Uniform approximation for diffractive contributions to the trace formula in billiard systems

Path integration over closed loops and Gutzwiller's trace formula

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of “Gutzwiller's trace formula” as a leading term in an asymptotic expansion around classical periodic orbits.

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

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.

Using the classical (CSC), primitive (PSC), and uniform (USC) semiclassical expressions for transition probabilities given by Miller and co-workers, we have calculated the reactive and nonreactive 0 → 0 and 0 → 1 transition probabilities for the collinear H + H2 exchange reaction. Comparison with previously calculated exact quantum and quasiclassical results for the reactive and nonreactive 0 → 0 transitions reveals that the semiclassical approximations are not very good, especially the CSC and PSC ones. All three semiclassical probabilities for the reactive 0 → 0 transition exceed unity in the collision energy range from 0.0 to 0.2 eV above the quasiclassical reaction threshold. This feature coupled with the failure of any of the semiclassical approximations to produce the marked quantum effects present in this transition causes these results to be less accurate than the corresponding quasiclassical ones. For the reactive and nonreactive 0 → 1 transitions the USC results are in qualitative agreement with the exact quantum ones and are better than the standard quasiclassical results. However, the reverse quasiclassical results are almost as good as the USC ones for these transitions. A probable reason for the inability of the USC expression to produce the strong oscillations observed in the exact quantum results is that the latter are due to interference between direct and resonant (i.e., compound state) processes whereas the present formulation of the semiclassical method does not encompass such phenomena. A comparison of the total reaction probabilities obtained by the USC and quasiclassical methods with the exact quantum one indicates that the USC result is more accurate than the quasiclassical one, except at collision energies less than 0.50 eV. This improved accuracy is due to a partial cancellation of errors in the contributing 0 → 0 and 0 → 1 USC reactive transition probabilities.

In an urban environment, radio wave interaction with buildings may affect considerably the propagation of radio waves, mainly because of multipath, diffraction, and shadowing effects. This paper outlines the effect of two-dimensional (2-D) (vertical and lateral) diffraction and compares a Fresnel-Kirchhoff scalar approach to a vectorial geometrical theory of diffraction (GTD) approach applied to a four-ray system. The former can be considered as a generalization of conventional knife-edge diffraction theory while the latter (GTD) is useful in the context of ray tracing. A very good agreement between the Fresnel-Kirchhoff and GTD results is found within the limits of validity of both methods. Moreover, it is shown that conventional knife-edge diffraction theory can largely overestimate attenuation (10-15 dB) behind tall buildings in the center of a city environment. Different frequency bands have been tested for the sake of comparison, and emphasis has been put on the radio communications frequency bands used for the global system for mobile communications (GSM) and DCS/DECT systems at 900 and 1800 MHz.

(1) What is semiclassical Analysis? (2) What is a Trace Formula? (3) Physical Motivations : “Quantum Chaos” (4) Gutzwiller Trace Formula. (Rigorous Approach)

This hybrid technique is a method for solving electromagnetic problems in which an antenna is located near a conducting body. The technique accomplishes this by casting the antenna structure in a moment method (MM) format, then modifying that format to account for the effects of the conducting body via the geometrical theory of diffraction (GTD). The technique extends the moment method to handle problems that cannot be solved by GTD or the moment method alone. Wire antennas are analyzed to find their input impedance when they are located near perfectly conducting circular cylinders, although the methods used are not restricted to circular cylinders. Three orthogonal orientations are identified, and antennas to match them are analyzed. For each case, the hybrid solution is checked with one of three independent solutions: an MM-eigenfunction solution, image theory, or experimental measurement. In almost all cases, excellent agreement is obtained due in large part to the fact that the moment method near fields are, for the first time, cast into a ray optical form.

One-particle, one-hole and two-particle, two-hole level densities are calculated in the local-density, i.e., Thomas-Fermi, approach. For the harmonic oscillator this leads to analytical expressions which are in excellent agreement with the exact quantum results.

A comparison is made between approximations using the geometrical theory of diffraction (GTD) and Kirchhoff theory and the exact solution using Maxwell's equations for the case of Fraunhofer diffraction of electromagnetic radiation incident upon a long, thin, perfectly conducting strip with the electric field vector polarized perpendicular to the strip axis. Strip widths from approximately 0.3lambda to 3lambda are considered. Glancing angles of incidence are taken from 4 degrees to 90 degrees . Irradiances are compared as a function of diffraction angle in the region on the same side of the strip as the incident radiation.

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.

We have developed a unified quantum molecular dynamics method, in which the centroid molecular dynamics including the quantum effect of atomic nuclei is unified with the semiempirical molecular orbital calculation for the atomic configuration generated in the molecular dynamics. This unified method has successfully been applied to a couple of diatomic molecular systems for examining its performance. The method has yielded excellent agreement with exact quantum results and taken sufficiently short CPU time for practical use. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

Part I presents several sets of comparisons of semi-classical, quasi-classical and exact quantum reactive scattering calculations for collinear chemical reactions. The possibility of modifying the standard quasi-classical method according to a quantum criterion is investigated. The systems studied are H + H_2, F + H_2, and F + D_2. In addition, a theoretical investigation of the semi-classical S matrix is made. Details of a quasi-classical current density analysis of the H + H_2 reaction are presented and a comparison with exact quantum results is made. A direct test of two versions of the vibrationally adiabatic theory of chemical reactions is made in Part II for the H + H_2 reaction. The adiabaticity of the symmetric stretch motion of the H_3 transition state is focussed upon. In addition, a determination of the completeness of adiabatic basis sets for scattering calculations is made. The theory of electronically non-adiabatic chemical reactions is presented in Part III. Quantum calculations of the collinear H^+ + H_2 → H_2 + H^+ reaction are described. A model and a realistic potential energy surface are employed in these calculations. A fictitious electronically non-adiabatic H + H_2 collinear chemical reaction is treated quantum mechanically. Two potential energy surfaces and a coupling surface are developed for this purpose. The reaction Ba(^1S) + ON_2(X^1Σ) → BaO(X^1Σ) + N_2(X^1Σ^+_g), BaO(a^3II) + N_2(X^1Σ^+_g) is studied quantum mechanically. The singlet and triplet potential energy surfaces are devised as is a spin-orbit coupling surface. Electronically adiabatic and non-adiabatic transition probabilities are calculated as a function of the initial translational energy of the reagents.

An efficient time-dependent (TD) Monte Carlo (MC) importance sampling method has recently been developed [G. Tao and W. H. Miller, J. Chem. Phys. 135, 024104 (2011)] for the evaluation of time correlation functions using the semiclassical (SC) initial value representation (IVR) methodology. In this TD-SC-IVR method, the MC sampling uses information from both time-evolved phase points as well as their initial values, and only the "important" trajectories are sampled frequently. Even though the TD-SC-IVR was shown in some benchmark examples to be much more efficient than the traditional time-independent sampling method (which uses only initial conditions), the calculation of the SC prefactor-which is computationally expensive, especially for large systems-is still required for accepted trajectories. In the present work, we present an approximate implementation of the TD-SC-IVR method that is completely prefactor-free; it gives the time correlation function as a classical-like magnitude function multiplied by a phase function. Application of this approach to flux-flux correlation functions (which yield reaction rate constants) for the benchmark H + H(2) system shows very good agreement with exact quantum results. Limitations of the approximate approach are also discussed.