Quantum mechanical calculation of spectral statistics of a modified Kepler problem

Periodic orbit (PO) theory can be used to make connections between the quantum energy eigenvalue spectrum and the closed orbits of the corresponding classical system. The two-dimensional annular billiard or circular disk system (namely, a particle in the plane confined between inner and outer infinite circular walls at fR≡Rin<Rout≡R where 0<f<1) is examined in the context of periodic orbit theory to illustrate several novel aspects of the PO analysis. Important features of this problem include (i) the appearance and disappearance of various features in the classical path length spectrum as a parameter (in this case f=Rin/Rout), is continuously varied, (ii) the presence of path length features which do not correspond to classical trajectories, but are rather due to purely wavelike phenomena (namely, diffraction around the inner annulus), and (iii) the study of the contribution of different regions in the energy eigenvalue space to different classes of classical trajectories. This last feature is a general observation about periodic orbit theory which can be usefully applied to a number of billiard systems. In addition, since the annular disk system is seldom studied in detail in either classical or quantum mechanics, we provide an appendix which reviews some of the more traditional connections between the classical and quantum probability densities for this geometry, exemplifying some of the important differences between one- and two-dimensional systems which arise from purely geometrical rather than kinematical effects.

Classical and quantum mechanics of a strongly chaotic billiard system

We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.

Vertical thermal convection exhibits weak turbulence and spatio-temporally chaotic behaviour. For this configuration, we report seven new equilibria and 26 new periodic orbits. These orbits, together with four previously studied in Zheng et al. ( J. Fluid Mech. , 2024 b , vol. 1000, p. A29) bring the number of periodic-orbit branches computed so far to 30, all solutions to the fully nonlinear three-dimensional Navier–Stokes equations. These new and unstable invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects. These interesting and important fluid mechanical processes in a small flow unit are shown to also appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Furthermore, these orbits are shown to be able to reconstruct statistically the core part of the attractor, so that these results may contribute to a quantitative description of transitional fluid turbulence using periodic orbit theory.

We investigate the resonance spectrum of the Hénon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller's trace formula, complemented by a uniform approximation for a co-dimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.

We investigate the resonance spectrum of the H\\\'enon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller\'s trace formula, complemented by a uniform approximation for a codimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.

We explore the transient and the stationary behavior of proposed measures for chaotic dynamical systems, with particular reference to thermostated nonequilibrium systems. The Kawasaki distribution function formalism, which is based on the Liouville equation, is shown to give a good description of the transient measure. An orbital measure with a support based on unstable periodic orbits is shown to give the stationary measure. Also, we emphasize the importance of determining the set of observables by which a nonequilibrium measure can be defined. The stationarity of these methods is considered in both the periodic orbit theory and the Liouville picture for the change in the distribution function. \textcopyright{} 1996 The American Physical Society.

We first give a short outline of the semiclassical periodic orbit theory that allows one to relate quantum shell fluctuations in a mean-field system to the periodic orbits of the corresponding classical system. Earlier applications to gross-shell effects in nuclei (ground-state deformations), metal clusters (ground-state deformations and supershells), and quantum dots (conductance oscillations in weak magnetic fields) are briefly summarized. We then report on a recent simple and transparent semiclassical interpretation of the origin of the mass asymmetry in the fission of heavy nuclei. Using only a few classical periodic orbits and a cavity model for the nuclear mean field, we explain the onset of left-right asymmetric shapes at the fission isomer minimum by the principle of constant action of the shortest orbits.

It is possible to measure in the laboratory the periods Tk or the classical actions Sk of classical periodic orbits of an atomic electron. (We discuss highly excited states of the atom, in which the electron behaves much like a classical particle, and classical orbits become observable.) We will examine situations in which classical dynamics of the electron makes a transition from orderly to chaotic. In this circumstance, periodic orbits proliferate—existing orbits bifurcate, sending out new ones, and new periodic orbits appear ‘‘out of nowhere.’’ In this paper we briefly review experimental measurements carried out by various groups that display these recurrences, and we review the theory that gives a quantitative description of these recurrences and their bifurcations.

A comparison between entropies of aromatic compounds from quantum mechanical calculations and experiment

We have conducted potential of mean force (PMF) analyses to derive the geometrical parameters of various types of hydrogen bonds on protein-ligand binding interface. Our PMF analyses are based on a set of 4535 high-quality protein-ligand complex structures, which are compiled through a systematic mining of the entire Protein Data Bank. Hydrogen bond donor and acceptor atoms are classified into several basic types. Both distance- and angle-dependent statistical potentials are derived for each donor-acceptor pair, from which distance and angle cutoffs are obtained in an objective, unambiguous manner. These donor-acceptor pairs are also studied by quantum mechanics (QM) calculations at the MP2/6-311++G** level on model molecules. Comparison of the outcomes of PMF analyses and QM calculations suggests that QM calculation may serve as an alternative approach for characterizing hydrogen bond geometry. Both of our PMF analyses and QM calculations indicate that C-H···O hydrogen bonds are relatively weak as compared to common hydrogen bonds formed between nitrogen and oxygen atoms. A survey on the protein-ligand complex structures in our data set has revealed that Cα-H···O hydrogen bonds observed in protein-ligand binding are frequently accompanied by bifurcate N-H···O hydrogen bonds. Thus, the Cα-H···O hydrogen bonds in such cases would better be interpreted as secondary interactions.

The semiclassical limit of the quantum few-body problem has not been studied in general terms from the point of view of periodic orbit theory. In a previous paper, we studied noninteracting two-body systems [Phys. Rev. A 62, 042109 (2000)] and discussed the fact that the periodic orbits occur in continuous families. Interactions destroy the periodic orbit families leaving a discrete set of isolated periodic orbits. In this paper, we consider the effect of weak two-body interactions, which can be thought of as symmetry-breaking perturbations and can thus be analyzed using a theory developed by Creagh [Ann. Phys. (N.Y.) 248, 1 (1996)]. The P\"oschl-Teller two-body system confined in a square well is analyzed to illustrate the use of the formalism. It is shown that the effect of the interaction can be evaluated for all two-particle periodic orbits, and that the coarse-grained quantum density of states can be fully reproduced from simply summing the perturbed contributions of each periodic orbit family. Good numerical estimates of the quantum singlet energies can actually be obtained, but it is found that that perturbed trace formulas cannot reproduce the multiplet splittings predicted from quantum mechanics. Several interesting properties are observed depending on the range of the interaction and on whether the interaction is attractive or repulsive.

How classical chaos manifests itself in microscopic systems where a quantum description must be used, is the subject of quantum chaos. In these lectures, I present the concepts and tools used in quantum chaos - random matrix theory, periodic orbit theory, localization theory - and show their relevance for studying complex atomic systems.

This study investigates the dynamics of an acoustically driven airbubble in water. Depending on the values of external parameters, theradial oscillations of the bubble can be either stable or chaotic. Thenecessary condition of chaotic behaviour is identified to be thenon-zero amplitude of the bubble's afterbounces at the beginning of thenext acoustic cycle, which brings memory into the system. We show thatfor some parameter values in the chaotic regime the dynamics can bereduced to a unimodal map. At these parameter values the periodic orbittheory is successfully applied to calculate averages of relevantphysical quantities, such as the air concentration at which the bubbleis in diffusive equilibrium with the surrounding liquid. Finally weinvestigate the convergence of the calculated quantities.

Manifestations of quantum chaos in resonant tunnelling