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Journal of Experimental and Theoretical Physics

Polarization of spin-1/2 particles in an axisymmetric magnetic field

Alexander J. Silenko

Published Oct 1, 1998• 2 citations
Publisher

Abstract

The present paper shows that the nature of the polarization of charged spin-1/2 particles moving in a uniform magnetic field changes dramatically in a relatively weak transverse axisymmetric magnetic field. The direction along which the spin projection is quantized has a fixed orientation with respect to the axes of a cylindrical coordinate system and can form a substantial angle with the direction of the uniform magnetic field. The presence of spin quantization is proved both by the fact that the commutator of the Hamiltonian operator and the projection of the polarization operator in the direction of quantization is zero and by analyzing the Bargmann-Michel-Telegdi equation for this given case. Finally, the possibilities of detecting this effect and utilizing it are discussed.

DOI
10.1134/1.558703

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The present paper shows that the nature of the polarization of charged spin-1/2 particles moving in a uniform magnetic field changes dramatically in a relatively weak transverse axisymmetric magnetic field. The direction along which the spin projection is quantized has a fixed orientation with respect to the axes of a cylindrical coordinate system and can form a substantial angle with the direction of the uniform magnetic field. The presence of spin quantization is proved both by the fact that the commutator of the Hamiltonian operator and the projection of the polarization operator in the direction of quantization is zero and by analyzing the Bargmann-Michel-Telegdi equation for this given case. Finally, the possibilities of detecting this effect and utilizing it are discussed.
The spin polarization of an electron moving nonrelativistically through a potential barrier V(z)V(z)V(z), with also a constant uniform magnetic field (0,0,B0,0,B0,0,B) applied, is calculated using the WKB approximation for the zzz dependence of the wave function. It is found that, in a classically allowed region, the polarization precesses at fixed angle $\ensuremath{\theta}$ about the magnetic field, with phase angle the same as a classical particle with a magnetic dipole moment. The particle penetrates through the barrier with no change in the phase of the precession but with a discontinuity in $\ensuremath{\theta}$.
Considered is a quantum system of N(≥2)N(\ge2)N(≥2) charged particles moving in the plane R2{\mathbb R}^2R2 under the influence of a perpendicular magnetic field concenrated on the positions where the particle exsists. The gauge potential which gives this magnetic field is called a winding gauge potential. Properties of the momentum operators with a winding gauge potential are investigated. The momentum operators with a winding gauge potential are represented by the fibre direct integral of Arai's momentum operators [1]. Using this fibre direct integral decomposition, commutation properties of the momentum operators are investigated. A notion of local quantization of the magnetic flux is introduced to characterize the strong commutativity of the momentum operators. Aspects of the representation of the canonical commutation relations (CCR) are discussed. There is an interesting relation between the representation of the CCR with respect to this system and Arai's representation. Some applications of those results are also discussed.
New degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the Pauli vector allows us to use the azimuthal angle of a second direction, denoted by unit vector I\mathbf II, with respect to the quantization direction to characterize the spin polarization. The rotation of I\mathbf II through an angle about the quantization axis leads to a rotation of the spin polarization vector through twice the angle about the same axis. Discussions are also made in Heisenberg picture as well. Upon utilizing this approach to a free electron and letting the quantization direction for each plane wave be the wave vector, we arrive at a representation in which the unit vector I\mathbf II functions as an independent index to characterize the spin polarization.
Equation for a spin 1/2 neutral particle possessing three additional characteristics is examined in uniform magnetic and electric fields. Solutions with cylindrical symmetry are searched. On solutions, operators of the energy and the third projection of the total angular momentum are diagonalized. After separating the variables, a system of 4 linked differential equations in partial derivatives with respect to cylindrical coordinates (r,z) is derived. With the use of a special algebraization method, the rrr-dependence of the searched solutions is fixed through the confluent hypergeometric functions. Then a system of 4 linked differential equations in z-coordinate is derived; which is solved as well with the use of algebraization method. The constructed wave functions depend substantially on three additional characteristics. Analytical results are studied numerically. tissue.
In the present paper, we will focus on the non-relativistic problem for a spin 3/2 particle in magnetic field, applying cylindrical coordinates and two tetrads: Cartesian and cylindrical. Here appear six different presentations for 4-component wave functions: three Cartesian ones L cart Ψ cart ,Ψ cart and provided by using the relevant gauge transformation L cyl = S (φ) L cart three different presentations in cylindrical tetrad L cyl Ψ cyl ,Ψ cyl First, we specify the non-relativistic equation for a spin 3/2 particle in magnetic field for Cartesian tetrad in bases with non-diagonal and diagonal matrix of the third spin projection. Solutions of two types are found: the first one is associated with the operator of the orbital angular momentum; the second solution relates to the third projection of the total angular momentum. Equations arising here are solved in terms of the confluent hypergeometric functions, and the corresponding energy spectra are found. The gauge transformation is introduced which relates two tetrads: Cartesian and cylindrical; it permits us to transform the system of equations in polar coordinate from Cartesian tetrad to cylindrical one. The rules for gauge transformations of diagonalized operators of the total angular and orbital momentums are found.
We make three remarks about the quantum theory of a charged particle moving in a time-independent, uni-directional magnetic field (which we take to be parallel to the z-axis). First, when the field is uniform, we comment on the relation between the wave-function on the one hand if it is chosen to be an eigenstate of momentum (leaving aside the trivial z-dependence) and hence non-normalizable, and on the other hand if it is chosen to be an eigenstate of angular momentum. Second, we discuss the form of the wave-function when the magnetic field had a weak z-dependence. Finally, we make a short comment on the derivation of the diamagnetism of a free electron gas.