ENSEÑAZA REVISTA MEXICANA DE FíSICA 47 (5) 480-488 OCTUBRE 2{X)I

Probability distributions in c1assical and quantum elliptic billiardsJulio C. Gutiérrez- Vega and Sabino Chávez-Cerda

Grupo de Fotónica y Ffsica Óptica, Instituto Nacional de Óptica y ElectrónicaApartado postal 51 y 216, 72000 Puebla, Pue .. Mexico

Ramón M. Rodríguez.DagninoCentro de Electr6nica)' Telecomunicaciones. ¡nsIituto Tecnológico y de Estudios Superior de Monterrey

Garza Sada 2501, Col. Tecnológico, MOlllerrey. N.L., Mexico

Recibido el 17 de mayo de 2001; aceptado el 14 de agosto de 2001

Advanccs in the fabricalion of nanocavi{ies lo confine electrons huve popularizcd ncwly the sludy of 20 quantum wells. Thc analogicsand difTcrcnccs bclwccn c1assical and quantum probability distributions and encrgy spcClr<lof a panic1e confined in an elliplic billiard areprcscnlcd. Clnssically, the probabilily den sities are charncterizcd by the eigenvnlues of an cquntion that involves elliptic intcgmls. whereas theordinary and modified Mathieu functions are applied te describe the quanlum distributions. The transition fram the elliptic geomelry towardthe circular geomctry is analyzcd as well. The problem is interesting ¡tscl!' because ir presents strong analogies with (he electromagneticpropagation in elliptic waveguides.

Ke)'words: Quantum and c1assical elliptic billiard; Mathieu functions; Hamilton-Jacobi thcory; e¡¡iptic integrals

Los avanccs en la fabricación de nanocavidades para confinar electrones han popularil.ado nuevamente el estudio de pozos cuánticos bidi-mensionales. En este trabajo se presentan las analogías y diferencias entre las distribuciones de probabilidad y espectros de energía deuna partícula conllnada en un billar elíptico. Clásicamente las densidades de probabilidad están caracterizadas por los eigenvalores de unaecuación que involucra integrales elípticas, mientms las ecuaciones radial y angular de 1\lathieu se aplican para describir las distribucionescuánticas. La transición de la geometría elíptica a circular es analizada también. El problema es interesantc pues presenta muchas analogíascon la propagación electromagnética en guías de onda elípticas.

Descriptores: Billar elíptico clásico y cuántico; funciones de Mathieu; teoría de Hamihon-Jacobi; integrales elípticas

PAes: 02.30.-f; 02.30.1r

1. Introduction

The study of energy spectea of physical systems plays a fun-damental role in quantum mechanics (QM). The problem of afree particle moving in an one-dimensional (10) infinite wellmay be the first and most frequenlly sol ved eigenvalue pro-blem in elementary QM lexlbooks [1,2]. The simplicily ofthe 10 Illodel perrnits us to get a first insight aboul importantresults ofQM (e.g., normalized wave functions, energy-Ievclspacings). Usually, Ihe two-dimensional (20) infinite rectan-gular billiard is considered lo show the existence of degene-racies {2], and infinite circular billiard is useful to introducethe concept of angular momentum [3] and also presents de-generacies. Evidently mosl of the textbooks are focused tostudy lhe quantum phenomena involving three-dimensionalsystems (e.g., probabilily distributions in an atom of hydro-gen).

The 20 systems have became more popular in recentyears because progress in nanotechnology have allo\\'ed tofabricate very small c10sed structures (i.e., quantum corrals),which can be used lo confine eleclrons [4], The boundary ofIhe nanodevices is sharp enough to consider that the electrotimay be regarded as a particle confined lo a 20 infinite bil-liard. Hence the energy spectrum of the electron can be des-cribed by solving lhe appropriate Schrodinger WJve equalioninside the billiard wilh lhe Dirichlet condition at the boun-dary. In addilion, the sludy of Irajeclories in 20 billiards is ofparticular interest in c1assical and quantum chaos as well [5].

As we mentioned aboye. the rectangular and circular bil-Iiards are well-treated in textbooks, however we think thatthis is not the case wilh the elliptic geometry. Oue to therenewed attention in 20 systems, the purpose of om workis lo present a study of the c1assical and quantum elliplicbilliard as an example of a 20 system with non-degeneratestales. Besides, the quantum elliptic billiard has exact analy-tieal salutions whieh allow us to write the eigenfunctions in aeloscd formo namcly Mathieu functions. In arder to establisha conncction between classical and quantum mechanics. wecompare the probability distributions to show that bOlh ap-proach each other as the energy increases. Our results arecompared with respect to known literatme of elliptic billiards(see Refs. 6-9, and 11, and references Iherein),

This work is inlendcd al so for researchers who are inle-resled in the solution of the Schrodinger equation in ellip-tic coordinales and lhe application of Mathieu functions. lnsuch a way, this papcr may be considered as a continuationof one preceding {lO], where we discuss the free oscillationsof an elliptic membrane, and include a more detailed treat.ment of the Malhieu function theory. Part of the results in-c1uded in this paper were already discussed in Ref. ti. On theother hand. the problem may resull interesting to researchersinvolved in the study of propagation of light inside ellipticwaveguides. due to the strong analogies between classicalmechanics solutions with lhe ray.trajectories in geomctricaloplics, and the QM solutions wilh the propagaling modes inelcctromagnetic thcory.

