The dynamics of a particle in square and circular billiards is studied within the framework of Bohm’s quantum mechanics. While conventional quantum mechanics predicts that the system shows no indication of chaotic behavior for these geometries from either the eigenvalue spectra distribution or the structure of the eigenfunctions, we find that in Bohm’s quantum mechanics these systems exhibit both regular and chaotic behavior, depending on the form of the initial wave packet and on the particle’s initial position

Bonfim, O. F. de Alcantara

Florencio, J.

Sá Barreto, F. C.

English

University of Portland

University of PortlandPilot ScholarsPhysics Faculty Publications and Presentations Physics9-1998Chaotic dynamics in billiards using Bohm’squantum mechanicsO. F. de Alcantara BonfimUniversity of Portland, bonfim@up.eduJ. FlorencioF. C. Sá BarretoFollow this and additional works at: http://pilotscholars.up.edu/phy_facpubsPart of the Quantum Physics CommonsThis Journal Article is brought to you for free and open access by the Physics at Pilot Scholars. It has been accepted for inclusion in Physics FacultyPublications and Presentations by an authorized administrator of Pilot Scholars. For more information, please contact library@up.edu.Citation: Pilot Scholars Version (Modified MLA Style)Bonfim, O. F. de Alcantara; Florencio, J.; and Sá Barreto, F. C., "Chaotic dynamics in billiards using Bohm’s quantum mechanics"(1998). Physics Faculty Publications and Presentations. 27.http://pilotscholars.up.edu/phy_facpubs/27Chaotic dynamics in billiards using Bohm’s quantum mechanicsO. F. de Alcantara BonfimDepartment of Physics, Reed College, Portland, Oregon 97202J. Florencio and F. C. Sa´ BarretoDepartamento de Fı´sica, Universidade Federal de Minas Gerais, 30.161-970 Belo Horizonte, MG, Brazil~Received 13 March 1998!The dynamics of a particle in square and circular billiards is studied within the framework of Bohm’squantum mechanics. While conventional quantum mechanics predicts that the system shows no indication ofchaotic behavior for these geometries from either the eigenvalue spectra distribution or the structure of theeigenfunctions, we find that in Bohm’s quantum mechanics these systems exhibit both regular and chaoticbehavior, depending on the form of the initial wave packet and on the particle’s initial position.@S1063-651X~98!50409-6#PACS number~s!: 05.45.1b, 03.65.Ge, 03.40.KfThe question of how the properties of nonintegrable clas-sical Hamiltonians are manifested in the corresponding quan-tum systems has been of great interest in recent years @1–4#.A variety of problems has been considered to investigate theconnections between classical and quantum systems underthe conditions where classical chaos is present @3,5–12#. Inparticular, two-dimensional billiards is among the problemsmost studied @5–7,9,12,13#.The stadium ~or noncircular billiard! is a planar table withcircular ends of radius a separated by parallel sides of length2b . The classical trajectories in such systems are found to beregular ~integrable! in the circular limit (b50), and chaotic~nonintegrable! for the noncircular case (b.0) @5#.The quantum version of those systems were studied byMacDonald and Kaufman @6#. They found that for the circu-lar billiard the energy level separation was a Poisson-likedistribution, while the noncircular billiard presented aWigner-like distribution, exhibiting mutual repulsion ofneighboring levels. They also found that the eigenfunctionnodal curves for the stadium exhibited a noncrossing irregu-lar pattern. For the circular billiard, the eigenfunction nodalcurves are concentric circles.A very interesting quantum manifestation of classicalchaos first observed in billiards @6,13# is that many of theeigenfunctions of the quantum problem appear to coalescearound the ~unstable! classical periodic orbits of the system.These larger than expected probability densities are localizedaround channels forming simple shapes, the so-called‘‘scars’’ of periodic orbits. The presence of scars is seen asan indication of non-integrability of these systems.After the seminal work of Bohigas et al. @7#, it was shown@2,3# that the level spacing statistics in a variety of quantumsystems with chaotic classical counterparts is well describedby random matrix theory ~RMT! @14#. That is, the level spac-ing distributions for those quantum systems are in excellentagreement with the spacing distribution between consecutiveeigenvalues of the random matrix. Although the energy levelspacing statistics for a variety of quantum systems that arechaotic when treated