Energy Spetrum and Eigenfuntions
through the Quantum Setion Method
J.S. EspinozaOrtiz
and R. Egydio de Carvalho y
Institutode Geoi^eniase Ci^eniasExatas
UniversidadeEstadual Paulista
Rua10, 2527,Santana,13500-230, Rio Claro,SP, Brazil.
Reeivedon16May,2001. Revisedversionreeivedon23July,2001.
WepresenttheQuantumSetionMethodasaquantizationtehniqueto omputetheeigenvalues
and theeigenfuntions of quantumsystems. As aninstrutiveexample we apply this proedure
to quantize the annularbilliard. The methoduses thesymmetry of the system to determinean
auxiliarysetionseparatingthesystemintopartialregionsandomputestheGreen'sfuntionsfor
Shroedinger'sequation,obeyingthesameboundaryonditionsimposedontheeigenfuntionsofthe
system. Theeigenvaluesare obtainedaszeroesofaniterealdeterminantandtheeigenfuntions
arealsodetermined.Thepresentanalytialandnumerialresultsareintotalagreementwiththose
obtainedbyotherproedures,whihshowstheeÆienyofthemethod.
I Introdution
The Green's funtion method applied to the
Shroedinger's equation assoiated to the Poinare's
setiontehniqueusedtostudylassialdynamis,have
originatedamethodtoomputetheeigenvaluesandthe
eigenfuntionsofquantumsystems. Thismethodis
de-notedbyQuantumSetionMethod(QSM).Itwasrst
introduedbyBogolmonyasanessentialingredientto
theresummationtheoryoflassialperiodiorbits
on-tributing to the quantum energy spetrum of haoti
systems[1,2℄.
Theonnetionbetweenlassialperiodi
or-bitsandquantumenergyspetrumomesfromthefat
that thesemilassiallimit ofstationary statesshould
beonnetedwithlassialinvariantmanifoldsandfor
ahaotiHamiltoniansystemthelatteraretheperiodi
orbits,besidesthetotalenergy. Forsystemswhihare
notfully haotibut mixed by haos and regular
mo-tion, the quantum energy spetrum arries also
inu-enesofinvarianttori.
Alass ofmodelswhihnielydesribe
regu-lar,mixed orhaotidynamisarethebilliards. These
orrespondto geometrialboundarieswhereapartile
movesfreelyinsidethem andreetsspeularlyatthe
boundaries. For these type of systems the motion is
stronglydependentontheshapeoftheboundariesthus
allowingthesimulationofanydesireddynamis.
Inthis work weonsider the annular billiard
(annulus) probing the appliability of the QSM. This
billiardonsistsoftwoeentri irleswhere onlythe
annular region onstitutes the aessible spae to the
motion. Changing the eentriityweanontrol the
seaof haosand alsotheregularityofthesystem. We
onsider lassial and quantum mehanial aspets of
theannularbilliard insetion II.InsetionIII we
de-velop the QSM. Finally in setion IV we present the
numerialalulationsandadisussionoftheseresults.
II The Model
AsdepitedinFig. (1a-1b)thebilliardisdened
by two eentri irles on the XY-plane, where the
outerirlehasradius R 'andtheinnerirleradius
r'. Theentre oftheouterirle isonsidered asthe
originof thesystemand the inner irleandisloate
horizontallyinrelationtoit. Thedistanebetweenthe
entres of the irles is alled by eentriity and we
denoteitby'Æ'. As'r'as'Æ'areonstrainedbythe
in-equality(r +Æ) R. Thetrajetoriesonthebilliard
are polygonal lines among ollisions with the
bound-aries. There are twodierent typesof motion
identi-edbytheletters'A'and'B'andwhenomposedan
desribe any possiblebehaviorof the system[3℄. The
A-typemotionorrespondsto thosetrajetorieswhih
ollidewiththeouterirlebeforeaollisionwithitself,
e-mail:ortizr.unesp.br
whiletheB-typemotionreportstothetrajetories
ol-lidingwiththeinnerirlebetweentwoollisions with
the outer one. There is also a partiular ase of
A-trajetories,alledWhisperingGalleryOrbits(WGO),
whih never attain the inner irle, what means they
neverrosstheausti-anauxiliaryirlewithradius
(r+Æ). The austihelps us toseparate bothkindof
motions and later we will assoiate to it an algebrai
tangeny ondition to distinguish the A-motion from
theB-motion.
