Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userComplex Trajectories in a Classical Periodic Potential
230
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper examines the complex trajectories of a classical particle in the potential V(x)=-cos(x). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories x(t) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position x(0) after some real time T. The second class consists of trajectories for which there exists a real time T such that $x(t+T)=x(t) pm2 pi$. These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.
rate research
Read More
According to Dirac, fundamental laws of Classical Mechanics should be recovered by means of an appropriate limit of Quantum Mechanics. In the same spirit it is reasonable to enquire about the fundamental geometric structures of Classical Mechanics which will survive the appropriate limit of Quantum Mechanics. This is the case for the symplectic structure. On the contrary, such geometric structures as the metric tensor and the complex structure, which are necessary for the formulation of the Quantum theory, may not survive the Classical limit, being not relevant in the Classical theory. Here we discuss the Classical limit of those geometric structures mainly in the Ehrenfest and Heisenberg pictures, i.e. at the level of observables rather than at the level of states. A brief discussion of the fate of the complex structure in the Quantum-Classical transition in the Schroedinger picture is also mentioned.
Missing bound-state solutions for fermions in the background of a Killingbeck radial potential including an external magnetic and Aharonov-Bohm (AB) flux fields are examined. The correct quadratic form of the Dirac equation with vector and scalar couplings under the spin and pseudo-spin symmetries is showed and also we point out a misleading treatment in the literature regarding to bound-state solutions for this problem.
Electromagnetic field interactions in a dielectric medium represent a longstanding field of investigation, both at the classical level and at the quantum one. We propose a 1+1 dimensional toy-model which consists of an half-line filling dielectric medium, with the aim to set up a simplified situation where technicalities related to gauge invariance and, as a consequence, physics of constrained systems are avoided, and still interesting features appear. In particular, we simulate the electromagnetic field and the polarization field by means of two coupled scalar fields $phi$,$psi$ respectively, in a Hopfield-like model. We find that, in order to obtain a physically meaningful behaviour for the model, one has to introduce spectral boundary conditions depending on the particle spectrum one is dealing with. This is the first interesting achievement of our analysis. The second relevant achievement is that, by introducing a nonlinear contribution in the polarization field $psi$, with the aim of mimicking a third order nonlinearity in a nonlinear dielectric, we obtain solitonic solutions in the Hopfield model framework, whose classical behaviour is analyzed too.
This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.
An omega-meson extension of the Skyrme model - without the Skyrme term but including the pion mass - first considered by Adkins and Nappi is studied in detail for baryon numbers 1 to 8. The static problem is reformulated as a constrained energy minimisation problem within a natural geometric framework and studied analytically on compact domains, and numerically on Euclidean space. Using a constrained second-order Newton flow algorithm, classical energy minimisers are constructed for various values of the omega-pion coupling. At high coupling, these Skyrmion solutions are qualitatively similar to the Skyrmions of the standard Skyrme model with massless pions. At sufficiently low coupling, they show similarities with those in the lightly bound Skyrme model: the Skyrmions of low baryon number dissociate into lightly bound clusters of distinct 1-Skyrmions, and the classical binding energies for baryon numbers 2 through 8 have realistic values.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