JULIO C. GUTIÉRREZ.VEGA. SABINO CHÁ VEZ.CERDA AND RAMÓN M. RODRÍGUEZ.DAGNINO 481

r¡ = 1tI2

(2)

where the contour integral is carried out over a complete pe-riod of the coordinate qi' The angle variable ei correspondsto the frequency w¡ of the periodic motion of the coordi-nate q¡ and is given by

_ dei aH(J, e)wi = ili = aJ '•

where H is the hamiltonian of the system. In our two-dimen-sional billiard, the periodic trajectories are determined by thecondition that wJw2 must be equal to r fn. A detailed treat-ment of Hamilton-Jaeobi theory can be found in Ref. 13.

Let us first introduce the eHiptic coordinates

x = fcosh~eos'J, y = fsinh~sin7).

r¡=O

~o

"" "--''\7" '-. ,-,

é, = o ," ,,".<'

----' ~, ": ' ..•'~,'"

,

,.,.' ,

,

" :,r -_

,,,-'-

,,,,

b

r¡=lt

2. Classical rncchanics solution

FIGURE t. Geornctry of the billiard in el1iptic coordinates. Theboundary is given by ( ;;;; (o and the runge of (he coordinatesare ( E [0,00) and ry = [0,2,,).

The paper is organized as follows. In Seco 2, we presentthe procedure to obtain the classical trajectories for periodicorbits in the billiard. The eigenfunctions corresponding toquanlum billiard are given in See. 3. The probability distribu-tians for classical and quantum cases are compared in Seco4,where the energy spectrum for elliptic billiard is comparedwith respect to circular billiard also. Finally, in the conclu-sions we point out same final comments.

(4)

(5)

where !tI and h2 are the metric factors of the generalized co-ordinales q1 = ~and q2 = 7)given by

Fh; = h~ = h2 = 2(cosh2~-cos2,,),

The boundary of the elliptie billiard is expressed as~= ~o = aretanh(b/a) = eonstant, where a and b arethe semi-major axis and semi-minor axis of the ellipse,respeetively, (b < a). The semifoeal distanee f is given by12 = a2 - b2, and the eccentricity e is defined as

f 1e = - = --o (3)a cosh~o

The velocity of the partide as a funetion of the elliptiecoordinates is

,r¡ =31t12

The geometry of the elliplie billiard is shown in Fig. l. C1as-sieally speaking, the particle moves freely inside the billiardand it is elastically bounced at the boundary. The systempresents two degrees of freedom and two constants of mo-tion [12], Ihe firsl ofthem is the total energy E and the seeondone is the dot product of the angular momenta with respectto the foei ofthe ellipse, A = L¡ . L2. It follows that elliptiebilliard is an integrable system whose dynamical behavior isregular and predietable. In general, the motion of the particleis characterized in terms of the canonical coordinates qi andcanonical momenta Pi' Because there exist two constants ofmation. the particle is restricted to move in a specific trajec-tory in the phase space (qi'P¡), where E and A are main-tained fixed.

In particular, we are interested in periodic trajectories inthe billiard. An effeetive method of handling these periodiesystems is provided by the application of action-angle varia-bles in the Hamilton-Jaeobi theory (HJT). In HJT termino-logy, the constants of motion are called the action variablesof the system and they are often represented as Ji (q¡, Pi)'For each J¡(qi,Pi) there exist a corresponding angle varia-ble designated by ei (qi' Pi)' The action variable Ji is deter-mined by

A = L1 . L2 = (r1 x Mv) . (r2 x Mv),

According to classical mechanics theory, the canonicalmomenta are expressed as

and t and f¡ are the elliptic unit vectors related to cartesianunit vectors by

(8)

(6)- eosh ~ sin r,) (~).smh ~ cos ry 7)

2 2H = E = p( +P'.

2Mh2

(i;) f (Sinh ~ cos 7)íi = h eosh ( sin "

aL 2d~p( = a~= M h dt'

aL 2dry1'" = ai¡ = Mh di' (7)

where L = (I/2)Mh2[(d~/dt)2 + (d7)/dtJ2] is the la-grangian and Al is the mass of the particle. For our conserva-tive billiard, the hamiltonian is equal to the total mechanicalenergy E, and it can be written in terms of canonical mo-menta as

As we mentioned previously, the second constant of mo-tioo is the product of angular momenta with respect to thefoci

(1)Ji = 21" f Pi dqi'

Rev. Mex. Fís. 47 (5) (2001) 48Q-488

-l82 PROBABILlTY OISTRIBUTlONS IN CLASSICAL ANO QUANTUM ELLlPTIC BILLlAROS

which can 3150 be expressed as a function of the canoniealmomenta by Eq. (7)

