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arXiv:1508.00266 · Journal of Surface Investigation X-ray Synchrotron and Neutron Techniques

Quantum-mechanical description of spin 1/2 particles and nuclei channeled in bent crystals

Alexander J. Silenko

Published Mar 1, 2015• 4 citations
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Abstract

A general quantum-mechanical description of relativistic spin 1/2 particles and nuclei channeled in bent crystals is presented using the cylindrical system of coordinates. The previously derived Dirac equation in this system is supplemented with terms characterizing anomalous magnetic and electric dipole moments. For the general case, transformation into the Foldy-Wouthhuysen representation is performed, quantum-mechanical equations of the motion of particles and their spins are obtained, and their classical limit is found. The physical nature of the main features of the description of particles and nuclei in the cylindrical system of coordinates is determined.

DOI
10.1134/s1027451015020172

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A general quantum-mechanical description of relativistic spin 1/2 particles and nuclei channeled in bent crystals is presented using the cylindrical system of coordinates. The previously derived Dirac equation in this system is supplemented with terms characterizing anomalous magnetic and electric dipole moments. For the general case, transformation into the Foldy-Wouthhuysen representation is performed, quantum-mechanical equations of the motion of particles and their spins are obtained, and their classical limit is found. The physical nature of the main features of the description of particles and nuclei in the cylindrical system of coordinates is determined.
The Foldy-Wouthuysen transformation for relativistic spin-1 particles interacting with nonuniform electric and uniform magnetic fields is performed. The Hamilton operator in the Foldy-Wouthuysen representation is determined. It agrees with the Lagrangian obtained by Pomeransky, Khriplovich, and Sen'kov. The validity of the Corben-Schwinger equations is confirmed. However, an attempt to generalize these equations in order to take into account the own quadrupole moment of particles was not successful. The known second-order wave equations are incorrect because they contain non-Hermitian terms. The correct second-order wave equation is derived.
The Foldy-Wouthuysen transformation for relativistic spin-1 particles interacting with nonuniform electric and uniform magnetic fields is performed. The Hamilton operator in the Foldy-Wouthuysen representation is determined. It agrees with the Lagrangian obtained by Pomeransky, Khriplovich, and Sen'kov. The classical and quantum formulae for the Hamiltonian agree. The validity of the Corben-Schwinger equations is confirmed. However, it is difficult to generalize these equations in order to take into account the quadrupole moment defined by a particle charge distribution. The known second-order wave equations are not quite satisfactory because they contain non-Hermitian terms. The Hermitian second-order wave equation is derived.
An examination is made of the consequences for the quantum mechanics of spinning particles of equations characteristic of Lorentz-covariant position variables. These equations are commutator analogs of the Poisson bracket equations that express the familiar transformation properties of space-time events in classical mechanics. For a particle of zero spin it is found that the usual canonical coordinate is the unique solution of these equations. For a particle with positive spin there is no position operator which satisfies these equations and has commuting components. For a particle and antiparticle there is a unique solution with commuting components which is valid for all values of the spin and reduces for zero spin to the canonical coordinate. For spin \textonehalf{} this is the Foldy-Wouthuysen transform of the position operator of the Dirac equation. A generalization of the inverse Foldy-Wouthuysen transformation, valid for any value of the spin, appears as a unique unitary transformation which takes this generalized Dirac position to the canonical coordinate. The application of this transformation to the canonical form of the Hamiltonian gives a generalization of the Dirac equation Hamiltonian. This is developed and compared with the literature for spin 1. It gives a nonlocal equation as the spin 1 analog of the Dirac equation.
We give a direct link between description of Dirac particles in the abstract framework of unitary representation of the Poincaré group and description with the help of the Dirac equation. In this context we discuss in detail the spin operator for a relativistic Dirac particle. We show also that the spin operator used in quantum field theory for spin s=1/2s=1/2s=1/2 corresponds to the Foldy-Woutheysen mean-spin operator.