Spectral oscillations, periodic orbits, and scaling (original) (raw)
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Periodic orbits, spectral oscillations, scaling, and vacuum energy: beyond HaMiDeW
Nuclear Physics B - Proceedings Supplements, 2002
Oscillations in eigenvalue density are associated with closed orbits of the corresponding classical (or geometrical optics) system. Although invisible in the heat-kernel expansion, these features determine the nonlocal parts of propagators, including the Casimir energy. I review some classic work, discuss the connection with vacuum energy, and show that when the coupling constant is varied with the energy, the periodic-orbit theory for generic quantum systems regains the clarity and simplicity that it always had for the wave equation in a cavity.
From Classical Periodic Orbits in Integrable pi\pipi-Rational Billiards to Quantum Energy Spectrum
arXiv: Quantum Physics, 2018
In the present note, we uncover a remarkable connection between the length of periodic orbit of a classical particle enclosed in a class of 2-dimensional planar billiards and the energy of a quantum particle confined to move in an identical region with infinitely high potential wall on the boundary. We observe that the quantum energy spectrum of the particle is in exact one-to-one correspondence with the spectrum of the amplitude squares of the periodic orbits of a classical particle for the class of integrable billiards considered. We have established the results by geometric constructions and exploiting the method of reflective tiling and folding of classical trajectories. We have further extended the method to 3-dimensional billiards for which exact analytical results are scarcely available - exploiting the geometric construction, we determine the exact energy spectra of two new tetrahedral domains which we believe are integrable. We test the veracity of our results by comparing them with numerical results.
Macroscopic quantum behaviour of periodic quantum systems
In this paper we introduce a simple procedure for computing the macroscopic quantum behaviour of periodic quantum systems in the high energy regime. The macroscopic quantum coherence is ascribed to a one-particle state, not to a condensate of a many-particle system; and we are referring to a system of high energy but with few degrees of freedom. We show that, in the first order of approximation, the quantum probability distributions converge to its classical counterparts in a clear fashion, and that the interference effects are strongly suppressed. The harmonic oscillator provides a testing ground for these ideas and yields excellent results. * Electronic address: alberto.martin@nucleares.unam.mx
Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Model
Physical Review Letters, 1994
We examine photodetachment of H in parallel electric and magnetic fields, hv + H H + e using semiclassical approximations. The fields cause the electron to return to the atom, producing recurrences that are visible as interference oscillations in the photodetachment cross section. As the energy is varied, new returning orbits are created through bifurcations.
Quantum-classical correspondence in the vicinity of periodic orbits
Physical review. E, 2018
Quantum-classical correspondence in chaotic systems is a long-standing problem. We describe a method to quantify Bohr's correspondence principle and calculate the size of quantum numbers for which we can expect to observe quantum-classical correspondence near periodic orbits of Floquet systems. Our method shows how the stability of classical periodic orbits affects quantum dynamics. We demonstrate our method by analyzing quantum-classical correspondence in the quantum kicked top (QKT), which exhibits both regular and chaotic behavior. We use our correspondence conditions to identify signatures of classical bifurcations even in a deep quantum regime. Our method can be used to explain the breakdown of quantum-classical correspondence in chaotic systems.
From classical periodic orbits in integrable $ \pi$ π -rational billiards to quantum energy spectrum
The European Physical Journal Plus, 2019
In the present note, we uncover a remarkable connection between the length of periodic orbit of a classical particle enclosed in a class of 2-dimensional planar billiards and the energy of a quantum particle confined to move in an identical region with infinitely high potential wall on the boundary. We observe that the quantum energy spectrum of the particle is in exact oneto-one correspondence with the spectrum of the amplitude squares of the periodic orbits of a classical particle for the class of integrable billiards considered. We have established the results by geometric constructions and exploiting the method of reflective tiling and folding of classical trajectories. We have further extended the method to 3-dimensional billiards for which exact analytical results are scarcely available-exploiting the geometric construction, we determine the exact energy spectra of two new tetrahedral domains which we believe are integrable. We test the veracity of our results by comparing them with numerical results.
Wave functions with localizations on classical periodic orbits in weakly perturbed quantum billiards
Physical Review E, 2006
We analytically investigate quantum wave functions with localizations on classical periodic orbits ͑POs͒ in the circular billiards. We construct the coherent states which are the coherent superpositions of nearly degenerate eigenstates and found to be localized around single classical POs. With the constructed coherent states, we can analytically express the mesoscopic eigenstates in the weakly perturbed systems, which are found to be localized on the ensemble of classical POs. Furthermore, these coherent states can be utilized to study the quantum vortex structures in quantum billiards.
European Journal of Physics, 2009
The dynamical behaviour of simple harmonic motion can be found in numerous natural phenomena. Within the quantum realm of atomic, molecular and optical systems, two main features are associated with harmonic oscillations: a finite ground-state energy and equally spaced quantum energy levels. Here it is shown that there is in fact a one-to-one mapping between the provision of equally spaced quantum energy levels and simple harmonic motion. The analysis establishes that the Hamiltonian of any system featuring quantized energy levels in an evenly spaced, infinite set must have a quadratic dependence on a pair of canonically conjugate variables. Moreover, specific physical inferences can be drawn. For example, exploiting this 'back-to-front' derivation, and based on the known existence of photons, it can be proved that an electromagnetic energy density is quadratic in both the electric and magnetic fields.
Intrinsic periodicity: the forgotten lesson of quantum mechanics
Journal of Physics: Conference Series
Wave-particle duality, together with the concept of elementary particles, was introduced by de Broglie in terms of intrinsically "periodic phenomena". However, after nearly 90 years, the physical origin of such undulatory mechanics remains unrevealed. We propose a natural realization of the de Broglie "periodic phenomenon" in terms of harmonic vibrational modes associated to space-time periodicities. In this way we find that, similarly to a vibrating string or a particle in a box, the intrinsic recurrence imposed as a constraint to elementary particles represents a fully consistent quantization condition. The resulting classical cyclic dynamics formally match ordinary relativistic Quantum Mechanics in both the canonical and Feynman formulations. Interactions are introduced in a geometrodynamical way, similarly to general relativity, by simply considering that variations of kinematical state can be equivalently described in terms of modulations of space-time recurrences, as known from undulatory mechanics. We present this novel quantization prescription from a historical prospective.