., 2 2[(dT/)2. 2 (d~)2. 2 ]A = .\1-j h di smh ~ - di ,m '1 ,

( 14)

( 15)

(16)

-y > O,

..,< O,

-y > O,

-y < O,

wheresiu<p, = )1- (J2/b2)-y.sin<p, = Jb2/(b2 - J2-y),and F(O, m) is Ihe elliptic inlegral of the first kind [14),given by

J Al Ej2 1.,;2J" = 4 ~ 'Te Jsin2

'/ + -y ,b/, (13)

where the lower Iimils of the integrals are differenl depend-ing on the kind of motion. Por rotational mOlion (¡ > O),~e = arccosh(l/ecJ and '/e = O. For oscil1aling mo.tion h < O). ~e = Oand '1""" = arceos (l/ce)'

The condition of pcriodic trajectories is reached when Ihequoticnt of the frequcncics associated to each coordinate is arational number. According to Eq. (2). we have

al! laJ'1aJ, _ aJ" _ a-y _,.al! - aJ, -laD'~I-;;aJ'T ,

By differentiating Eqs. (12) and (13) with respect to -y,and after sorne manipulatiolls, we arrive lo

J¡\tEj2 (" /. ,J, = 2 2,,2 J,c V smh ~ - -yd~, (12)

By insening the momenta I',(E, -y) and I',,(E,..,) InEq. (1), the actions are given by

18 dOF(O, m) = --;0====

o JI-rnsin20

Finally, by substituting Eqs. (15) and (16) in Eq. (14). thecharacteristic equations for periodic trajeclories are

(9)_ ¡z ( 2. 2 2 .' 2 ).\ - ---:2 Pr¡ slIlh (- p( SIIl 7/ .h

where TI (X - f)i: + yf¡ and T2 = (X + f)i: + yf¡. Byusing Eqs. (-1.)and (6), we can obtain A as a function of (he

elliptical coordinales

where the lowcr Iimil Amin = -2J\1 Ej'l corresponds to thernotion along the y-axis and the upper limit Am1\x = 21\1 Eb2

represents the Iirniting motion over the boundary. Because Ais cnergy-dependent, it is preferable to define a new energy-independent pararneter, say /' given by

The conserved quantity A is a useful parameter 10 de-fine the kind of mation of lhe particle. In gcncfui, there aretwo types of moticn depending on lhe sign of .\. The firstcase,.\ > O. corresponds lO trajectories thal have an ¡nncrelliptic caustic ~ = ~e = constan! > O. confocal to theboundary and, therefore, lhe particle never crosses lhe x-axisbel\\'een lhe foei. In this case, lhe E, coordinate is restrictcd looscillate in lhe range E, E [f.c. £'0] whercas lhe 1/ coordinatedocs nol have any restrictions. We will identify this motionas rollltiollal lype (R), (see Fig. 3).

On Ihe olher hand, in lhe second typc of rnolion, A < 0,lhe trajcclorics gencrate two caustics in lhe forrn of cofocalhypcrbolas, TI = 'lc and 1] = 11" - TIc' where lle < 7f /2. Now,the particle always cross the x-axis between the foci and thebounces happen alternately in the upper and lower parts ofthe billiard. The T} coordinate of the particle is Iimited to vi-brate in lhe range T} E [17e,1I" - lle]' whereas the E, coordinateoscillales in the whole range f. E [O, f.o]. As above, \\!e willidentify this motion as oscillatiTlg type 0, (see Fig. 5).

The separalrix occurs when .•\ = O, and the particle justcrosses over Ihe focal points. It should be mentioned thal therange of A is Iimited to

In terrns of /. the eccentricity of the caustic ec is deter-mined by

(" 2) ,. ( 2)F '2' Ce = 271 F f/JI' ec '

(" 1) ,. ( 1 )F -'2 = -F cP212 '2 ec 2n ec

-y > O,

-y < O.

( 17)

(18)

(11 )j 1Ce = Oc = JI + ,,'

The canonical mornenta Eq. (7) can be written as a functionof -yand E as fol1ows

pi = 2M Ej2( sinh2 ~ - -yj,

I'~ = 2.\1Ej2( sin2'/ + -y).

The root5 of the characteristic equations are the possiblcvalues of / to have pcriodic orbits in the billiard.

2.1. Periodic trajectorics

\Ve firsl discuss the rolational trajectories characlerized byEq. (17).l3y applying thejacobian elliptic function s¡¡(u) lo

Re". Mex. Fís. 47 (5) (2001) 48Q.-488

JULIO C. GUTIÉRREZ.VEGA. SABINO CHÁ VEZ.CERDA AND RAMÓN M. RODRiGUEZ.DAGNINO 483

••

"0:1 sn(2nK(if¿ Ir)

O'

11,~1 ¡~~.(IJ)

13,1I_¡6.1I

1"1_(1.2)

IUI

IS,')

(1,'1

",

\'2.~,I'O~',

'1.3l

" I"~l

11.11_112.')