Figure 1. a) The annular billiard sheme exhibiting the
outerirleofradiusR ;theinnerirle ofradiusr,the
e-entriityÆ;bothangleoordinates and ;and atypial
trajetoryofA-motion;b)atypialtrajetoryofB-motion;
)anauxiliaryirularsetionofradius,onentriwith
theinnerirle,deningtwopartialregions.
Thedynamisonphasespaeisnaturally
de-sribedbyBirkho'soordinates,these aredened on
the outer boundary at the positions where the
olli-try,twoangleoordinatessuÆetodenetheollision
points: ' assoiated with the ar-length ounted
from the positive absissaaxis and ', the reeted
angle formed between the trajetory and the normal
at theollision point, see Fig. (1a-1b). Hene we
de-ne the phase spae oordinates as S = sin() and
L==(2),sothatphasespaereduestothe
retan-gle: 1 S 1, 1=2 L1=2. Thusthemotion
anbe desribed by adisrete dynamis, or merelya
map,involvingonlykikswiththeexternalirle. The
map an be easily obtained through pure geometrial
onsiderations. Hereafterweonsider R=1andfora
trajetorystarting at anyinitial ondition (
0 ;
0 ) we
requirethetangenyondition
jsin()+Æsin( )j > r (1)
whih means that at the nextollisionwith the outer
irle the trajetory had not previously hit the inner
irle. This denes the A-motion (Fig. 1a) given by
themap equations
1
=
0
1
=
0
+( 2
0 ):
(2)
Ifthetangenyonditionisnotsatised,whihmeans
thatat thenextollisionwiththeouterirle the
tra-jetoryhaspreviouslyreahedtheinnerirleone,the
B-motion(Fig. 1b)isdenedviathemapequations
sin() = ( sin(
0
)+Æsin(
0
0 ))=r;
(
1 +
1
) = 2 (
0
0 );
sin(
1
) = rsin() Æsin(
1 +
1 );
(3)
where isthe angle ofthe reetion at theinner
ir-le (see Fig. 1b). It is easy to verify from the rst
formulaaboveas thetangeny onditionemerges. To
generatethedynamis of atrajetoryonphase spae,
thetangenyonditionshouldbehekedat eah
iter-ation to selet what map has to be used. The set of
Figs. (2a-2d)exhibits thedynami rihnessof the
an-nular billiarddepending ontherelationshipbetweenr
andÆ. From these plots we an observesomegeneral
results[3℄,theoriginofthephasespae(L;S)=(0;0)
orresponds to an ellipti xed point (stable
equilib-rium) while r > Æ and it beomes a hyperboli xed
point (unstableequilibrium) if r <Æ passingthrough
a neutral stability when r = Æ. Looking at the Fig.
1bthe trajetory(L;S)=(0;0) orrespondsto aline
over the positive absissa starting at the outer irle
andkikingtowardtheinnerirle. Ontheotherhand
wenotethatthepoint(L;S)=(1=2;0)always
orre-spondstoahyperbolixedpoint(exeptwhenÆ=0).
From Fig. 1b, the orresponding trajetory starts on
thenegativeabsissa at the outer irle and kiks
to-wardtheinnerirle. Numerialobservationsindiates
that the aessible phase spae beomes almost fully
haotiatÆ 3r[3℄. Inthatontextitisimportantto
establishonemoreidentiationoftheWGO:onphase
spaethey are thestraightlines whih haveordinates
Figure2. Thephasespae(L;S)forr=0:2andfourvaluesofÆ: (a)Æ=0:02,thereisathinlayerofhaosandtheWGO
areloatedatjSj > 0:22;(b)Æ=0:15,theoriginstillorrespondstoastablexedpointbutthehaotiseahasinreased,
the WGOare loated at jSj > 0:35 ; ()Æ =0:30, theorigin is a hyperbolixed point andhaos ontinues inreasing,
theWGOareloated atjSj > 0:5; (d)Æ=3r=0:60,theaessible phasespaeisalmostfullyhaoti,andtheWGOare
loatedatjSj > 0:8.
