vibrating soap ﬁlms: an analog for quantum chaos on...

Vibrating soap films: An analog for quantum chaos on billiardsE. Arcos, G. Baez, P. A. Cuatlayol, M. L. H. Prian, R. A. Mendez-Sanchez,a)

and H. Hernandez-SaldanaLaboratorio de Cuernavaca, Instituto de Fı´sica, University of Mexico (UNAM), A. P. 20-364, C. P. 01000,Mexico, D. F., Mexico and Facultad de Ciencias, UNAM, University of Mexico (UNAM), 04510,Mexico, D. F., Mexico

~Received 25 June 1996; accepted 10 November 1997!

We present an experimental setup based on the normal modes of vibrating soap films which showsquantum features of integrable and chaotic billiards. In particular, we obtain the so-calledscars—narrow linear regions with high probability along classical periodic orbits—for the classicallychaotic billiards. We show that these scars are also visible at low frequencies. Finally, we suggestsome applications of our experimental setup in other related two-dimensional wave phenomena.© 1998 American Association of Physics Teachers.

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I. INTRODUCTION

In recent years, there has been increasing interest inproperties of quantum systems whose classical analogschaotic.1–4 Part of the work in this new field, called quantuchaos,4 refers to essentially two-dimensional cavitieswells of infinite potential called billiards. These billiards catake different forms, such as rectangles, circles, and omore complicated geometries~see Fig. 1!. The circle and therectangle correspond to integrable systems.4 Furthermore, weinclude in Fig. 1 the so-called Bunimovich stadium and tSinaı billiard, which are completely chaotic.5 However, in-

601 Am. J. Phys.66 ~7!, July 1998

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termediate situations, i.e., systems with both integrablechaotic behaviors, are the most common type of dynamsystems.6

The quantum analog of a classical billiard is called a qutum billiard and its eigenfunctions are closely related toclassical features of the billiard.7 Quantum billiards obey theHelmholtz equation with vanishing amplitude on the bord~homogeneous Dirichlet boundary conditions!. Such systemscan be simulated as a drum or any other membrane vibrain a frame. The principal purpose of this paper is to shthat the quantum behavior of classically chaotic and in

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Fig. 1. Integrable billiards:~a! rectangular and~b! circular. Chaotic billiards:~c! the Bunimovich stadium, formed by two semicircumferences of radiurjoined by two segments of lengthd; ~d! the Sinaı¨ billiard, formed by a square with a circle of radiusR inside.
601© 1998 American Association of Physics Teachers

Fig. 2. Typical trajectories in billiards:~a! in the rectangle,~b! in the circle~this trajectory shows a caustic!, and~c! in the Bunimovich stadium.

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grable billiards can be modeled in a classroom with an alog experiment. The experimental setup basically containfunction generator and a mechanical vibrator and is basethe vibrations of a soap film.8 Thus, as Feynman said i1963, ‘‘the same equations have the same solutions.’’9

In the next section we briefly discuss classical and qutum billiards. In Sec. III we introduce our experimental setand show the analogy with the quantum billiard. Other uof our experimental setup are discussed in the same secSome remarks are given in the conclusion.

II. CLASSICAL AND QUANTUM BILLIARDS

In order to make a more explicit definition of the classicbilliard we take a two-dimensional region denoted byR anddefine the potentialV for the particle as

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in Rotherwise. ~1!

This means that insideR the particle is free and moves istraight lines. When the particle collides with the boundait bounces following the law of reflection. Under this dynamics, the rectangle and circle billiards are regular. Typical tjectories inside are shown in Fig. 2. On the other hand,dynamics of a particle in the stadium as well as in the Si¨billiard, are chaotic. Almost all trajectories for these billiarare ergodic and exponentially divergent. Roughly speakthis is so because in the Sinaı¨ billiard, two very close trajec-tories are separated when one collides~or both! with thecentral circle. After some time, the separation betweentrajectories~in phase space! is exponential. In the stadiumtwo particles with very close initial conditions are focus

602 Am. J. Phys., Vol. 66, No. 7, July 1998

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when they collide with one semicircle. After this focusinthey began to separate until they bounce again but nowthe other semicircle. The exponential divergence appearscause the separation time is greater than the focusing tApart from these trajectories there also exist periodic orbThese trajectories are unstable and typically isolated. T

Fig. 3. Periodic orbits in the Bunimovich stadium. Notice that in some camore than one orbit is present and that all obey the reflection law.

