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Technical Physics Letters

The harmonic oscillator equation for ultracold neutrons moving in an inhomogeneous magnetic field

T. Tchen

Published May 1, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.1134/1.1482748

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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.
Abstract The problem of the interaction of ultracold neutrons with a neutron interference filter oscillating in space is investigated. In the problem, the evolution of the wave packet is considered by means of numerically solving the non-stationary Schrödinger equation by splitting the evolution operator. The filter oscillating in space acts as a quantum modulator of the ultracold neutron flux. The resulting spectra of the transmitted and reflected states are obtained depending on the motion parameters of the interferometer.
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.
In connection with Bitter–Dubbers experiment, an exact solution for the quantum dynamics of neutrons in magnetic field which is harmonically inhomogeneous (helical) and changes harmonically with time is obtained by invoking a time-dependent transformation with spin–space coupling. It predicts that a beam of neutrons with monotonous momentum will be split into two beams by the magnetic field. As its limiting case, the Berry's geometric phase in moving reference frame appears while the field is very strong.
A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.