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Semiclassical description of neutron polarization in an inhomogeneous magnetic field

O. N. Naǐda, A.G. Prudkovskii

Published Dec 1, 1978• 0 citations
OpenAlex

Abstract

A method is obtained for constructing the semiclassical solutions to the Pauli equation for the neutron in a magnetic field which describe both the Stern-Gerlach beam splitting and spin-flip processes. The accuracy of these solutions makes it possible (in contrast to the known semiclassical methods) to describe interference phenomena, which can be interpreted as corrections to the Larmor precession.

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A method is obtained for constructing the semiclassical solutions to the Pauli equation for the neutron in a magnetic field which describe both the Stern-Gerlach beam splitting and spin-flip processes. The accuracy of these solutions makes it possible (in contrast to the known semiclassical methods) to describe interference phenomena, which can be interpreted as corrections to the Larmor precession.
This chapter discusses the quasi-isotropic approximation (QIA) with the semiclassical asymptotics of the Pauli equations for spin 1/2 particles in a magnetic field. The case of spin 1/2 and the Pauli equation are taken to avoid complex manipulations and demonstrate the essence of novel features brought in by the QIA to this well-studied problem. The role of quantum constraints ensuring the description of the Stern-Gerlach splitting is played by the special initial conditions. The latter would imply that particle world lines are dependent on the states of particle spin. This is exactly what occurs in the Stern-Gerlach effect. However de Broglie was unable to find the expression for that additional term. Precession of the magnetic moment in this mode comes about due to interference mainly arising from the difference in eikonals corresponding to these modes. The solutions admit all the best known solutions as particular cases, in particular, the Stern-Gerlach particle trajectories and solutions of the Pauli-Galanin-Rubinow-Keller type.
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.
The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a semiclassical framework. The concept of coherent internal states is used in the formulation of the semiclassical approximation from the full quantum-mechanical expression. The classical trajectories are defined only for certain spin states, that satisfy the conditions for being coherent internal states. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
The coherent superposition of particle quantum states resulting from different spatial paths shows well-known interference effects. Another type of interference is Larmor precession, which can be described as neutron wave interference in spin space, i.e. interference on the same spatial path. Here we show several new manifestations of interference phenomena of the latter type. The interference is produced by coherent population of different I z = ±½ spin states in a polarized neutron beam between two resonant radio-frequency (RF) spin flippers. Their spin flip probability, and thus the occupation numbers of the spin states, can be tuned between 0 and 1 by changing the RF amplitude. This leads to interference effects in the polarization since the energy exchanged between the RF field and spin yields different phase evolution in the spatial parts of the neutron states. Depending on the preparation of the initial neutron states, five values for the time-independent phase difference between the final neutron states can be observed.
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.