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Fig. 4. Eigenfunctions of the Bunimovich stadium calculated by the finite element method for a quadrant of stadium with Dirichlet boundary conditionmaxima and minima of each normal mode correspond to the darkest zones;~a! a typical state scarred by a short classical orbit connecting the two extremof the stadium,~b! a state scarred by an orbit of large period,~c! a typical state scarred by a ‘‘bouncing-ball’’ orbit,~d! a ‘‘whispering gallery’’ state, and~e!a typical noisy state if one looks only at a quadrant.

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number increases exponentially as a function of their lenbut they are of measure zero in phase space. We mayalso families of unstable and non-isolated periodic orbsuch as the ‘‘bouncing-ball’’ orbits in which the particbounces between the two parallel segments of stadium. Speriodic orbits for the Bunimovich stadium are shownFig. 3.

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The time-independent Schro¨dinger equation for the potential defined in Eq.~1! is

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with the wave numberk5(2mE/\)1/2. Here,E and m arethe energy and mass of the particle and\ is the Planck con-stant. The homogeneous Dirichlet boundary condition istained because ifV5` the wave function vanishes. ThHelmholtz equation inR and the Dirichlet boundary condition define the quantum billiard. Note that this is just tequation for the normal modes of a membrane if we interpthe functions as vibration amplitudes.

For classically integrable billiards, the eigenfunctions awell-known. For example, the solutions for circular and reangular billiards are Bessel functions and sinusoidal futions, respectively.10 On the other hand, the features of wafunctions for classically chaotic billiards have been westudied numerically by Heller.7 Recently, experiments in microwave cavities have been performed.11 In Fig. 4 we showeigenfunctions of the Bunimovich stadium we calculatedmerically using the finite element method.12 However, theycan alternatively be calculated by standard software.eigenfunctions of Fig. 4~a!–~c! show certain similarities withthe orbits of Fig. 3. Following Heller, we say that the eigefunctions are ‘‘scarred’’ by the orbits. Figure 4~d! showswhat is called a ‘‘whispering gallery’’ state, because theexist certain galleries13 in which the sound travels insidthem, following the border. This kind of state is associawith orbits also close to the boundary. The eigenfunctshown in Fig. 4~e! resembles noise4 when we see only aquarter of stadium.

The appearance of the scars is quite well understoo14

based on the theoretical work by Selberg, Gutzwiller, aBalian.15 Due to the low density of the short periodic orbitthey may well be seen in quantum experiments and simtions either as dominant features in the Fourier spectrumas scars. While the Fourier analysis of experimental datatomic16 and molecular physics17 is quite striking, directdemonstrations of scars are difficult even in microwave caties. A simple explanation of scarring is based on de Brogwaves. Close to the periodic orbit there exist standingBroglie waves whose wavelengthl is associated with thelengthL of the periodic orbit:

2L5nl, n51,2,3,... . ~3!

These de Broglie waves are localized around periodic ordue to the exponential divergence of nearing trajectoriesthe next section we will show how a simple demonstratsetup can display the most interesting features of the eigfunctions on a soap film.

III. SOAP FILM ANALOGY

The normal modes of a rectangular and circular soaphave been well studied.18–20 A textbook showing theseeigenfunctions is French’s book entitledVibrations andWaves.21 The governing equation is the time-independewave Eq.~2!, but now with a different interpretation:C isthe membrane vibration amplitude and the wave numbenow k5v/v, with v the angular frequency andv the speedof the transverse waves on the membrane.

A problem arises for the experimental setup of this analAt high frequencies~corresponding to the semiclassiclimit !, the damping is large. To solve this problem we feenergy into the system permanently with an external resotor of a well-defined but variable frequency. We chosefeed the external frequency into the system by vibratingwire that delimits our soap film. We use a mechanical vib


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tor ~PASCO Scientific model SF-9324, see Fig. 5! connectedto a function generator~we used a generator Wavetek mod18022! and used wires with different shapes~rectangle, circle,Bunimovich stadium, Sinaı¨ billiard, and some others ointerest23!. Alternatively a speaker could be used to transmthe frequency through the air.19,20,24 The chemical formulafor a soap film with large duration is given by Walker,24 butcan be made up with soap, water, and glycerin by trial aerror.

We can now start the demonstration. In Fig. 6~a! we showa normal mode of the rectangle, in agreement withknown result. Figure 6~b! shows a normal mode for a circldisplaying a Bessel function. Roughly speaking, the shinand dark zones establish a periodic pattern associatedthe normal mode. Although the pattern established cangive a quantitative measure of the amplitude, it is sufficieto give an idea of the form of the normal mode and to idetify it. However, the more interesting normal modes asome of the classically chaotic billiards. In Fig. 6~c! we showa ‘‘bouncing-ball’’ state at low energies for a Bunimovicstadium. The alternating dark and shining zones in thisin the following figures make evident the presence of staing waves in the membrane. These waves are associatedde Broglie waves in the quantum billiard and at the satime they are associated with periodic orbits in the classbilliard. In Fig. 6~d! and ~e! scarred eigenfunctions are displayed, the latter in the high-frequency regime. In ordershow that they are scars and not some spurious effect ofvery simple experimental setup, we calculate numericallynormal modes at frequencies near the experimental onesobserve in Fig. 4 some of these eigenfunctions and theresponding orbits in Fig. 3.

We want to mention that the normal modes in soap filcan also be used for other two-dimensional wave phenena, such as the search for normal modes of the clay lacorresponding to the old Tenochtitlan lake, which playscrucial role in the earthquake damage patterns of MexCity.25 The application in this case comes from the fact ththe upper clay layer of the Mexico Valley, as well as tsoap films, are practically two-dimensional.25 A wire mayreadily be shaped to the corresponding boundary andresults are shown in Fig. 6~f!.

Moreover, the experimental setup which we presmay be used to show other related wave phenomena. Typexamples are scars on liquids,26 Faraday waves—crystallographic patterns in large-amplitude waves27—quasicrystalline patterns on liquids,28 sand dynamics,29 or

Fig. 5. Experimental setup used to show the quantum features of chNotice that the membrane is excited in a way that breaks the symmetr

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Fig. 6. Normal modes for different shapes.~a! Low mode for the rectangle at a frequency of 22.48 Hz.~The frequency changes slightly for different soasolutions and different thicknesses of the soap film.! The rectangle is of size 0.215 m30.155 m.~b! Bessel function for a circular-shape wire. The radiusthe wire is 0.092 m and the excitation frequency is around 33.5 Hz.~c! A ‘‘bouncing-ball state’’ close to 40 Hz.~d! A scarred state corresponding to thnumerical one shown in Fig. 4~b! at a frequency close to 110 Hz.~e! A scarred state by a short classical orbit connecting the two extremes of the stadiua Bunimovich stadium withr 50.035 m andd50.035 m. The corresponding frequency is about 478 Hz.~f! The normal mode for a wire with ‘‘oldTenochtitlan lake’’ form at frequencyn548.88 Hz.

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normal modes of Chladni’s plates.30 These can be done breplacing the wire in our experimental setup with water cotainers or thin plates. If we want to see scars on surfwaves, we must use a stadium-shaped tank filled with waFaraday waves can be obtained by adding shampoo to


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water and increasing the amplitude and frequency butcreasing the level of the liquid to several millimeters. Itnot necessary to use tanks with different shapes becauspatterns do not depend on the boundary. If we want toserve quasicrystalline patterns with this experimental se

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we must excite the mechanical vibrator with—at least—tfrequencies.28 This is easily obtained by changing the sinsoidal time dependence of the driving force to a trianguone.31 Another application is the demonstration of the modof thin plates. In this case, as well as in the water tanks,whispering gallery states are easily visible. We may uselipses, pentagons, or any other shape. All these platesexcited at a point at which the mechanical vibrator loathem. Figure 7~a! shows a whispering gallery state forstadium-shaped iron plate. Figure 7~b! shows a scarred pattern for the same plate. Finally, if we consider a tank withlayer of sand of variable thickness, we may study sevetopics on dynamics of granular media. As an example westudy ‘‘standing waves’’ on sand.29

IV. CONCLUSIONS

The analog model of soap films for the quantum billiagives a demonstration of quantum chaos features. For igrable regions the normal modes correspond to the eigfunctions of a two-dimensional square box and a circle.classically chaotic billiards the normal modes correspondscarred eigenfunctions in the semiclassical limit. The expmental setup presented here is particularly cheap, sim

Fig. 7. Nodal patterns observed on iron plates with stadium shape. Thewas of 1 mm thickness and we usedr 50.1 m andd50.1 m. ~a! A whis-pering gallery at 5.79 kHz. Notice that this pattern is better defined neaboundary and between the two segments of the stadium. This meansthe amplitudes are higher in these regions.~b! A scar which reflects theorbit connecting the two extremes of the stadium. In this case the pattebetter defined on the periodic orbit. The frequency for this case was aro4.22 kHz.