" n03]"..," 1','1.(1,2)_112.3)

11'31

" 1'0.21

" 11."'1'211

I"ZI,.,1.'1

" 1'0.'1

<.Sn(2nK('/~/rJ

FIGURE 2. Plots of characterislic cqualion fOf R.¡rajcctories for <Incllipsc with !JIu = ~ill (11'/3) and f = 0,5.

1811

(10.4)

16.21

1143)

(10,31

16\l

114.21

(10,2118,31

(1251

FIGlIRE 4. PIOlS 01' the charaClcristic cquaLÍon for O-lrajcctoriesfor an cllipsc with b/a = :-in(n-¡8). Thc curve (8,1) corres-ponJs to ¡;ul.off condilion /l/r = 1/8 ando thcrcfore, intersect atlimil') = -1. Observe thal curve (10, 1) docs nol intersect bccauseir is out of the valid rangc 1/8 $ ll/r < 1/2.

P,3)

15.2'

(7,2)

15'l

161)

(311

00(7.1)OOg8

18,1) (8.3) (9,11 19,2)O~OOFIGURE 3. PcrioJic R.rrajeclOrics corrcsponding lOdala in Fig. 2.

both sides of the Eq. (17), we can oblainFIGURE 5. Pcriodic O-trajcctories corresponding lOdata in Fig.-l.

n osin -7[ >

r - a'

3. Quantum mcchanics solution

The quantum energy spectrum of a particle in the infi-nite billiard is obtained by solving the time-independentSchrodinger equation

f¡<P(~,II)= [- ;~!\72+U(~,rl)]<p(~,'/)=E<P(~"I), (21)

where Ji is the energy operator (i.e. the hamiltonian), <p(~,~)is the eigenfunction corresponding to the characteristic ener-gy E, and U(C 1/) is the pOlential cnergy expressed by

{D, for ~ ::;~"'

U(~, T/) =00, for ~ > ~o.

where the equality corresponds to motion along the minor-axis of the ellipse as it is shown in Fig. 5. 80th sidesof the characteristie equation (Eq. (20)] are plotted inFig. 4, where we have chosen (l = Jsin(rr/3)/sin(rr/8)"nd 01(l = sin (rr18), [I.e., e = ros (rr18) 1 with two purposes:

a) 1'0 match the trajectory (8, 1) with the cut.off condi.tion in order to visualize this situation.

b) 1'0 maintain the same area ofthe ellipse (rrao) from theaboye rotating case (Fig. 2) in order to make appropri-ate comparisons with the quantum solutions.

( 19)

(20)'Y < O.

'Y> O,

J 02 ~2j2'Y = sn [2; 1( C~)],The aboye equation has real solutions for r 2: 4 and

n < r /2. where r must be an even integer in order tohave c10sed oscillating trajectories. Furthermore, a certainQ-trajectory (r, n) can be present in the ellipse only if it sa-tisfies the nexl cut.off condition

where 1((111) = F(rr/2,rn) is Ihe complete eJliplie inte-gral of Ihe "rSI kind. Equalion (19) has real solulions forr ~ 3 and n < r /2. The pararneters r and H are the nUITI-ber af bounces al (he boundary and the rotation number res-peelively. In Fig. 2 we plot both sides of the Eq. (19) for thefirst values of r and 11. The intersections correspond 10 theeigenvalues af 1 to have closed trajectories ¡nside the bil-¡¡ardoThe paths assoeiated 10 the eigenvalues of 'Y are plottedin Fig. 3. We should mention lhal we have taken advantageof MATLAB software ulililies [15]10 implement aJl plots inthis work.

On the other hand, Eq. (18) is inverted as foJlows

Re". Mex. Fís. 47 (5) (2001) 4R()..488

(27)

," ,,

" " ,. " " "•

"""

'o

,,.,_"_"_"_"o

¡ Ce,(~,q,) = k~oAk(r,q,)eOShk~,R,(O = _

SC'+l (~, '1,) = L Bdr, '1,) sinh k~.k=l

Thc Dirichlel condition at lhe boundary is satisfied when

R,(~o) = { Cc,(~o, '1)= O, (28)Se,+1 (~" '1) = O.

On lhe olher hand, lhe solulions for MME [Eq. (23)]is obtained from lhe OMP by setting the change of varia-ble '/ = i~

FIGURE 6. Plots oC0r((J) (salid curves), {3r+l ('1) (dashed curves),pf/m (n) (dotleJ curves) and of/m ({3) (dash-dOl curves) in the(q, u)-planc. Thc square markers correspond lOeqr,m eigenvalues(cross-points hctween O'r and eqm) and circle markers correspondlo uf/r, ••• eigcnvalues (cross-points between (:Jr and oqm) Cor an el-!ipse with d.lta as in Fig, 2, (j.e., e = 0.5). \Ve can appreciale that¡J,(q) -> 0,(") -> r' asq -> O and ¡ha. ¡J,+,(q) -> 0,(,,)as q -t oo.

