Semiclassical Quantization Scheme Research Articles

Adiabatic formation of high-Q modes by suppression of chaotic diffusion in deformed microdiscs

Resonant modes with high-Q factors in a two-dimensional deformed microdisc cavity are analyzed by using a dynamical and semiclassical approach. The analysis focuses particularly on the ultra-small cavity regime, where the scale of a resonant free-space wavelength is comparable with that of the microdisc size. Although the deformed microcavity has strongly chaotic internal ray dynamics, modes with high-Q factors in this regime show unexpectedly regular distributions in configuration space and adiabatic features in phase space. By tracing the evolution process of such high-Q modes through the deformation from a circular cavity, it is uncovered that the high-Q modes are formed adiabatically on cantori. Due to the openness of microcavities, such adiabatic formation of high-Q modes around cantori is enabled, in spite of strong chaos in ray dynamics. Since the cantori are in close contact with short periodic orbits, their influence on the modes, such as localization patterns in phase space, can be also clarified. In order to quantitatively analyze the spectral range where high-Q modes appear, the phase space section of the deformed microcavity is partitioned by partial barriers of short periodic orbits, and the semiclassical quantization scheme is applied to the partitioned areas and their action fluxes. The derived spectral ranges for the high-Q modes show a good agreement with a numerically observed spectrum. In the course of semiclassical quantization, it is shown that the chaotic diffusion in the system that we investigate can be resolved by the scale of a quarter effective Planck's constant, and the topological structure of the manifolds in phase space allows for this resolution higher than a Planck constant scale. By analyzing flux Farey trees, the role of short periodic orbits in chaotic diffusion and their connection to cantori are verified.

Semiclassical quantization of hyperbolic map on torus

We consider the semiclassical quantization of two hyperbolic maps on torus, i.e., the baker's map and the sawtooth map. We demonstrate that for both maps, the semiclassical quantization scheme based on the Riemann–Siegel lookalike formula fails to yield its eigenvalues. The reason for this failure is that the truncation of the infinite series w.r.t. the periodic orbits cannot work for the quantized hyperbolic map on a torus. We show that for the baker's map, an alternative semiclassical quantization scheme including all periodic orbit contribution yields its eigenvalues with reasonable accuracy, although they are, in general, complex-valued. The same scheme for the sawtooth map is constructed.

Triaxial Ellipsoidal Quantum Billiards

The classical mechanics, exact quantum mechanics and semiclassical quantum mechanics of the billiard in the triaxial ellipsoid are investigated. The system is separable in ellipsoidal coordinates. A smooth description of the motion is given in terms of a geodesic flow on a solid torus, which is a fourfold cover of the interior of the ellipsoid. Two crossing separatrices lead to four generic types of motion. The action variables of the system are integrals of a single Abelian differential of second kind on a hyperelliptic curve of genus 2. The classical separability carries over to quantum mechanics giving two versions of generalized Lamé equations according to the two sets of classical coordinates. The quantum eigenvalues define a lattice when transformed to classical action space. Away from the separatrix surfaces the lattice is given by EBK quantization rules for the four types of classical motion. The transition between the four lattices is described by a uniform semiclassical quantization scheme based on a WKB ansatz. The tunneling between tori is given by penetration integrals which again are integrals of the same Abelian differential that gives the classical action variables. It turns out that the quantum mechanics of ellipsoidal billiards is semiclassically most elegantly explained by the investigation of its hyperelliptic curve and the real and purely imaginary periods of a single Abelian differential.

Numerical accuracy of Bogomolny's semiclassical quantization scheme in quantum billiards

We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of section (SOS) and use a finite approximation for the transfer operator in coordinate space. Computation of the eigenvalues of this matrix combined with a quantization condition, determines a set of semiclassical eigenvalues which are compared with those obtained by solving the Schr\odinger equation. The classical dynamics of this billiard system undergoes a smooth transition from integrable (circle) to completely chaotic motion, thus providing a test of Bogomolny's semiclassical method in coordinate space in terms of the morphology of the wavefunction. We analyze the results for billiards which exhibit both soft and hard chaos.

Energy spectra of the hydrogen atom and the harmonic oscillator in two dimensions from Bogomolny's semiclassical transfer operator.

The semiclassical quantization scheme formulated by Bogomolny [E.B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)], employing a suitably chosen Poincar\'e surface of section, has been used to calculate the energy eigenvalues of the hydrogen atom in one and two dimensions and the anisotropic harmonic oscillator in two dimensions. For the two-dimensional systems it was found to be advantageous to decompose Bogomolny's transfer operator into two "half-mapping" operators. This approach, developed by Haggerty [M.R. Haggerty, Ph.D. thesis, Massachusetts Institute of Technology, 1994 (unpublished); Phys. Rev. E 52, 389 (1995)], leads to an analytical solution for the energy eigenvalues of the hydrogen atom. However, the energies are found to depend on the quantum number $n$ as ${(n\ensuremath{-}\frac{1}{4})}^{\ensuremath{-}2}$, unlike the exact quantum energies, which go as ${(n\ensuremath{-}\frac{1}{2})}^{\ensuremath{-}2}$. An attempt to explain this one-quarter shift on the basis of the Langer-modified WKB approximation is only partly successful. For the two-dimensional harmonic oscillator, numerical calculations yield results close to the exact quantum energies.

Electromagnetic form factors of the nucleon in the chiral quark soliton model

In this paper we present the derivation as well as the numerical results for the electromagnetic form factors of the nucleon within the chiral quark soliton model in the semiclassical quantization scheme. The model is based on semibosonized SU(2) Nambu -- Jona-Lasinio lagrangean, where the boson fields are treated as classical ones. Other observables, namely the nucleon mean squared radii, the magnetic moments, and the nucleon--$\Delta$ splitting are calculated as well. The calculations have been done taking into account the quark sea polarization effects. The final results, including rotational $1/N_c$ corrections, are compared with the existing experimental data, and they are found to be in a good agreement for the constituent quark mass of about 420 MeV. The only exception is the neutron electric form factor which is overestimated.

