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Motion of a wave packet in an external Yang-Mills field

H. Arodź

Published Jan 1, 1983• 0 citations
OpenAlex

Abstract

Two further examples of motion of a wave packet in external Yang–Mills fields are investigated. We find that because of the interaction with the gauge field the wave packet splits into a number of parts which remain spatially separated in the ħ $\to$ 0 limit. Therefore, these examples confirm our point of view that quantum mechanics of colored particles does not have simple classical limit.

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Two further examples of motion of a wave packet in external Yang–Mills fields are investigated. We find that because of the interaction with the gauge field the wave packet splits into a number of parts which remain spatially separated in the ħ →\to→ 0 limit. Therefore, these examples confirm our point of view that quantum mechanics of colored particles does not have simple classical limit.
The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. Wave packet is often taken as the best approximation of quantum theory to a classical particle. If the wave packet is regarded as a point, with zero amplitude everywhere except in a very small region, then its motion can be interpreted as the motion of a single particle. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron wave packets. The localized wave packet solution disperses or spreads in time, however, at a later time, it reforms and appears as its initial state. Revivals in quantum systems were first studied in the Jaynes–Cumming model, describing a two-level atom interacting with a resonant monochromatic field.
It is studied the classical limit of a quantum particle in an external non-abelian gauge field. It is shown that the unitary group describing the quantum fluctuations around any classic phase orbit has a classical limit when h tends to zero under very general conditions on the potentials. It is also proved the self-adjointness of the Hamilton's operator of the quantum theory for a large class of potentials. Some applications of the theory are finally exposed On etudie la limite classique d'une particule quantique dans un champ exterieur non abelien. On demontre que l'existence de la limite classique du groupe unitaire qui decrit les fluctuations quantiques autour d'une orbite classique quelconque est assuree par des conditions peu restrictives sur les potentiels. On demontre que l'operateur hamiltonien de la theorie quantique est auto-adjoint pour une large classe de potentiels
We investigate the dynamics of a charged particle interacting with a multimode quantized electromagnetic field and obtain an analytic solution for the full electron--field system. This framework enables the calculation of position expectation values and uncertainties for arbitrary wave packets and field states, allowing us to identify quantum corrections to the corresponding classical motion. While the corrections to the position expectation value are weak and largely insensitive to the quantum state of the field, the wave packet broadening exhibits a pronounced dependence on the field state. In particular, the quantum uncertainty of the radiation is directly imprinted onto the spatial uncertainty of the particle. We illustrate these effects for Gaussian wave packets interacting with coherent, Fock, and squeezed states, including bright squeezed vacuum. The interaction with a finite-duration laser pulse is also analyzed as a multimode example. Our results provide a transparent analytic route toward understanding how quantum fluctuations of light influence electron dynamics in strong-field settings.
Quantum fluctuations of some systems vanish not only in the limit ℏ→0\hbar\to 0ℏ→0, but also as some other parameters (such as 1N1\over NN1​, the inverse of the number of `colors' of a Yang-Mills theory) vanish. These lead to new classical limits that are often much better approximations to the quantum theory. We describe two examples: the familiar Hartree--Fock-Thomas-Fermi methods of atomic physics as well as the limit of large spatial dimension. Then we present an approach of the Hecke operators on modular forms inspired by these ideas of quantum mechanics. It explains in a simple way why the spectra of these operators tend to the spectrum of random matrices for large weight for the modular forms.
We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets.