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arXiv:2206.14901 · Modern Physics Letters A

Exact quantum-mechanical equations for particle beams

Alexander J. Silenko

Published May 30, 2022• 8 citations
arXivHTMLPDF

Abstract

The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.

DOI
10.1142/s0217732322500973

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The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.
In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.
In the traditional prescriptions, the Fermat’s principle of least time or/and the Helmholtz equation are used to derive the beam-optical Hamiltonian. Now, we know that the Maxwell’s electromagnetism gives rise to the correct theory of light. So, we developed a formalism of beam optics starting with an exact matrix-representation of Maxwell’s equations. The resulting beam-optical Hamiltonian is not only exact but also has the mathematical structure of the Dirac equation, enabling us to employ quantum methodologies. The resulting formalism of Maxwell optics successfully describes beam optics along with polarization of light.
General guidelines for constructing a quantum theory of charged-particle beam optics starting ab initio from the basic equations of quantum mechanics, appropriate to the situation under study. In the context of spin-1/2 particles, these guidelines are used starting with the Dirac equation. The spinor theory just constructed is used to obtain the transfer maps for normal and skew magnetic quadrupoles respectively. As expected the traditional transfer maps get modified by the quantum contributions. The classical limit of the quantum formalism presented, reproduces the well-known Lie algebraic formalism of charged-particle beam optics.
The quantum theory of a magnetic electron lens with rotational symmetry about its straight optic axis has recently been studied entirely on the basis of the Dirac equation [R. Jagannathan, R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Lett. A 134, 457 (1989)]. Following the same type of algebraic approach, the present paper elaborates on the quantum mechanics of beam transport through a general electron-lens system with a straight axis. The basic theory is developed in a general framework, and examples of applications are treated under the paraxial approximation and thin-lens assumption. Extension of the theory to systems with curved optic axes is dealt with very briefly. The present formalism based on a ``beam-optical'' representation of the Dirac equation goes over directly to the Hamiltonian formalism of geometrical electron optics upon classicalization.