(22)

(25)

PROBABlLlTY DISTRIBUTIONS IN CLASSICAL AND QUANTUM ELLlPTIC BILLlARDS

The eigenfunclions 'P( ~, 'J) musl salisfy lhe DiriehJelboundary condilion 'P(~o' ,/) = O, and lhe periodicily condi-lion 'P( ~, ,/) = 'P(e '/+ 27r). The lapJaeian operalor expressedin eJliptic coordinates is written as [10]

484

2102102V = ir2 Oe + ir2 Ory2 '

where ir is given by Ee. (5). By repJaeing lhe laplacian inEq. (21), and laking lhe fael lhal U(e ,/) = O in lhe billiard,lhe 2D Sehrodingerequalion (2D SE) in elliplicaJ coordinalesis wr¡Hen as follow5

where () is the constant of separation and q is a dimensionlessparamcter given by

R"(O - (o - 2qcosh20R(O = O, (23)

0"(,/) + (o - 2qeos2/J)0(ry) = O, (24)

k2F MF'1 = -4- = 2ir2 E.

The Eqs. (23) and (24) are known as modified Malhieuequalion (MME) and ordinary Malhieu equalion (aME), res-pectively. Thc solutions 10 these Equations are known asmodified Malhieu funelions (MMF) and ordinary Malhieufunclions (OMF) [IG], respeclively.

\Ve first consider lhe soJulion of aME [Eq. (24)]. Sincethe O!\1F must be periodic. we can express them in terl11 ofFourier series

[0

20

2k

2 F ]oe + 01/2 + 2(C05h 2~ - cos 2'1) 'P(~, ,/) = O,

where k2 = 2M EI,,2.Supposing lhe solulion 'P(~, 'J) = R(00(,/), lhe 2D SE

is separatcd into the nex! two ordinary differential equations

where lhe order (1' ~ O) is relaled lo Ihe r-Ih eigenvalue o, ofthe equation and CCr(1J,lJ) and SCr+1(11,r¡) are known as theeven and odd ~,1athicu functions, respectively. These eigen-values are those values of o which. for a certain fixed para-meter fJ, the O~lEadmits solutions with period ¡r or 2n.

For simplicity, we denote the eigenvalues associatedlo ee,('J, '1,) and se,+, ('J, '1,) as 0,('1) and 13,+1 ('1), respee-lively. Aeeording lo lhe Slurm-Liouville lheory, all eigenva-lues are real and (lo < 131 < Cl:1 < (32'" The plolsof 0,('1) and 13,+,('1) in lhe (q,o)-pl.ne are Iines lhal donol inlcrscct (Fig. 6). Tlle even-order solulions have period n,whereas odd-ordcr solulions have period 2n. The Fourier co-efficients Ak(1',fJr) amlLJdr,fJr) can be computed by usinglhe well-known recurrence relations for the Mathieu egua-lions [10, 1,1].

The solulion of lhe MME [Eq. (23)] can also be conlem-platcd aS an independent eigenvalue problem, bUI now, fJ islhe lInknown cigenvalue and Q is the fixed pararneler. Asaboye, we can express Ihe eigenvalues associated lo O' and (3aS I'fJm(o) and ofJm({3), respectively, where TTl > 1 andlhe slIbindex e or o refers to even or odd MMF. The plotsof ,'1",(0) and 0'1",(13) are shown in Fig. 6. Sinee lhe solu-lion 'P(e ,/) is lhe produel of solUlions of Eqs. (23) and (24)for rile same \'lllue o/ Q (/3) alld q, then there exist a validsolution for each cross-point between the two farnilies ofcurves. In other words, lhe MtvlF are decreasing-oscillatorynon-pcriodic funclions similarly to Bessel functions, there-fore, jf we choose a ccrtain order r we ha\'e an infinite set ofpossible values of '1 lhal salisfy Eq. (28).

Let qr,fJlbe lhe m-th zero of l\.lMF of r-order. Accordingto Eg. (25) for each fJr,m there exiSI a corresponding energyeigenvaluc Er.m associated to the eigenfunction <Pr.",(~,11)

¡ ee,(//, '1)0,(1/) =

se'+l ('/, '1)

== L Ad1', '1) cos kr/,k=O== L Bk(r,q) sin krJ,k=l

(26)

(/2) (2 )E = ~ = l!::..- fJr,m

e 1",711 .A112 qr,m Al a2 e2 1(29)

Rev. Mex. Fís. 47(5) (2001)4~0-4~~

JULIO C. GUTIÉRREZ-VEGA, SABINO CHÁ VEZ-CERDA AND RAMÓN M. RODRfGUEZ-DAGNINO 485

where e is the eccentricity of the ellipse and a is the semi-majer axis. The energy eigenvalues are proportional 10 qr,m'therefore, the q~axis in Fig. 6 may be considered as an E-axisin 2h2 1Al J' units. The energy eigenvalues are the projee-liaos of the intersection-points on the E-axis.

