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arXiv:quant-ph/9805004

Wigner transform and quantum-like corrections for charged-particle beam transport

Renato Fedele, Vladimir I. Man’ko

Published May 1, 1998• 0 citations
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Abstract

It is shown that charged-particle beam transport in the paraxial approximation can be effectively described with a quantum-like picture in semiclassical approximation. In particular, the classical Liouville equation can be suitably replaced by a von Neuman-like equation. Relevant remarks concerning the standard classical description of the beam transport are given.

DOI
10.48550/arxiv.quant-ph/9805004

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It is shown that charged-particle beam transport in the paraxial approximation can be effectively described with a quantum-like picture in semiclassical approximation. In particular, the classical Liouville equation can be suitably replaced by a von Neuman-like equation. Relevant remarks concerning the standard classical description of the beam transport are given.
It is shown that the standard classical picture of charged-particle beam transport in paraxial approximation may be conveniently replaced by a Wigner-like picture in a semiclassical approximation. In this effective description, the classical phase-space equation for electronic rays is replaced by a von Neumann--like equation, where the transverse emittance plays the role of $\ensuremath{\Elzxh}$. Relevant remarks concerning the quantumlike corrections for an arbitrary potential in comparison with the standard classical description of the beam transport are given.
An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
An alternative procedure to the one by Gloge and Marcuse [J. Opt. Soc. Am. 59, 1629 (1969)] for performing the transition from geometrical optics to wave optics in the paraxial approximation is presented. This is done by employing a recent "deformation" method used to give a quantumlike phase-space description of charged-particle-beam transport in the semiclassical approximation. By taking into account the uncertainty relation (diffraction limit) that holds between the transverse-beam-spot size and the rms of the light-ray slopes, the classical phase-space equation for light rays is deformed into a von Neumann-like equation that governs the phase-space description of the beam transport in the semiclassical approximation. Here, Planck's constant and the time are replaced by the inverse of the wave number, not lambda, and the propagation coordinate, respectively. In this framework, the corresponding Wigner-like picture is given and the quantumlike corrections for an arbitrary refractive index are considered. In particular, it is shown that the paraxial-radiation-beam transport can also be described in terms of a fluid motion equation, where the pressure term is replaced by a quantumlike potential in the semiclassical approximation that accounts for the diffraction of the beam. Finally, a comparison of this fluid model with Madelung's fluid model is made, and the classical-like picture given by the tomographic approach to radiation beams is advanced as a future perspective.
A recent proposal (see quant-ph/9803068) to simulate semiclassical corrections to classical dynamics by suitable classical stochastic fluctuations is applied to the specific instance of charged beam dynamics in particle accelerators. The resulting picture is that the collective beam dynamics, at the leading semiclassical order in Planck constant can be described by a particular diffusion process, the Nelson process, which is time-reversal invariant. Its diffusion coefficient Nλc\sqrt{N}λ_{c}N​λc​ represents a semiclassical unit of emittance (here NNN is the number of particles in the beam, and λcλ_{c}λc​ is the Compton wavelength). The stochastic dynamics of the Nelson type can be easily recast in the form of a Schroedinger equation, with the semiclassical unit of emittance replacing Planck constant. Therefore we provide a physical foundation to the several quantum-like models of beam dynamics proposed in recent years. We also briefly touch upon applications of the Nelson and Schroedinger formalisms to incorporate the description of collective coherent effects.
In this paper we briefly review the main aspects of a recent proposal to simulate semiclassical corrections to classical dynamics by suitable classical stochastic fluctuations, and we apply it to the specific instance of charged beams in particle accelerators. The resulting picture is that the collective beam dynamics, at the leading semiclassical order in Planck constant can be described by a particular diffusion process, the Nelson process, that is time--reversal invariant. Its diffusion coefficient p N c represents a semiclassical unit of emittance (here N is the number of particles in the beam, and c is the Compton wavelength). The stochastic dynamics of the Nelson type can be easily recast in the form of a Schroedinger equation, with the semiclassical unit of emittance replacing Planck constant. Therefore we provide a physical foundation to the several quantum--like models of beam dynamics proposed in recent years. We also briefly touch upon applications of the Nelson and Schro...
In molecular dynamics applications there is a growing interest in mixed quantum-classical models. The quantum-classical Liouville equation (QCL) describes most atoms of the molecular system under consideration by means of a classical phase space density but an important, small portion of the system by means of quantum mechanics. The QCL is derived from the full quantum dynamical (QD) description by applying the Wigner transform to the \\classical part" of the system only. We discuss the conditions under which the QCL model approximates the full QD evolution of the system. First, analysis of the asymptotic properties of the Wigner transform shows that solving the QCL yields a rst order approximation of full quantum dynamics. Second, we discuss the adiabatic limit of the QCL. This discussion shows that the QCL solutions may be interpretated as classical phase space densities, at least near the adiabatic limit. Third, it is demonstrated that the QCL yields good approximations of non-adia...