Regardingthese mainlassialaspetsofthe
annulus,the orresponding quantum problem onsists
onsolvingtheHelmholtzequation
r 2
+ 2
(;)=0; (4)
obeying Dirihlet's boundary onditions, where =
p
2E=~(m = 1) with and the radialand angular
polarvariablesrespetively. Itisanalogoustothe
eigen-modeproblemforanannularvibratingmembrane,also
withaninnitenumberofdisretevibrational
frequen-ies. FortheaseÆ=0theeigenvalues
nm
assoiated
with the Helmholtzs equation are obtainedas zeroes
oftheequation
J
n (
nm R )
J
n (
nm r)
Y
n (
nm r)
Y
n (
nm R )
=0 (5)
andtheorrespondingeigenfuntionsaregivenby
nm
(;)=A
nm
J
n (
nm )
J
n (
nm r)
Y ( r) Y
n (
nm )
os(n) ;n=0;1;2;::::::evenstates;
sin(n) ;n=1;2;3;::::::oddstates;
where J
n andY
n
aretheBessel'sfuntions ofrstand
seond kindrespetivelyand A
n;m
is anormalization
onstant. Exeptforn=0whihisasinglestate,the
remainingstatesaredegeneratedoublets. Forthease
Æ6= 0theangularmomentumisnotonserved,however
thesystemstillpossessesadisretesymmetryunder
re-etion with respet to theline joining the entres of
bothirles. Consequentlytheeigenstatesofeq.(4)
on-tinuetobeseparatedintoevenandoddstates,butnot
longer degenerate due to an eet of dynamial
tun-nelling'whihsplitsthem. Thisphenomena,inuened
byhaos,isattributedtolassialtransportonneting
WGO regions[3, 4℄. These quantum billiardhasbeen
onsidered by dierent theoretial methods in the
lit-erature [5,6,7, 8℄and also experimentally [9℄,paving
thediretionsforusto applytheQSM.
III The Quantum Setion
Method
Thismethodonsidersthepossibilitytoseparate
any system into partial regions through oneauxiliary
setionin ordertoexploreertainloalsymmetries,as
forexamplebilliardboundariesorpotentialsymmetries
(in theaseofopensystems)[10℄.
Theunertaintypriniplepreludesthe
simul-taneousknowledgeofmomentaandpositionofa
quan-tum partile, henethe quantum setion is developed
only on position spae. We onstrut partial Green's
funtions,G
1 (q;q
0
;E)and G
2 (q;q
0
;E),in bothpartial
regionswiththefollowingproperties: a)theysatisfythe
non-homogeneous Shroedinger's equation inside eah
partial region r 2 + 2 G j (q j q 0 j
;)=Æ(q
j q
0
j
);j=1;2; (7)
b)ontheboundariestheseGreen'sfuntionsshould
sat-isfythesameboundaryonditionsastheeigenfuntions
oftheoriginal system, ) these Green'sfuntions may
bearbitraryonthesetion,buttheyshouldmath
on-tinuously. Wewritedowntheansatzforsolutionsinthe
formofasinglelayerpotential
j;E (q)= Z G j (q;Q 0 ;E) j (Q 0 )dQ 0 ; (8) whereQ 0
is denedon thesetionand
j
isthesingle
layer density whih is initially an unknown funtion.
Theideanowistotakeasolutioninonepartialregion
andextenditto theother region,imposing ontinuity
forthepartialfuntions
j;E
(q)andtheirderivativeson
thesetion. Then thequantization onditionemerges
andtheeigenvaluesareobtainedaszeroesofamatrix
determinantdenedonthesetion. Thisallowsthe
sin-glelayerdensitytobeobtainedandfurthertoalulate
theeigenfuntionsoftheoriginalsystem.
For the annular billiard the additional
par-tialseparability of the Shroedinger's equation in
po-laroordinates makespossibleto onstrutexat
par-tialGreen'sfuntionsineigenmodesrepresentationand
fromtheretheonstrutionofpreisepartialfuntions
in eah side of the setion asdepited in Fig.1. We
dene the setion as an auxiliary irle of radius ,
onentri with the outer irle, whih separatesboth
partial regions: the region (1) outer to the irle
andtheregion(2)innertoit.