1/Nc rotational corrections to gA in the Nambu-Jona-Lasinio model and charge conjugation.

We show that the 1/${\mathit{N}}_{\mathit{c}}$ rotational corrections of ${\mathit{g}}_{\mathit{A}}$, derived using the semiclassical quantization scheme in the Nambu--Jona-Lasino model, possess correct properties under charge conjugation.

Classical and semiclassical zeta functions in terms of transition probabilities

We propose a semiclassical quantization scheme for bound hyperbolic systems based on the properties of a single ergodic trajectory. The dynamics of the system is approximated by transition probabilities between cells of a partition of the phase-space. We construct a transfer matrix of the corresponding Markov graph which approaches the classical Frobenius-Perron (transfer) operator in the limit of infinitesimal tesselations of the phase-space. A semiclassical zeta function may be obtained as the determinant of an appropriately weighted transfer operator and leads to a product over the closed paths of the graph in close analogy to the Gutzwiller-Voros zeta function which is a product over periodic orbits.

Studies of Bogomolny's semiclassical quantization of integrable and nonintegrable systems

The semiclassical quantization scheme formulated by Bogomolny, employing a suitably chosen Poincare surface of section (PSS), has been used to calculate approximate energy eigenvalues for the quantum analogues of several Hamiltonian systems. Using a finite approximation to the transfer operator in coordinate space, we have carried out calculations of the energy eigenvalues for the circle billiard and the 45 degrees wedge billiard, both of which are integrable systems. Calculations have also been performed for the 49 degrees and 60 degrees wedges, which classically exhibit hard chaos, and for the 41 degrees and 30 degrees wedges which classically exhibit soft chaos or mixed behaviour. In all cases, the low-lying energy eigenvalues are in excellent agreement with the exact quantum energy eigenvalues and, in the case of the 49 degrees wedge, are better than the results obtained using any other semiclassical quantization scheme. We have also studied a variant of Bogomolny's approach which employs the symbolic dynamics to construct a representation of the transfer operator. We show explicitly for the 49 degrees wedge billiard how the accessible part of the PSS in phase space is divided into cells labelled by sequences of n symbols. This leads to a systematic way of finding the pruning rules for this system. Results are presented for the 49 degrees wedge based on schemes employing 2-, 3-, and 4-symbol cells. Because of the extensive pruning in this system, this approach cannot be easily implemented for the general case of n-symbol cells. Furthermore, the numerical results are not as good as those obtained from finite approximation to the transfer operator in coordinate space.

Semiclassical quantization of the periodic Toda chain

Gutzwiller's semiclassical quantization scheme for the trace of Green's function is applied to the periodic Toda chain. We obtain a set of algebraic equations that determine the energy levels arising from a special periodic orbit, namely the single cnoidal wave solution. Our formulae show a simple dependence on the number of particles N in the chain. N merely occurs as a parameter. We perform the soliton limit of our equations and get a semiclassical correction to first order in h to the dispersion relation E=E(p) of a soliton on the infinite chain which is in remarkable agreement with the Bethe ansatz result. The classical data which enter into the semiclassical quantization formula are of interest in their own right. We give a complete treatment of the linear stability analysis of a single cnoidal wave and also some new expressions for its dispersion relation which expresses the frequency v as a function of the wavenumber k.

Rotational corrections to axial currents in semibosonized SU(3) Nambu-Jona-Lasinio model

We examine O(1/ N c) rotational corrections to axial couplings g A (3) and g A (8) in the framework of the semibosonized SU(3) Nambu-Jona-Lasinio model. A novelty is due to the observation that the rotational (cranking) velocity and the rotation matrix itself do not commute within the semiclassical quantization scheme. If time ordering in the quark loop is maintained new contributions, which have no analogue in the Skyrme model, appear. They substantially improve numerical results for the axial couplings which without the present corrections are badly underestimated.

Semiclassical mechanics of the quadratic Zeeman effect.

A comprehensive approach to semiclassical quantization of the quadratic Zeeman effect in the hydrogen atom (QZE) is presented which utilizes the connection between the Kepler system and the four-dimensional harmonic oscillator. Past attempts to quantize the QZE have encountered problems associated with singularities arising because of a classical separatrix. Here, a uniform semiclassical quantization scheme that passes smoothly through the separatrix is developed using classical perturbation theory. This method accounts for the topologically distinct kinds of dynamics that arise in the QZE, and provides good estimates of tunneling splittings. Extension to the quantization of arbitrarily-high-order classical perturbation expansions is straightforward.

Generalized periodic-orbit sum rules for strongly chaotic systems

The periodic-orbit theory of Gutzwiller is the only known semiclassical quantization scheme that can be applied to non-integrable systems. We present a generalized version of this theory that leads to absolutely convergent periodic-orbit sum rules, using as an example strongly chaotic billiard systems.

Collective dynamics for the bag: Semiclassical approach to the breathing motion

Starting from the bag Lagrangian, we derive a collective Lagrangian which describes the breathing motion of the bag in a microscopic manner. The essential point of our procedure is to adopt a parametrized quark wave function promoted by the local Lorentz-boost so that the linear boundary condition of the bag model is exactly satisfied. After evaluating the mass parameter and the potential energy, we calculate quantized spectra of the breathing mode of nucleon and delta by using the semiclassical quantization scheme. The calculated mass of Roper resonance is comparable to the recent estimate by using the Skyrme model.