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Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences

5.—Mott Scattering and Stern-Gerlach Effect

Péter Faragó

Published Jan 1, 1972• 0 citations
Publisher

Abstract

Synopsis It is shown that, from an operational point of view, the production and detection of spin-polarised electrons by Mott scattering is equivalent to the performance of a Stern-Gerlach type experiment with heavy spin-one-half particles. The argument is based on standard quantum mechanical formalism using density matrices and Stokes vectors for the description of polarised assemblies of particles.

DOI
10.1017/s0080454100009213

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Synopsis It is shown that, from an operational point of view, the production and detection of spin-polarised electrons by Mott scattering is equivalent to the performance of a Stern-Gerlach type experiment with heavy spin-one-half particles. The argument is based on standard quantum mechanical formalism using density matrices and Stokes vectors for the description of polarised assemblies of particles.
It is suggested that a thorough treatment of electron spin has advantages as a method of presenting many of the concepts and techniques of intermediate quantum mechanics. The starting point is the Stern-Gerlach experiment, from which the anticommutation relations for the spin operators may immediately be deduced. The commutation relations, which demonstrate that the spin is an angular momentum, are derived from these and it is shown that the anomalous gyromagnetic ratio is a natural consequence of the properties of the spin operators. The Pauli theory, which is the simplest example of the use of matrices in quantum mechanics, is briefly summarised, and employed in an analysis of the double Stern-Gerlach experiment. This example of a change of representation is used to introduce Dirac's transformation theory which, together with a variant of the double Stern-Gerlach experiment, is made use of in a discussion of the EPR paradox, Bell's inequality and hidden-variable theories. It is emphasised that the quantum mechanical description of these phenomena is consistent and that, from the viewpoint of quantum theory, no action-at-a-distance is required to explain them.
We now have all the tools we need to describe quantum systems using a formal mathematical language. Our strategy is to apply this formalism to quantum systems you are already familiar with—the Stern-Gerlach experiment and light. We start by returning to our old friend the Stern-Gerlach experiment, which turns out to be the simplest quantum system to describe. We also use the proper language to describe these systems—we say we are working with particles that have spin (or sometimes we are working with just spins), rather than saying we are working with quantum objects that “act like current loops.” This is our first foray into formalism. We fully develop the formalism for spin-one-half including Pauli matrices, the generalized Euler identity, the Leibniz product rule for commutators, and the exponential disentangling identities for SU(2). We then revisit much of the conceptual ideas covered in the first four chapters and we reanalyze them using Dirac notation and our new formalism.
The Stern–Gerlach (SG) experiment (1922) was one of the earliest experiments for which the result could not be explained using classical physics by any means. Hence it demonstrated unambiguously the importance of quantum effects. The SG experiment can be explained using the quantum theory for the spin-½ particle, which is a quantum system that can be represented simply by 2 × 2 matrices. We will use the spin-½ particle to introduce some of the basic concepts of quantum mechanics, such as state space, operators, measurement, the uncertainty principle, and the evolution equation. Furthermore, the theory for the spin-½ particle can be readily generalized to finite dimensional quantum systems, i.e., systems that can be represented by finite dimensional matrices.
Abstract The Stern–Gerlach (SG) experiment is a fundamental experiment for revealing the existence of “spin”. In this experiment, beams of silver atoms are sent through inhomogeneous magnetic fields to observe their deflection. Thus, the conventional SG experiment can be viewed as a magnetic-type spin effect. In this work, we successfully generalize the SG effect from magnetic-type to electric-type by solving Dirac’s equation with a potential barrier, revealing an extraordinary spin effect. Beams of Dirac particles can be regarded as matter waves. Based on Dirac’s equation, we obtain the explicit forms of the incident, reflected, and transmitted waves. The electric-type SG effect shows that the reflected and transmitted waves can exhibit notable spatial shifts, which depend on the spin direction and the incident angle of the wave. The electric-type SG effect has potential applications for separating Dirac particles with different spin directions and for estimating the spin direction of Dirac particles. Some discussions related to the interaction between spin and the electric field are also presented.
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.