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arXiv:physics/0112085 · Quantum Aspects of Beam Physics

QUANTUM MECHANICAL FORMALISM OF PARTICLE BEAM OPTICS

Sameen Ahmed Khan

Published May 1, 2002• 4 citations
arXivHTMLPDF

Abstract

A general procedure for construction of the formalism of quantum beam optics for any particle is reviewed. The quantum formalism of spin-1/2 particle beam optics is presented starting {\em ab initio} with the Dirac equation. As an example of application the case of normal magnetic quadrupole lens is discussed. In the classical limit the quantum formalism leads to the well-known Lie algebraic formalism of classical particle beam optics.

DOI
10.1142/9789812777447_0042

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A general procedure for construction of the formalism of quantum beam optics for any particle is reviewed. The quantum formalism of spin-1/2 particle beam optics is presented starting {\em ab initio} with the Dirac equation. As an example of application the case of normal magnetic quadrupole lens is discussed. In the classical limit the quantum formalism leads to the well-known Lie algebraic formalism of classical particle beam optics.
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.
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.
This chapter reviews quantum mechanics required to pursue the quantum charged particle beam optics, the central theme of this book. All physical phenomena are quantum mechanical at the fundamental level. Classical mechanics observed in the macroscopic world is an approximation. That is why it raises a curiosity when classical mechanics is found to be very successful at a microscopic level in the design and operation of beam devices like an electron microscope or a particle accelerator. This chapter gives an introductory review of nonrelativistic and relativistic quantum mechanics just at the level necessary for presenting the formalism of quantum charged particle beam optics. All the basic equations of quantum mechanics, Schrödinger’s time-evolution equation, the Klein-Gordon equation, and the Dirac equation, are given in detail. The Feshbach-Villars formalism of the Klein-Gordon equation and the Foldy-Wouthuysen transformation of the Dirac equation, forming the basis for the development of quantum charged particle beam optics, are described.
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.