The eigenvalues of the second constant of motioo A canbe direelly obtained from Eq. (9) by subslituling Ihe momen-lum operalors 1'( = -il¡[) la( and 1'" = -il¡[) la'l

J'/¡2 (a2 &2 )-2- sin2 'I-a2 - sinh2 (& 2 '1' = '\'1'." ( '1

By supposing the solution '1'«(, '1)= R(00(7)), the aboyeEquation is separaled inlo two Mathieu equations (Eqs. (23)and (24)J, whereo = A/"2 -C/2 andq = -C/4, and C is aconstant of separation. It follows that the eigenvalue problemof A is equivalent to eigenvalue problem of H if A is chosensueh thal A = (o - 2q)f¡2, where a and q are the eigen-values of the hamiltonian problem. Aeeording 10 Eqs. (10)and (29), "1 is given by

The separalrix "1 = O is represenled on Ihe (a, q)-planeby the straight ¡¡ne o: = 2ql and it divides the plane intotWQ zcnes, the rotating region I > Oand the oscillating re-gion "1 < O (see Fig, 6). The vertical dislance belween anintersection and the separatrix coincide with the value of Ain 1t2 units. The c1assical Iimit for oscillating motion is givenby a = -2q (Le., "1 = -1) and il is plotted in Fig, 6 as well.

The energy eigenvalues of the firsl 22 slationary slateseorresponding 10 dala of Fig, 6 (i,e" e = 0.5) are listed inTable 1, where we can see that oEr,m >e Er,lIl' The oscilla.ting modes are always non-degenerate, whereas the rotatingmodes become more degenerate as '"'(increases.

In Fig. 7, we plot e,oEr,m for ellipses with geometry fromI'igs. 2 and 4, (j.e., e = 0.5 and e = 0.92.1 respeetively)in increasing order. Since the energy depends on the fo-cal dislance [Eq. (29)J, lhe eigenvalues musl be s"aled ae-eordingly 10 make applieable eomparisons. The even andodd eigenvalues are plotted separately in order to be awareof their different behavior. The straight lines connect eEr 1

with oEr,l' (r = 1,2,3, ... ) as a form to visualize the te~-dency eEr,Ol ::::::oEr,Ol as r increases. This trend is more no-torious as e 4 Oand, in the circular limit, U.e., e = O) theeigenvaluessatisfy f'Er•m = oEr,m' This behaviorcan beap-preciated also in the Fig. 6 where the square and cirele mar-kers get closer as the energy (i.e., q value) bereases.

FIGURE 7. [ncrcasing ordcred energy eigcnvalues (in 2h2/J.\/u2

units) ror clliptic billiards with e = 0.5 alid e = 0.924. We canappreciatc that oEr,m > eEr,m' (see Table I).

IV

•

0.2394

0.0964

0.10400.1477

0.00130.15950.1318

0.05070.14130.1462

0.10910.08120.1510

0.08580.1009

0.09330.1500

0.0507

0.04420.10030.0985

0.1034

oO

OR

O

OR

R

OOR

R

O

ORRO

OOR

R

O

Motion-y

-0.5517-0.0332

-0.56290.1897

-00186-0.7068

0.35890.2918

-0.3202-0.76330.52130.5039

-0.1198-0.40840.6721

0.6682-0.8052

-0.0051-0.1720

0.80500.8042

-0.5000

TABLE I. Eigenvalues of cnergy (in 2ñ2lM a2 units) and "r corres-ponding to first 22 stationary states.

State E (r, m)

1 1.6858 ,O, 12 3.9729 ,1,1

3 4.5843 01,14 7.2693 ,2,1

5 7.6806 02,16 9.2967 ,0,2

7 11.4389 ,3,18 11.6409 03,19 13.7318 ,1,210 15.5806 01,211 16.3332 ,4,1

12 16.4067 04,113 19.3709 ,2,2

14 20.9897 02,215 21.9001 e5,1

16 21.9219 05,1

17 23.5324 l'0,318 26.1874 ,3,2

19 27.4137 03,220 28.1425 ,6,1

21 28.1482 06,122 30.0605 ,1,3

(30)0- 2q

"1- --- 4q -

4. Probability distributions

We first discuss the probability distributions in the quantummechanics billiard. Exeepting for the case r = O,eaeh eigen-funetion t.pr .••• can be split into an even mode ecpr .••• and anodd mode o"Pr .•••. By using this notation, the eigenfunctions

associated to Er•m in the elliptic quantum billiard are ex-pressed in lhe following form

Even modes:,'1' •._«(,7)) = Ce.«("q.,m)ce.(7)"q.,m)' (31)

Odd /IIades:.'1'.,_«(,7)) = Be. (e oq.,m)se.(7), oq.,m)' (32)

Rev. Mex. f"ls. 47 (5) (2001) 48Q-488

486 PROBABILlTY OISTRIBUTIO:-lS IN CLASSICAL AND QUANTUM ELLlPTIC BILLlARDS

" ,."

