Kin'ya Takahashi

Wigner and Husimi Functions in Quantum Mechanics

The quantum dynamics in Husimi representation is studied for the commutation relation, time development of distribution function and eigen distribution function. In the integrable system, the Husimi eigen distribution function is confined almost on the classical torus and can be approximated by Gaussian distribution with the variance \((\Delta\xi_{l})^{2}{\sim}\hbar/2\) around the torus. Further, considering the correspondence between quantum and classical mechanics, the Husimi distribution function is a better representation than the Wigner distribution function, because coarse graining is usually involved in observational process.

Applicability of symplectic integrator to classically unstable quantum dynamics

Applicability of symplectic integrator (SI) to classically unstable (chaotic) quantum systems is examined, and accuracy and efficiency as a numerical integrator of Schrodinger equation are demonstrated. The second order SI is well known as the split operator method to molecular scientists. Recently, construction of higher order SIs has been developed by several authors. In the present paper, we compare systematically various higher order SI schemes by applying them to a simple quantum chaos system and emphasize the necessity of introducing higher order schemes. Although the higher order SIs have originally been invented for high precision computation of classical trajectories, it will more promisingly be applied to quantum systems. This is because the exponential instability is absent in quantum systems, but an extensive numerical test reveals that the accumulation of error accompanying a long time integration by SI reflects the stability of the system in the classical limit, and closer attention must be ...

Chaos and Time Development of Quantum Wave Packet in Husimi Representation

Time development of quantum wave packet is studied in the Hisumi representation for regular and irregular case. The Husimi distribution function spreads on tori or into irregular region for regular or irregular case, respectively and coincides to classical probability distribution function. However by the quantum mechanical interference effects, small packets appear in the process of time development and thus the Husimi distribution function distributes inhomogeneously in contrast to the classical uniform probability distribution function. These small packets are shown to be more stable than classical one.

APPLICATION OF SYMPLECTIC INTEGRATOR TO STATIONARY REACTIVE-SCATTERING PROBLEMS : INHOMOGENEOUS SCHRODINGER EQUATION APPROACH

The FFT-symplectic integrator (SI) scheme devised for solving the wave packet propagation problem is applied to stationary reactive-scattering problems. In order to relate the stationary problem to the time-dependent problem, a class of Schrodinger equation with an inhomogeneous wave source term is introduced. By using the equivalence between the stationary scattering eigenstate and the equilibrium state of the inhomogeneous Schrodinger equation, the scattering eigenstates can be computed by integrating the inhomogeneous Schrodinger equation with the FFT-SI scheme. A Gaussian wave source is proposed as an efficient wave source exhibiting rapid relaxation toward the eigenstate. Our method is tested by a one-dimensional example which has an analytical solution, and great numerical accuracy is confirmed. It is further examined by an example of time-dependent scattering and by a two-dimensional example of chaotic tunnel-scattering.

Complex-classical mechanism of the tunnelling process in strongly coupled 1.5-dimensional barrier systems

The fringed tunnelling, which can be observed in strongly coupled 1.5-dimensional barrier systems as well as in autonomous two-dimensional barrier systems, is a manifestation of intrinsic multi-dimensional effects in the tunnelling process. In this paper, we investigate such an intrinsic multi-dimensional effect on the tunnelling by means of classical dynamical theory and semiclassical theory, which are extended to the complex domain. In particular, we clarify the underlying classical mechanism which enables multiple tunnelling trajectories to simultaneously contribute to the wavefunction, thereby resulting in the formation of the remarkable interference fringe on it. Theoretical analyses are carried out in the low-frequency regime based upon a complexified adiabatic tunnelling solution, together with the Melnikov method extended to the complex domain. These analyses reveal that the fringed tunnelling is a result of a heteroclinic-like entanglement between the complexified stable manifold of the barrier-top unstable periodic orbit and the incident beam set. Tunnelling particles reach the real phase plane very promptly, guided by the complexified stable manifold, which gives quite a different picture of the tunnelling from the ordinary instanton mechanism. More fundamentally, the entanglement is attributed to a divergent movement of movable singularities of the classical trajectory, namely, to a singular dependence of singularities on its initial condition.

Husimi Function around Hyperbolic Point

Husimi functions for eigen states around a hyperbolic point are studied. Husimi functions for eigen states of wave forms with even and odd parity are strictly concerned with the hyperbolic point and the separatrix in classical limit (\(\hbar{\rightarrow}0\)), respectively. It is interesting that classical behavior is affected by quantum symmetry.

