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arXiv:1711.03546 · Physical Review A

Spin effects on the semiclassical trajectories of Dirac electrons

R. Gutiérrez-Jáuregui, R. Pérez-Pascual, Rocío Jáuregui

Published Nov 8, 2017• 1 citation
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Abstract

The relativistic semiclassical evolution of the position of an electron in the presence of an external electromagnetic field is studied in terms of a Newton equation that incorporates spin effects directly. This equation emerges from the Dirac equation and allows the identification of scenarios where spin effects are necessary to understand the main characteristics of the electron trajectories. It involves the eigenvalues of the non-Hermitian operator ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}$, with ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\mu}\ensuremath{\nu}}$ and ${F}^{\ensuremath{\mu}\ensuremath{\nu}}$ as the spin and electromagnetic tensors. The formalism allows a deeper understanding on the physics behind known analytical solutions of the Dirac equation when translational dynamics seem to decouple from spin evolution. As an illustrative example, it is applied to an electron immersed in an electromagnetic field which exhibits chiral symmetry and optical vortices. It is shown that the polarization of intense structured light beams can be used to suppress or enhance spin effects on the electron semiclassical trajectory; the latter configuration yields a realization of a Stern-Gerlach apparatus for an electron.

DOI
10.1103/physreva.96.052109

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The relativistic semiclassical evolution of the position of an electron in the presence of an external electromagnetic field is studied in terms of a Newton equation that incorporates spin effects directly. This equation emerges from the Dirac equation and allows the identification of scenarios where spin effects are necessary to understand the main characteristics of the electron trajectories. It involves the eigenvalues of the non-Hermitian operator \ensuremathΣ\ensuremathμ\ensuremathνF\ensuremathμ\ensuremathν{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\mu}\ensuremath{\nu}}{F}^{\ensuremath{\mu}\ensuremath{\nu}}\ensuremathΣ\ensuremathμ\ensuremathν​F\ensuremathμ\ensuremathν, with \ensuremathΣ\ensuremathμ\ensuremathν{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\mu}\ensuremath{\nu}}\ensuremathΣ\ensuremathμ\ensuremathν​ and F\ensuremathμ\ensuremathν{F}^{\ensuremath{\mu}\ensuremath{\nu}}F\ensuremathμ\ensuremathν as the spin and electromagnetic tensors. The formalism allows a deeper understanding on the physics behind known analytical solutions of the Dirac equation when translational dynamics seem to decouple from spin evolution. As an illustrative example, it is applied to an electron immersed in an electromagnetic field which exhibits chiral symmetry and optical vortices. It is shown that the polarization of intense structured light beams can be used to suppress or enhance spin effects on the electron semiclassical trajectory; the latter configuration yields a realization of a Stern-Gerlach apparatus for an electron.
The time evolution of the density corresponding to the polarization operator, originally constructed to commute with the Dirac Hamiltonian in the absence of an external electromagnetic field, is investigated in terms of the time-dependent Dirac equation taking the presence of an external electromagnetic field into account. It is found that this time evolution leads to ‘tensorial’ and ‘vectorial’ particle current densities and to the interaction of the spin density with the external electromagnetic field. As the time evolution of the spin density does not refer to a constant of motion (continuity condition) it only serves as auxiliary density. By taking the non-relativistic limit, it is shown that the polarization, spin and magnetization densities are independent of electric field effects and, in addition, no preferred directions can be defined.
Most textbooks introduce the concept of spin by presenting the Stern-Gerlach experiment with the aid of Newtonian atomic trajectories. However, to understand how both spatial and spin degrees of freedom evolve over time and how the latter influence experimental outcomes, it is essential to employ a quantum approach. In this paper, we offer two simple methods, the Baker-Campbell-Hausdorff formula and the direct integration of the Schrödinger equation in an interaction picture, to determine the corresponding evolution operator. We not only provide an interpretation of the individual terms within this operator but also establish connections with semiclassical calculations, when feasible. Moreover, we compute the wave function and touch upon the concept of position-spin entanglement to illustrate how a full quantum description of the Stern-Gerlach experiment can open doors to topics like quantum measurement and nonlocality.
It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order $\\hbar $ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.
We investigate the spin of the electron in a non-relativistic context by using the Galilean covariant Pauli-Dirac equation. From a non-relativistic Lagrangian density, we find an appropriate Dirac-like Hamiltonian in the momentum representation, which includes the spin operator in the Galilean covariant framework. Within this formalism, we show that the total angular momentum appears as a constant of motion. Additionally, we propose a non-minimal coupling that describes the Galilean interaction between an electron and the electromagnetic field. Thereby, we obtain, in a natural way, the Hamiltonian including all the essential interaction terms for the electron in a general vector field.
A relativistic spin operator cannot be uniquely defined within relativistic quantum mechanics. Previously, different proper relativistic spin operators have been proposed, such as spin operators of the Foldy-Wouthuysen and Pryce type, that both commute with the free-particle Dirac Hamiltonian and represent constants of motion. Here we consider the dynamics of a relativistic electron spin in an external electromagnetic field. We use two different Hamiltonians to derive the corresponding spin dynamics. These two are: (a) the Dirac Hamiltonian in the presence of an external field, and (b) the semirelativistic expansion of the same. Considering the Foldy-Wouthuysen and Pryce spin operators we show that these lead to different spin dynamics in an external electromagnetic field, which offers possibilities to distinguish their action. We find that the dynamics of both spin operators involve spin-dependent and spin-independent terms, however, the Foldy-Wouthuysen spin dynamics additionally accounts for the relativistic particle-antiparticle coupling. We conclude that the Pryce spin operator provides a suitable description of the relativistic spin dynamics in a weak-to-intermediate external field, whereas the Foldy-Wouthuysen spin operator is more suitable in the strong field regime.