/I~,l~l """ " .~.,(t,l

o oo " o, o.•

(a) (b)

Tow-0633 06T 00.0- -o.n~

" TCl-06!>2 Tel.- -0.752

PCl:(16,5) 04 PCl:(30,~)

"02

o • .R~,i~) .~.2('l)

o oO, o, " "o pV5 pV' 4pi15, ,

(e) (d)

FIGURE 10. Comparison of quantum and c1assical probabilityucnsitics. a) Transitiun uf the radial probability [.....•Ce;(.!.qr."Olfrom oscillatory motion to rotalional motion. ln b), e), andd) PCL: (1',71) is associ<ned to a 1'.bounces c1assicaltrajeclOry.

_"'1 _".11 _IUJ """,7'1- ••-- - ••--,;,) _(2,J) _,.l.,) _tJ2¡.- '~f .'. '•.•....... -o- ...•:•...'.- . -:~""""'1 _'"7) _i~l) <>0<1(6')

•• •• '::: ::; ."'. "". ..,.. ~,..~ • •.. . _.,.--

_,O " .....,'0.1) •••..•••'0 l ""'(11)- -"' •. [' 11 ....",], """'17.1) _(22)~ .1. ~...~•• .., -_,JI, _13;» •••••1.') •••.••• ,"511.::. "Ih, .'. ." .•.• • • •'0. .,,-

FIGURE 8. Probability dislributions for (he first cvcn modcs corres-ponding lo data in Table 1.

FIGURE 9. ProbabililY dislrihutions for [he firs[ odd modcs corres-ponding lo dala in Table 1.

In Figs. 8 and 9 we provide the probabilily densilies(o.,p,.m =•.•1':._) of the first even and odd modes from Ta.ble l. Tú compare with the c1assical solution. plOlS of (hecorresponding caustics are al50 included. The eccentriciticsof the caustics are related 10 'Y (fourth caJumn in Table 1)by Eq. (11).

By observation of the probability distributions in Figs. 8and 9 we can recognize that the mode 'Pr,m. has r angular no-dal-lines corresponding to TJ == constant curves, and In radialnodal-Iines corresponding to ~ = constant curves includingthe boundary nodal-line.

The behavior of the radial density R,.l (O as r inereasesis shown in Fig, IDa. The plots correspond to data from Ta-ble I. whcre we can observe'that the first two modes Ceo.}and Cel,l present oscillating motion. As r increases, the ave-rage value of the coordinate ~ --+ ~o and the motion be-carne from oscillating to rotational kind. In Figs. ~Ob-IOd,we compare the classical and quantum densities for motionswith a very similar '"Yvalue. The c1assical distributions wereobtained by dividing the whole range for ~ (or ry) in 50 sub.divisions and assuming the probability as proportional to thetime spent by the particle in each subinterval. The verticaldashed I¡ne is the c1assical Iimit of motion. A bidimensionalcomparison belween classical and quantum densities is givenin Fig. 11.

In Figs. 8, 9, and 10 it is evidentlhat quantum probabili-ly distributions penetrate the forbidden zone defined by lhec1assical I¡mit (i.e., caustics). In arder to quantify lhis pene-tration. we calculate the quotient

J ['P•. _(~, 7))]' daU' == ?fz == forbiddell i!Oll{'

Pi J ['P •. ",(~, 1/)]' dae1lipsc

FIGURE 11. Prohability distributions for the first odd modes corres.ponding tu dala in Tahle 1.

where Prz is the probability inside the farbidden c1assical zo-ne, Pi is the probability in the whole ellipse, da = h2d~dll =(1'/2)(eosh 2~ - eos 2'1) is the differential element of area.and 1'•.",(~, '1) are the eigenfunetions given by Eqs. (31)and (32). ,\fter replaeing the aboye expressions in lI', weobtain for even mades

• 1\ Jo'e J~'e Ce;(Oce;(II)(eosh 2~ - eos 2,/) d~dllH' == 27t 1

Jo'oJo Ce;(~)ee;(II)(eosh 2~ - cas 21/) d~dry

where the uppcr limits and the constant f( depend on theforbiddcn zone (i,C., on the sign of 1). For rotational mo-lion (-y > O). ~e = ar('eosh (l/ee), rye = 27r, and[( = 1. For oseillating motion (-y < O), ~e = ~o, '/m'" =arceos (l/e,), and [( = 4. The double integrals in the aboyeequation can be separated into one-dimensional integrals, theresulting expression is written as

Re" Mex. Fís. 47 (5) (2001) 480-488

-

JULIO C. GUTIÉRREZ.VEGA, SABINO CHÁ VEZ.CERDA AND RAMÓN M. RODRÍGUEZ.DAGNINO

f<c Ce' cash (20 d( f"C ce' d'l - f.o'" Ce; d( Jo"C ce; cos (2'1) ,11111' - }" Jo 1'" Jo r 1

- \ ( (21f1r JooCe; cosh (20 d( - JooCe; d( Jo ce; cos (211)d'l

487

(33)