The Green's funtions are onstruted
satis-fying the non-homogeneous Helmholtz's equation (7),
whereq
1
=(;)andq
2
=(;)arethepositionsin
re-lationtotheR -irle'andther-irle',respetively. As
fortheeigenfuntions,werequireG
1
tosatisfy
Dirih-let's boundary onditions on the outer irle. In the
samefashion G
2
anelsontheinner irle,withboth
Green'sfuntionsarbitraryoverthesetion.
Due to partial separability in polar
oordi-natesinrelationtothe R(r)irle',wemaywrite
G 1 ( 0 ; 0 ;) = P m g 1;m ( 0
;)hjmihmj 0
i; 0
<<R;
G 2 ( 0 ; 0 ;) = P n g 2;n ( 0
;)h jnihnj 0
i; r<< 0
;
(9)
where h mji = hjmi / os[sin℄(m)for theGreen'sfuntion G
1
andfor theGreen'sfuntion G
2
oneexhanges
m! nand!toobtainhnji = h jni / os[sin℄(n). Forintegersmorntheangularfuntions 0 os 0 and 0 sin 0
willyieldsolutionspossessingevenandoddsymmetries,respetively. Nowitishosenapartiularone-dimensional
Green's funtiong
j
in radialoordinates,exludinganypossibilityforG
1 andG
2
toaneloverthesetion
g
1;m (
0
;) = H (1)
m
(R )H (2) m ( 0 )fY m (R )J
m
() J
m (R )Y
m ()g ; g 2;n ( 0
;) = H (1) n ( 0 )H (2) n
(r)fY
whereJ;Y;H (1;2)
are theBesselfuntions oftherst,
seondandthirdkindrespetively. TheseGreen's
fun-tionsaregiven,inAppendix A,byapplyingtheimage
method for asoure in 0
( 0
) with respet to = R
( =r)in suhawaythatthequantumdynamis
or-responds to twoquantum paths between 0
and ( 0
and ). Onepath is diret while the other reets in
R (r). There isnopossibility forthepartileto ome
bak sineit leavesthe setion. Inthis sense there is
ananalogywiththesatteringformalism[11℄.
Both Green's funtions are diagonal in eah
eigenmodes representation. We write down anygiven
partialwavefuntionineahpartialregionintheform
ofasinglelayerdensity
1 (;;) = R 2 0 d0( 0 )G 1 ( 0 ; 0 ;); 2 (;;) = R 2 0 d 0 ( 0 )G 2 ( 0 ; 0 ;): (11)
These funtions satisfy Shroedinger's equation with
appropriated boundary onditions in eah partial
re-gion. Theonditionfor these funtions andtheir
nor-mal derivativesto math over the setion is the
exis-teneofonlyonedensity. Thisonditionimposes
Z 2 0 d 0 ( 0 ) ~ G (q 0 2 q 0 1 ;)= Z 2 0 ~ G (q 0 2 q 0 1 ;)( 0 )d 0
= 0; (12)
where ~ G(q 0 2 q 0 1 ;)= Z 2 0 dfG 2 ( 0 ; 0 ;) G 1 ( 0 ; 0
;) G
1 ( 0 ; 0 ;) G 2 ( 0 ; 0
;)g: (13)
Thisequationmayberewrittenas
~ G (q 0 2 q 0 1 ;) = P mn < 0
jn><nj ~
Gjm><mj 0
>; (14)
where,
<nj ~
Gjm> = R 2 0 dfG 1 (;;) G 2
((;Æ;);(;Æ;);)+
G
2
((;Æ;);(;Æ;);)
G
1
(;;)g;
(15)
andtheonditiongivenin eq.(12)beomes
X m hjmi D mj ~ Gjn E = X n D mj ~ Gjn E
h nji = 0: (16)
The eigenvaluesare omputed asthezeroes of detj <nj ~
G jm> j, where the integralsare evaluated over
thesetion. Before omputingthe integralsit isneessaryto drive 0
ontoR aswell as 0
onto r andto expand
G
2
(;)in termsof(;). Thisexpansionusestheaddition theoremforBessel'sfuntions
J n () os(n) sin(n) =os(n) P l J l+n ()J l (Æ) os(l) sin(l) sin(n) P l J l+n ()J l (Æ) sin(l) os(l) ; (17) and Y n () os(n) sin(n) =os(n) X l Y l+n ()J l (Æ) os(l) sin(l) sin(n) X l Y l+n ()J l (Æ) sin(l) os(l) (18)
Finallytheintegralsareomputedonthesetiononsidering
