arXiv:1108.2527

Semiclassical wave functions in billiards built on classical trajectories. Energy quantization, scars and periodic orbits

Stefan Giller, Janiak, Jarosław

Published Aug 11, 2011• 5 citations

Abstract

A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SWF's as well as to calculate energy spectra for the circular and rectangular billiards as well as to construct the simplest SWF's and the respective spectrum for the Bunimovich stadium. The scar phenomena are considered and a possibility of their description by the skeleton method is discussed. PACS number(s): 03.65.-w, 03.65.Sq, 02.30.Jr, 02.30.Lt, 02.30.Mv Key Words: Schrödinger equation, semiclassical expansion, Lagrange manifolds, classical trajectories, chaotic dynamics, quantum chaos, scars

DOI: 10.48550/arxiv.1108.2527

Related papers

Classical-quantum correspondence for chaotic motion

Eric J. Heller

We examine semiclassical propagation of nonstationary states. The fundamental object is the 1928 time dependent Van Vleck semiclassical Green`s function, as modified by Gutzwiller in 1967. This direct approach to semiclassical quantum mechanics has received very little attention over the years. The problems of evaluating the Green`s function (finding all trajectories which go from x{prime} to x in time t, for every x, x{prime}, and t sounds foreboding) and the development of caustics at an alarming rate in chaotic systems, may have suppressed its use. We have developed several strategies for evaluating it efficiently, including for chaotic dynamics. We found it can be used to accurately propagate smooth wave functions in spite of the caustics. The smooth wave function has the effect of strongly damping the caustics and leaving an accurate propagated state. Moreover many of the contributions to the semiclassical amplitude are legal, and allow a much longer time scale for breakdown of the semiclassical propagation than had previously been thought. This has allowed us to extract accurate spectra and eigenstates for chaotic dynamics in the stadium billiard and to calculate a complete molecular electronic absorption spectrum for two nonlinearly interacting vibrational modes, including line intensities and positions, usingmore » only classical trajectory input.« less

Bulletin of the American Physical SocietyMay 1, 19930 citations

Quantization condition of scarring states in complex soft-wall quantum billiards

Xiaoliang Li, Xianzhang Chen, Chen-Rong Liu +1 more

Quantum scar is an intriguing phenomenon in quantum or wave dynamics that the wavefunction takes an exceptionally large value around an unstable periodic orbit. It has attracted much attention and advances the understanding of the semiclassical quantization. Most of previous researches involving quantum scars focus on hard-wall quantum billiards. Here we investigate the quantum billiard with a smooth confinement potential which possesses complex classical dynamics. We demonstrate that the semiclassical quantization approach works well for both the stable and unstable classical periodic orbit, besides the fact that the shape of the orbits varies as the energy increases or even the stability switches. The recurrence rule of the quantum scars in this complex solf-wall billiard differs from that of the hard-wall nonrelativistic quantum billiard, such as being equally spaced in energy instead of being equally spaced in the square root of energy. These results implement the previous knowledge and may be used for understanding the measurements of density of states and transport properties in two-dimensional electron systems with random long-range impurities.

Acta Physica SinicaJan 1, 20200 citations

Multidimensional wave functions from classical trajectories

Michael J. Davis, Eric J. Heller

A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.

The Journal of Chemical PhysicsOct 15, 198198 citations

Exploring Quantum, Classical and Semi-Classical Chaos in the Stadium Billiard

Chris C. King

This paper explores quantum and classical chaos in the stadium billiard using Matlab simulations to investigate the behavior of wave functions in the stadium and the corresponding classical orbits believed to underlie wave function scarring. The simulations use three complementary methods. The quantum wave functions are modeled using a cellular automaton simulating a Hamiltonian wave function with discrete (square pixel) boundary conditions approaching the stadium in the classical limit. The classical orbits are computed by solving the reflection equations at the classical boundary thus giving direct insights into the wave functions and eigenstates of the quantum stadium. Finally, a simplified semi-classical algorithm is developed to show the comparison between this and the quantum wave function method. Quanta 2014; 3: 16–31.

QuantaJan 14, 20145 citations

Asymptotic-boundary-layer method for unstable trajectories: Semiclassical expansions for individual scar wave functions

Alexei Vagov, Henning Schomerus, Victor V. Zalipaev

We extend the asymptotic boundary layer (ABL) method, originally developed for stable resonator modes, to the description of individual wave functions localized around unstable periodic orbits. The formalism applies to the description of scar states in fully or partially chaotic quantum systems, and also allows for the presence of smooth and sharp potentials, as well as magnetic fields. We argue that the separatrix wave function provides the largest contribution to the scars on a single wave function. This agrees with earlier results on the wave-function asymptotics and on the quantization condition of the scar states. Predictions of the ABL formalism are compared with the exact numerical solution for a strip resonator with a parabolic confinement potential and a magnetic field.

arXiv:0907.1626Physical Review ENov 3, 200917 citations

Elliptic Quantum Billiard

Holger Waalkens, Jan Wiersig, Holger R. Dullin

The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phase space into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schrödinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the The semiclassical quantization of a Hamiltonian system is deeply connected to the structure of its phase space. The generic Hamiltonian system contains a complicated mixture of near-integrable and

Jan 1, 199740 citations