Complex-Domain Semiclassical Theory: Application to Time-Dependent Barrier Tunneling Problems

Semiclassical theory based upon complexified classical mechanics is developed for periodically time-dependent scattering systems, which are minimal models of multi-dimensional systems. Semiclassical expression of the wave-matrix is derived, which is represented as the sum of the contributions from classical trajectories, where all the dynamical variables as well as the time are extended to the complex-domain. The semiclassical expression is examined by a periodically perturbed 1D barrier system and an excellent agreement with the fully quantum result is confirmed. In a stronger perturbation regime, the tunneling component of the wave-matrix exhibits a remarkable interference fringes, which is clarified by the semiclassical theory as an interference among multiple complex tunneling trajectories. It turns out that such a peculiar behavior is the manifestation of an intrinsic multi-dimensional effect closely related to a singular movement of singularities possessed by the complex classical trajectories.

Numerical Study on Acoustic Oscillations of 2D and 3D Flue Organ Pipe Like Instruments with Compressible LES

Acoustic oscillations of flue instruments are investigated numerically using compressible Large Eddy Simulation (LES). Investigating 2D and 3D models of flue instruments, we reproduce acoustic oscillations excited in the resonators as well as an important characteristic feature of flue instruments – the relation between the acoustic frequency and the jet velocity described by the semi-empirical theory developed by Cremer & Ising, Coltman and Fletcher et al. based on experimental results. Both 2D and 3D models exhibit almost the same oscillation frequency for a given jet velocity, but the acoustic oscillation as well as the jet motion is more stable in the 3D model than in the 2D model, due to less stability in 3D fluid of the rolled up eddies created by the collision of the jet with the edge, which largely disturb the jet motion and acoustic field in the 2D model. We also investigate the ratio of the amplitude of the acoustic flow through the mouth opening to the jet velocity, comparing with the experimental results and semi-empirical theory given by Hirschberg et al.. PACS numbers: 43.75.Qr,43.75.Np,43.75.Ef,43.75.-z,43.28.Ra

Numerical Study on Multi-Stable Oscillations of Woodwind Single-Reed Instruments

Single reed instruments with cylindrical and conical bores were studied numerically, focusing on the dependence of the acoustic mechanism on the bore geometry. As experimentally demonstrated by Idogawa et al. [J. Acoust. Soc. Am., Vol.98, p.540 (1993)], reed instruments can be characterized as multi-attractor systems, which exhibit multi-stable oscillations and hysteretic transitions among oscillating states with change of a control parameter, such as blowing pressure in the mouth. We numerically analyzed dynamical models of single reed instruments to clarify differences in the acoustic mechanisms between cylindrical and conical bore instruments. We analyzed a dynamical model of a cylindrical pipe fitted with a clarinet mouthpiece(CPCM), as a simplified model of the clarinet, and a dynamical model of a truncated cone with a clarinet mouthpiece (TCCM), as a model of conical-bore instruments, together with intermediately shaped models. We found a clear difference in the global structure of transition diagrams (i.e., the overall picture showing the transitions among all the excited states with change of the blowing pressure) between CPCM and TCCM, together with interesting structural changes of the transition diagrams in the intermediate models.

Sounding Mechanism of a Cylindrical Pipe Fitted with a Clarinet Mouthpiece

The sounding mechanism of a cylindrical pipe fitted with a clarinet mouthpiece (CPCM) is studied. The main aim of this paper is to introduce a reliable and minimal model which satisfactorily reproduces nonlinear vibrations excited in the air column and associated hysteretic transitions between them when the blowing pressure is varied as a control parameter. Such phenomena are commonly observed for woodwind reed instruments blown artificially, as reported by Idogawa et al. The noteworthy point of our model is that a reflection function consists of two main inverted peaks, one with a long delay, which represents the reflection from the open pipeend, and the other with a short delay, which originates from the irregular bore geometry of the mouthpiece. Our numerical calculation is carried out using the Schumacher model [Acustica 48 (1981), 71], including such a reflection function. Many kinds of nonlinear vibrations, periodic and quasi-periodic vibrations, as well as the hysteretic transitions between them are obtained numerically. Our results reproduce those observed experimentally quite well. They strongly suggest that the sound wave reflection due to the irregularity of the mouthpiece is one of the essential properties of woodwind sounding.