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arXiv:2412.02115 · Nuclear Physics B

Quantum and classical dynamics of neutron in a magnetic field

A. Bogomyagkov, В. П. Дружинин, Eugene B. Levichev, Alexander I. Milstein, I.N. Okunev, Sergey Yu. Taskaev

Published Feb 11, 2025• 2 citations
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Abstract

The paper presents solution of quantum problem of neutron propagation in the magnetic field with multipole field expansion. Rigorous solution of the Pauli equation for neutron reveals existence of two solutions, finite and infinite, for any multipole configuration. As an example, we present detailed study of neutron motion in quadrupole and sextupole magnets. Our predictions agree with the results of Stern-Gerlach experiment for neutrons. To verify existence of finite and infinite motion, we discuss an experiment which could be performed in the Budker Institute of Nuclear Physics using existing equipment. We conclude with considerations of neutron storage ring with straight section and discrete magnets focusing the beam.

DOI
10.1016/j.nuclphysb.2025.116833

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The paper presents solution of quantum problem of neutron propagation in the magnetic field with multipole field expansion. Rigorous solution of the Pauli equation for neutron reveals existence of two solutions, finite and infinite, for any multipole configuration. As an example, we present detailed study of neutron motion in quadrupole and sextupole magnets. Our predictions agree with the results of Stern-Gerlach experiment for neutrons. To verify existence of finite and infinite motion, we discuss an experiment which could be performed in the Budker Institute of Nuclear Physics using existing equipment. We conclude with considerations of neutron storage ring with straight section and discrete magnets focusing the beam.
We present a comprehensive simulation study of the Newtonian and quantum model of a Stern-Gerlach experiment with cold neutrons. By solving Newton's equation of motion and the time-dependent Pauli equation for a wide range of uniform magnetic field strengths, we scrutinize the role of the latter for drawing the conclusion that the magnetic moment of the neutron is quantized. We then demonstrate that a marginal modification of the Newtonian model suffices to construct, without invoking any concept of quantum theory, an event-based subquantum model that eliminates the shortcomings of the classical model and yields results that are in qualitative agreement with experiment and quantum theory. In this event-by-event model, the intrinsic angular momentum can take any value on the sphere, yet, for a sufficiently strong uniform magnetic field, the particle beam splits in two, exactly as in experiment and in concert with quantum theory.
The motion of ultracold neutrons in a magnetic field featuring a quadratic inhomogeneity is considered. A partial solution to the stationary Schrödinger equation is found, which leads to a harmonic oscillator equation for the X coordinate. It is demonstrated that, under certain conditions, it is basically possible to provide for a spatial focusing of neutrons with the formation of either a single point focus or two side foci. Dependence of the focusing on the degree of monochromaticity of the neutron beam is considered.
The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a quantum mechanical framework. The validity of the semiclassical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
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The model of the motion of particles in the Stern-Gerlach apparatus in the classical and quantum mechanics was developed. The data simulation of particle trajectories and distribution of silver atoms on the surface of the plate in their deposition are discussed. It was found that for the experimentally observed distribution of two-dimensional shapes of the atoms must be assumed that the atoms are not involved in the precession motion in a magnetic field, while maintaining the direction of the magnetic moment, for example, parallel to the induction vector of the magnetic field during the time of motion in the apparatus. To obtain a realistic picture of the figure of the scattering of atoms used a classical model of movement and expression of forces compatible with the quantum picture of the motion of particles with spin ½. The magnetic field is simulated based on the original Stern-Gerlach data describing the distribution of the gradient of the induction components related to the splitting of the beam. Quantum model of particle motion is based on the Pauli equation in the boundary layer approximation. It is found that in this model, depending on the initial polarization of the particle, beam is split into either two or is deflected towards the magnet blade or in the opposite direction. It is shown that if the initial conditions for the task are reproducing the geometric dimensions and the magnetic field in the Stern-Gerlach apparatus, the figure of the scattering particles in the shape of the outline is similar to the experimentally observed shape