classically are described by RMT, itwas found recently that two systems which are chaotic clas-sically, namely, the hydrogen atom in a magnetic field and atwo-dimensional quartic oscillator, have in the quantum re-gime an energy level spacing distribution drastically differ-ent from the expected Wigner distribution @11#In spite of some progress in the theoretical developmentsconcerning the signatures of chaos in quantum systems, arigorous procedure to fingerprint chaos in such systems isstill lacking. One fundamental reason for this problem is thatchaos in classical mechanics is defined in terms of the expo-nential divergence of neighboring trajectories, while the veryconcept of a trajectory is absent in conventional quantummechanics. One way to cope with this problem is to adoptBohm’s formulation of quantum mechanics @15–17#, inwhich particle trajectories are well-defined and, conse-quently, the definition of chaos in classical mechanics can benaturally extended to the quantum domain. In fact, Bohmand Hiley @16# were the first to propose the application ofBohm’s theory to the problem of quantum chaos. Theyspeculated on the possibility of chaotic behavior for a singleparticle in a two-dimensional box. Since then some papershave appeared dealing with applications of Bohm’s quantummechanics to the study of chaos @18#.One should note that when a statistical ensemble of par-ticles trajectories is incorporated into Bohm’s theory, the re-sults will be identical to those of conventional quantum me-chanics. That is, by averaging over the initial positions of theparticle, one obtains the same results as those of conven-tional quantum mechanics. In the averaging procedure theremay be contributions from chaotic as well as from noncha-otic trajectories arising from different initial conditions. Bo-hm’s theory has the advantage that it can separate chaoticfrom non-chaotic behavior arising from distinct initial con-ditions. Such an insight cannot be inferred from conventionalquantum mechanics alone.In the present paper, we investigate the quantum problemsof square and circular billiards within the deterministicframework of Bohm’s quantum mechanics. We would like todiscover whether or not the statistical properties of the eigen-value spectra have anything to do with the actual motion of aquantum particle when its dynamics is governed by Bohm’smechanics. That is, we want to know whether the WignerPHYSICAL REVIEW E SEPTEMBER 1998VOLUME 58, NUMBER 3PRE 581063-651X/98/58~3!/2693~4!/$15.00 R2693 © 1998 The American Physical Society~Poisson! distribution of energy levels necessarily impliesthat the system will behave chaotically ~regularly! or othercriteria apply.The deterministic interpretation of Bohm’s quantum me-chanics @15# arises when we express the wave function in theform c5R exp(2iS/\) and rewrite the Schrodinger equationas conditions on both the phase S(r,t) and amplitude R(r,t).The real and imaginary parts of the equations can be sepa-rated, yielding a pair of equations for the squared amplituder5R2 and phase S ,]r]t1S rSM D50, ~1!]S]t1~¹S !22M 1V1Q50, ~2!where M is the mass of the particle and Q52(\2/2M )/(¹2R/R) is the so-called quantum potential.The first equation represents the usual conservation of prob-ability. The similarity between Eq. ~2! and the Hamilton-Jacobi equation led Bohm to define the momentum of thequantum particle, just as in classical mechanics, as MV5S @15#. The velocity of the particle is then given in termsof the wave function byV~x ,y ,t !5\2Mi ~c*c2cc*!/~c*c!. ~3!The particle is assumed to have a well-defined position and avelocity, given by Eq. ~3!, which upon integration yields thetrajectories. The presence of the quantum potential indicatesthat the particle is guided by a wave that is the solution to theSchrodinger equation, just as in the pilot wave picture pro-posed earlier by de Broglie @19,20#. In this way, the quantumdynamics is completely understood as the motion of a par-ticle experiencing forces from both classical and quantumpotentials. Newton’s second law, modified by the presenceof the quantum force, can be written asMd2rdt2 52~V1Q !ur5r~t! . ~4!Hence, quantum mechanics can be described in a way similarto classical mechanics, and it seems reasonable to use thesame criteria to characterize the dynamical state of the sys-tem. The quantum trajectories of the particle are obtained byfirst solving the time dependent Schrodinger equation, thenby determining the quantum potential, and finally by inte-grating the modified Newton’s equation, Eq. ~4!. An equiva-lent, but more practical, way to