider a Stern-Gerlach type model, where a spin- 1/2 particle is in an inhomogeneous magnetic field, the whole set up being in contact with a large environment. Here we find that the density matrix of the combined system and apparatus becomes diagonal and the momentum of the particle becomes correlated with a spin operator, selected by the choice of the system-apparatus interaction. This allows for a measurement of spin via a momentum measurement on the particle with associated probabilities in accordance with quantum principles.
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
CM system is monitored by the nucleus "relative system (
R)". There appear the effective (the electrons-mediated) interaction between
CM and
R that is possibly responsible for decoherence. For larger atoms, the interaction scales as
Z2 (
Z is the "atomic number"), being totally independent on the atomic mass. The interaction selects the
CM-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 "
CM + spin"-state "reduction" ("collapse"), i.e. for the irreversible retrieval of the classical information from the quantum world.
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.
While the Stern–Gerlach experiment is an old and familiar problem, no analysis of this experiment is presented in the pedagogical literature using modern quantum mechanical techniques. Expositions of the Stern–Gerlach experiment are usually based on ‘‘old quantum theory,’’ i.e., semiclassical Bohr–Sommerfeld quantum mechanics. The experiment is also a popular one in discussions of the postulate of quantum measurement, which asserts that the process of measuring an observable forces the state vector of the system into an eigenvector of that observable, and the value measured will be the eigenvalue of that eigenvector. The most enduring of the philosophical arguments associated with quantum mechanics ultimately revolves around the mechanism of quantum measurement. Since the postulate of quantum measurement is expressed in terms well outside the realm of the old quantum mechanics, the traditional analysis of the Stern–Gerlach experiment cannot reveal any information about the relationship between dynamics and measurement. Therefore, a modern analysis of the Stern–Gerlach experiment must be made if this experiment is to reveal any information about the relationship between dynamics and the mechanism of quantum measurement. This paper presents such an analysis, and develops some of the implications for the theory of quantum measurement.