27T •where lhe orthogonality property Jo ce; dl] = 7r. has becnused in Ihe denominalor. \Ve apply a Newton-Cotes high-order method to evaluate the inlegrals in Eq. (33), The resultsfor the firSl22 modes are given in the sixth column ofTable L

4.1. Comparison with lhe circular billiard

In this subsection we compare lhe solutions of elliptic bil-liards with respecl to known results of lhe circular billiards.The solution foc lhe circular quantum billiard is well-knownin lhe literaturc [3]. For a circular biJliard of radius a, lhedegenerate eigenfunctions are given by

'P, .• (p,B) = J,(k"",p) (~~~;~),

where Jr is lhe r.th crder Bcssel function, (p, B) are rhecircular cylindrical coardinates, and kr,m is dctcnnincd bysalisfying lhe Dirichlet condilion at boundary, J,(k,a) = O.Then

2i.\f Er,1Il'Ir,m == kr(1 = (l h2

where qr,m is the m-th zero of the Jr Bcsscl function. Fina-lIy, the encrgy eigcnvalues are given by

E - ( 2'" ) '1;,,,, (34)r.1Il - .\1(12 ..J'

In Table Il we compare the first eigenvalues of energy fora circular billiard of radius a, and for an elliptic billiard ofmajor semi-axis a with e = 0.2 (j.('., b ::::::0.98a). According10 Eqs. (29) and (34), lhe elliptic and circular energy eigen-values are proportional to fJr,lIl/e2 and q~.IIl/'l, respectively.Despite the axes ofthe cllipse still have very similar yalues, itis enough to show that each circular mode splits into an evenmode and an odd mode. In Table 11we can aIso see that theeven modes have lower energy eigenyalues than odd modes.By comparing Table 11with respect to Table I. it is notori.ous that decreasing the eccentricity implies a reduction in thecorresponding energy eigenvalues.

5. Conclusions

We haye compared the probability distributions of periodictrajectories inside a classical and a quantum elliptic billiard.The most important results are sUllllllarized as follows

i) Thc' periodic trajcctories in c1assical elliptic billiardsare characterized by eigcnvalues of cquations involvingelliptic integrals. There are two kinds of motion, cor-responding to elliptic and hypcrbolic caustics insidelhe billiard. The inlroduetion of the parameter -y,Eq. (10), is appropriale because it allows us to makeanaJogies between the c1assical and quantum case.

ii) By excepting the fundamental hannonics (1' = O), eachcigenfunction 'P,.,•., in the qllantum elliptic billiard can

TABLE 11.Comparison 01'(he cigenyalucs of energy (in 2/¡2 j.\fa2

units) for a circular and ao ellipüc billiard.State Circular Elliplic

E (r, m) E (r, m)1 1.4458 0,1 1.4759 ('0,12 3.6705 1, 1 3.7087 ('1, 13 3.7851 01, 14 6.5937 2,1 6.7210 ('2,15 6.7308 02,16 7.6178 0,2 7.7863 ('0,27 10.1766 3,1 10.3823 e3,l8 10.3830 03,19 12.3046 1,2 12.4385 ('1, 210 12.6942 01,211 14.3957 4,1 14.6881 e4, 112 14.6861 04,113 17.7125 2,2 18.0231 e2,214 18.0879 02,215 18.7218 0,3 19.1766 ('0,316 19.2347 5,1 19.6260 ('5,117 19.6260 05,1

be separated into an even eigenfunction e'Pr .•.. and anodd eigenfunction o'P•..•,,' This separation is irrelevantin a circular billiard.

iii) The odd energy eigenvalues are larger than even eigen-values. The difference in energy increases with the ec.centricity ofthe ellipse. In this rnanner, as e -7 O, bothenergy eigenvalues get c10ser and tend to the eigen-value of the circular billiard, the motion becomes morerotational (elliptic caustics in the c1assical billiard). Onthe contrary, the / values in even eigenfunctions arelarger than odd eigenfunclions. The oscillating modesare always non-degenerate, whereas the rotating modesbecome more degenerate as / increases.

The eigenfunctions in the elJiptic quantum billiard can beexpressed as alinear combination of Mathieu functions. \Vehave shown graphically Iypical probability-pattems for dif-ferent eceenlricities of lhe elliplie boundary. The effeel of el-liplicily of lhe boundary has been found to be of great impor-tance on the eigenfllnction structure. We have also presentedinteresting differences between the circular and elliptical bil-liards.

Acknowlcdgrncnts

The authors are grateful to Dr. Eugenio Ley Koo forconstruc.tive suggestions and discussions concerning the manuscriptand Mathieu functions. This work was partially supported byCONACyT.

R,,'. Mex. Fis. 47 (5) (2001) 48G-488

488 PROBABILITY OISTRIBUTIONS IN CLASSICAL ANO QUANTUM ELLIPTIC BILLIAROS

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Rev. Mex. Fís. 47 (5) (2001) 48ü-488