accomplish that is by inte-grating the guidance formula Eq. ~3! directly, for a giveninitial position (x0 ,y0).Let us first consider a particle in a two-dimensionalsquare box of side L . The wave function can be built fromcomponents of the set of eigenfunctions of the energy,umn~x ,y !5~2/L !sin~kxx !sin~kyy !, ~5!where kI5nIp/L . The complete time dependent solutionwill involve a linear combination of these eigenstates:c~x ,y ,t !5(m ,nCmnumn~x ,y !exp~2iEmnt/\!, ~6!where Cmn are complex coefficients, and Emn5(\2/2M )(kx21ky2) are the energies of the system. At this point,we would like to mention that a qualitative discussion on thedynamics of a particle in a two-dimensional box was givenearlier by Bohm and Hiley @16#. They noticed that from Eqs.~3! and ~6! the expression for the velocity of the particle inthe box contains a large number of small terms involvingcombinations of sines and cosines. As time evolves, theseterms change rapidly, and more so the higher the quantumnumbers m and n . Although the ratio of the frequencies forthe x and y components is rational, it will approach the caseof incommensurability in the limit of large quantum num-bers. Consequently, the expression for the velocity consistsof many terms having complex phase relations with oneother. Bohm and Hiley argued that the particle motion wouldbe, in a way, similar to a Lissajous figure. They claimed thatin a real box ~where irregularities are present! the frequen-cies and wave numbers would be related incommensurably,and the particle motion would be chaotic-like. We shall seebelow that the motion of the particle in the ~ideal! box can bequite complex even when we consider an initial wavepacketconsisting of just a few eigenfunctions with low quantumnumbers, and that chaotic behavior is manifested even if thewalls have no irregularities.We now discuss the results of our calculations for theparticle in the square box. As the initial wavepacket, weconsider linear combinations involving only a few of thelowest eigenfunctions. In all of our numerical analysis, weset \52M51. The linear size of the box is taken as L51.The integration of Eq. ~3! is carried out by using a fourthorder fixed step Runge-Kutta routine with an integration stepdt50.001. We find that the particle dynamics is stronglydependent on the form of the initial wavepacket. Notice thatbecause of the absence of any position dependent phase fac-tor in the eigenfunctions of the particle in a square box @Eq.~6!#, if a particle was in a single eigenstate it would remain atrest for all times. Consider the following initial wavepacket,c(x ,y ,0)5u11(x ,y)1u12(x ,y)1iu21(x ,y), with the particleinitially at (x0 ,y0)5(0.8,0.5). The trajectory of the particlein the (x ,y)-plane is depicted in Fig. 1~a!.That particular choice of initial conditions led the particleto pass through the center of the box on every turn. Oneshould notice that the particle never hits the walls; this is dueto the presence of a strong repulsive quantum potential nearthe walls. A different choice of the initial position can giverise to an entirely new trajectory, depending on whether ornot that point lies inside the basin of attraction of the originalattractor. On the other hand, a change in the initial wave-function will generate an altogether different quantum poten-tial and, consequently, an entirely new trajectory for the par-ticle. The type of motion shown in Fig. 1~a! can becharacterized by looking, for example, at the Poincare´ sec-tion, plotted in Fig. 1~b!. The distribution of points on thatcurve indicates that the motion is quasi-periodic. This is con-firmed by both the power spectrum F(v) shown in Fig. 1~c!,where only a few sharp peaks are present, and by the factthat largest Lyapunov exponent is zero.R2694 PRE 58de ALCANTARA BONFIM, FLORENCIO, AND Sa´ BARRETOFigure 2~a! shows a new particle trajectory resulting fromusing c(x ,y ,0)5u12(x ,y)1iu21(x ,y)1gu23 as the initialwave packet, (x0 ,y0)5(0.5, 0.25) as the initial position,with g51. The particle’s trajectory looks quite irregular,even for such a simple wave function. The scattered points inthe Poincare´ section plot @Fig. 2~b!# and the broad spectrumshown @Fig. 2~c!