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and elegant—i.e, the alternative experiments do not satthe Helmholtz equation and/or boundary condition. Micrwave cavities are expensive and do not display the scarsdirectly visible manner. Thus the simplicity of the expemental setup and the facility to use any shape makes it vsuitable for the undergraduate laboratory. Furthermorenormal modes of soap films can be used to demonstrate otwo-dimensional analog phenomena. Finally, we wantmention that the proposed experimental setup can be quichanged to study other related wave phenomena. A wvariety of highly nontrivial wave-like phenomena can be dplayed with minimal experimental requirements.

ACKNOWLEDGMENTS

We want to thank T. H. Seligman, F. Leyvraz, and J.Heras for their valuable comments. Also we would likethank the C. O. F., Fı´sica General and Fı´sica Moderna Labo-ratories of the Facultad de Ciencias U.N.A.M., especiaFelipe Cha´vez. Finally, we thank L. M. de la Cruz for hiuseful help with some of the figures. The numerical wowas done with the Cray supercomputer of U.N.A.M.

a!Electronic mail: [email protected]. V. Chirikov, ‘‘A universal instability of many-dimensional oscillatosystems,’’ Phys. Rep.52, 263–379~1979!.

2O. Bohigas, M. J. Giannoni, and C. Schmit, ‘‘Characterization of chaoquantum spectra and universality of level fluctuation laws,’’ Phys. RLett. 52, 1–4~1984!. See also ‘‘Spectral fluctuations of classically chaoquantum systems,’’ inQuantum Chaos and Statistical Nuclear Physic,edited by T. H. Seligman and H. Nishioka~Springer, Berlin, Heidelberg,New York, 1986!, pp. 18–40.

3T. H. Seligman, J. J. M. Verbaarshot, and M. R. Zirnbauer, ‘‘Quantspectra and transition from regular to chaotic classical motion,’’ PhRev. Lett.53, 215–217~1984!. See also ‘‘Spectral fluctuation propertieof Hamiltonian systems: The transition between order and chaos,’’ J. PA 18, 2751–2770~1985!.

4For a pedagogical paper on quantum chaos, see M. V. Berry, ‘‘Semicsical mechanics of regular and irregular motion,’’ inChaotic Behavior ofDeterministic Systems, Les Huoches Lectures XXXVI, edited by R. HHelleman and G. Iooss~North-Holland, Amsterdam, 1985!, pp. 171–271.

5M. C. Gutzwiller,Chaos in Classical and Quantum Mechanics~Springer-Verlag, New York, 1990!, pp. 196.

6Eric J. Heller and Steven Tomsovic, ‘‘Postmodern quantum mechanicPhys. Today46 ~7!, 38–46~1993!.

7E. J. Heller, ‘‘Qualitative properties of eigenfunctions of classically chotic Hamiltonian systems,’’ inQuantum Chaos and Statistical NucleaPhysics, edited by T. H. Seligman and H. Nishioka~Springer, Berlin,Heidelberg, New York, 1986!, pp. 162–181, and references therein.

8Both the Schro¨dinger equation for a billiard and the wave equation formembrane are reduced to the Helmholtz equation when the tiindependent part is taken. Nevertheless, to deduce the wave equatiothe membrane air, gravity, and damping in the high frequency regimeusually neglected.

9Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Fman Lectures on PhysicsMainly Electromagnetism and Matter, Vol. II~Addison–Wesley, Reading, MA, 1964!, pp. 12-1.

10K. F. Graff, Wave Motions in Elastic Solids~Dover, New York, 1991,reprint!, pp. 225–229.

11S. Sridhar, ‘‘Experimental observation of scarred eigenfunctions of chamicrowave cavities,’’ Phys. Rev. Lett.67, 785–788~1991!.

12M. Mori, The Finite Element Method and its Applications~MacMillan,New York, 1983!.