# indicate that the motion of the particle ischaotic. As the coefficient g is lowered towards zero thepower spectrum gets less noisy ~the largest positiveLyapunov exponent gets smaller!, indicating that the particlemotion is less chaotic, until g reaches zero, when the particledescribes a circular motion about the center of the box.These results are in disagreement with the criteria previouslyused to characterize quantum chaos, based on the distributionof nodes @6# or on the distribution of energy states @6,7#.We have also studied the case of a particle in a circularbilliard. The analysis is similar to the case of the squarebilliard. We found that the particle can have either regular orchaotic types of motion, depending on the initial wavepacket and initial position, just like in case of the squarebillard. Therefore, the details of our calculations for the par-ticle in the circular billiard will not be reported here, sinceFIG. 1. Quantum particle trapped in a two-dimensional box ofsides L . The system of units is such that \52M51 and the lengthunit is the linear size L of the box. The wave packet for t50 waschosen to be c(x ,y ,0)5u11(x ,y)1u12(x ,y)1iu21(x ,y) and the ini-tial position of the particle was ~0.8, 0.5!. ~a! Actual trajectory ofthe particle in the (x ,y) plane; ~b! Poincare´ plot in the (y ,vy) plane;~c! power spectrum P( f ) for the time series of x(t). The powerspectrum is shown in arbitrary units.FIG. 2. Quantum particle moving in the square box whose linearsize is L . The initial wave packet is c(x ,y ,0)5u12(x ,y)1iu21(x ,y)1u23(x ,y), with starting position at ~0.5, 0.25! in unitsof the box size L . In the system of units used, \52M51. ~a!Actual particle’s trajectory in the (x ,y) plane; ~b! phase portrait inthe (y ,vy) plane; ~c! power spectrum P( f ) from the time series ofx(t) showing a broadband, which indicates chaotic motion.PRE 58 R2695CHAOTIC DYNAMICS IN BILLIARDS USING BOHM’S . . .the outcomes are qualitatively the same as those of thesquare billiard.To summarize, we have discussed the dynamics of aquantum particle in square and circular billiards. We findthat for both geometries the motion of the particle can beeither regular or chaotic, depending on the initial form of thewave packet and on the particle’s initial position. This is asurprising result, even from Bohm’s point of view, in that alinear combination containing only a few eigenfunctions isalready sufficient to produce very complex trajectories. Weconclude that the dynamics of the particle, as described byBohm’s quantum mechanics, is not determined by the distri-bution of the eigenvalue spectra: In both cases investigated,the level spacing follows a Poisson-like distribution, whichwould suggest regular behavior, yet we found instanceswhere the motion is clearly chaotic. Moreover, the chaoticnature of Bohmian trajectories is not dictated by whether ornot the underlying classical Hamiltonian counterpart is cha-otic. In the two cases we studied, the classical versions arenot chaotic, yet we find instances where the Bohmian orbitsdo show chaos. That can be understood from the fact that inBohm’s picture the wave function introduces an additionalinteraction, the quantum potential, into the system. It followsthat by studying Bohmian trajectories one cannot distinguishsystems with chaotic classical counterparts from systemswith nonchaotic classical analogues.One of us ~O. F. A. B.! would like to thank the Departa-mento de Fisica, UFMG, where part of this work was done,for their hospitality. We also thank David Griffiths for acritical reading of the manuscript. This work was partiallysupported by FAPEMIG, CNPq, MCT, and FINEP ~Brazil-ian agencies!.@1# M. C. Gutzwiller, Chaos in Classical and Quantum Systems~Springer-Verlag, Berlin, 1990!.@2# F. Haak, Quantum Signature of Chaos ~Springer-Verlag, Ber-lin, 1990!.@3# Chaos and Quantum Physics, Les Houches Lecture Series 52,edited by M.-J. Giannoni, A. Voros, and J. Zinn-Justin ~North-Holland, Amsterdam, 1991!.@4# L. E. Reichl, The Transition to Chaos in Conservative Classi-cal Systems: Quantum Manifestations ~Springer-Verlag, Ber-lin, 1992!.@5# G. Benettin and J.-M. Strelcyn, Phys. Rev. A 17, 773 ~1978!.@6# S. W. McDonald and A. N. Kaufman, Phys. Rev. Lett. 42,1189 ~1979!.@7# O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett.52, 1 ~1984!. See also O. Bohigas, in Chaos and QuantumPhysics, Ref. @3#.@8# T. 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A 53, 2059~1996!; O. F. de Alcantara Bonfim, J. Florencio, and F. C. Sa´Barreto ~unpublished!.@19# L. de Broglie, C. R. Acad. Sci. URSS, Ser. A 183, 447 ~1926!;185, 580 ~1927!.@20# L. de Broglie, Nonlinear Wave Mechanics ~Elsevier, Amster-dam, 1960!.R2696 PRE 58de ALCANTARA BONFIM, FLORENCIO, AND Sa´ BARRETO

Chaotic dynamics in billiards using Bohm’s quantum mechanics

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Chaotic dynamics in billiards using Bohm’s quantum mechanics

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