13St. Pauls in U.K. is an example.14E. B. Bogomolny, ‘‘Smoothed wave functions of chaotic quantum s

tems,’’ Physica D31, 169–189~1988!.15These authors developed a formula according to which a solution to q

tum problems may be written as a sum over all periodic orbits ofclassically chaotic counterpart. This formula is exceedingly complicadue to the exponential increase of the number of such orbits as a funof their length. On the other hand, the shortest orbits are few and dis

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some typical features of the problem. There are textbooks on classicaquantum chaos containing discussions of the Selberg–Gutzwiller–Baperiodic orbits sum expression. See, for example, L. E. Reichl,The Tran-sition to Chaos~Springer-Verlag, New York, 1992!, pp. 318–381. Seealso M. C. Gutzwiller, Ref. 5, pp. 282–321.

16A. Holle, J. Main, G. Wiebusch, H. Rottke, and K. H. Welge, ‘‘QuasLandau spectrum of the chaotic diamagnetic hydrogen atom,’’ Phys. RLett. 61, 161–164~1988!.

17M. Lombardi and T. H. Seligman. ‘‘Universal and nonuniversal statistiproperties of levels and intensities for chaotic Rydberg molecules,’’ PRev. A 47, 3571–3586~1993!.

18Cyril Isenberg,The Science of Soap Films and Soap Bubbles~Tieto,Clevedon, Avon, England, 1978!. This book gives an excellent account othe many properties of soap films and soap bubbles. It also containextensive list of references on the subject.

19D. T. Kagan and L. J. Buchholtz, ‘‘Demonstrations of normal modes obubble membrane,’’ Am. J. Phys.58, 376–377~1991!.

20L. Bergmann, ‘‘Experiments with vibrating soap membranes,’’ J. AcouSoc. Am.28 ~6!, 1043–1047~1956!.

21A. P. French,Vibrations and Waves~Norton, New York, 1971!, pp. 185–186.

22The generator and mechanical vibrator can be used for other vibrasystems. See, for instance, Christopher C. Jones, ‘‘A mechanical rnance apparatus for undergraduate laboratories,’’ Am. J. Phys.63, 232–236 ~1995!.

23The possibility to introduce a billiard of any shape easily is the gradvantage of the experimental setup which we present. Domainsdifferent shape but the same spectrum may be studied. See, for inst


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H. Weidenmu¨ller, ‘‘Why different drums can sound the same,’’ PhyWorld ~22–23 August 1994!. Other billiards associated with classicallchaotic or integrable billiards may be studied. This is the case ofellipse billiard, the oval billiard, the triangle billiard~all kinds of tri-angles!, polygonal billiards, etc., and in general, billiards constructedorder to obtain some special feature of classically chaotic systems suspecial discrete symmetries.

24J. Walker, ‘‘Music and ammonia vapor excite the color pattern of a sofilm,’’ Sci. Am. 257 ~2!, 92–95~1987!.

25J. Flores, O. Novaro, and T. H. Seligman, ‘‘Possible resonance effecthe distribution of earthquake damage in Mexico City,’’ Nature~London!326, 783–785~1987!.

26R. Blumel, I. H. Davidson, W. P. Reinhart, H. Lin, and M. Sharno‘‘Quasilinear ridge structures in water surface waves,’’ Phys. Rev. A45,2641–2644~1992!.

27G. M. Zaslavsky, R. Z. Sagdeev, D. A. Usikov, and A. A. ChernikoWeak Chaos and Quasi-Regular Patterns~Cambridge U.P., New York,1991!, pp. 170–187.

28S. Fauve, ‘‘Parametric instabilities,’’ inDynamics of Nonlinear and Dis-ordered Systems, edited by G. Martı´nez-Meckler and T. H. Seligman~World Scientific, Singapore, 1995!, pp. 67–115.

29F. Melo, P. Umbanhowar, and H. L. Swinney, ‘‘Transition to parametwave patterns in a vertically oscillated granular layer,’’ Phys. Rev. L72, 172–175~1994!.

30J. Stein and H.-J. Sto¨ckmann, ‘‘Experimental determination of billiardwave functions,’’ Phys. Rev. Lett.68, 2867–2870~1988!.

31J. Bechhoefer and B. Johnson, ‘‘A simple model for Faraday waves,’’ AJ. Phys.64, 1482–1487~1996!.

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vibrating soap ﬁlms: an analog for quantum chaos on...

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