R
2
0
os(n)os(m)os(l) = Æ
l; (m n) +Æ l;(m+n) +Æ l;(m n) +Æ l; (m+n) =4 R 2 0
sin(n)os(m)sin (l) = Æ
l;(m+n) +Æ
l; (m n) Æ l;(m n) Æ l; (m+n) =4 R 2 0
os(n)sin(m)sin (l) = Æ
l;(m+n) +Æ
l;(m n) Æ
l; (m n) Æ l; (m+n) =4 R 2 0
sin(n)sin(m)os(l) = Æ
l; (m n) +Æ l;(m n) Æ l;(m+n) Æ l; (m+n) =4 (19)
yieldingthegoal
<nj ~
Gjm> = 1 2 f[Y n (r)J m () J n (r)Y m ()℄ [Y m (R )J
m
() J
m (R )Y
m ()℄
[Y
m (R )J
m
() J
m (R )Y
m ()℄ [Y n (r)J m () J n (r)Y m ()℄ gj
=
fJ
m n
(Æ)( 1) n
J
m+n (Æ)g:
In this form the eigenvalues are obtained as
zeroesof the determinant of the real matrix given in
eq.(20), where the refers to the evenand odd
solu-tions,respetively.
IV Results and Disussion
Without loss of generality by taking the setion
overto theouterirle, whatreads=R ;the
math-ing ondition leads to the known Nagaya & Sing &
Kothary's eigenvalues expression[5℄. The ase Æ = 0
impliesthat thesinglepossibilityprovidingnon-trivial
solutions foreq.(20) holds when m = n. In this ase
thematriesbeomediagonalforbothparitiesandthe
eigenvaluesareexatlydegenerate(exeptwhenmand
n are null). When Æ > jm nj; for a xed , the
Bessel's funtionsbeomenon-osillatingand they
de-rease exponentiallyasjmnj inreases. Soitis
ne-essary to be areful when enlarging the determinant
dimension beause it will beome numerially
unsta-ble. Hene as anie and interestingonsequene, the
dimension of detf<nj ~
G jm >g is alwaysnite. With
the eigenvaluesin hand the eigenfuntionsare readily
determined going bak to eq.(16) and omputing the
density.
Theoperator given in eq.(14) is alled
quan-tum transferoperator andin thesemilassiallimit it
an be expressed in term of lassial paths providing
a wayto interpret the quantum dynamial tunnelling
phenomenain termsoflassialtransport.
Wehaveomputedtheeigenvalueswith14
dig-its of preision for any given energy sale. Figs. (3)
and (4) onern to an energy spetrum with the
fol-lowing set of parameters: ~ 0:4583; r = 0:40 and
Æ=0:19. InFig.(3)itisplottedthenumerial
umula-tivedensityof statesfortherst2500eigenvaluesand
in Fig.(4)its deviation in relation to thesoft
umula-tivedensityofstatesforbilliards,givenbytheformula:
N()= 1
4~ 2
fA 2
L g[12℄, wherefor theannular
billiard, A isthe netarea (=(R+r)(R r)) andL
thenetperimeter(=2(R+r)). Frombothplotsitis
inferred that nolevelismissing from theenergy
spe-trum and that there is no lak of onvergene. A
se-queneofeigenvaluesofbothparitiestogetherisshown
in Fig.(5) for dierent values of the eentriity, with
xed~ 0:2731;andr=0:20. Thisplotpresentsthe
levelsdynamisasafuntionof Æ exhibiting avoided
rossing' as it is expeted in haoti/mixed systems.
Note also the degeneraies breakdown for some levels
as haos inreases(remember thehaotisea enlarges
with the inreasing of Æ; see Fig.(2a-2d)). These
en-ergy splittingsrelatinglevelsofunoupled parities are
attributedtoakindoftunnelingdenotedasdynamial
tunneling. Finallysomesarredeigenfuntionsare
pre-sentedinFig.(6)forsomeseletedeigenvaluesobtained
withthesameparametersofFig.(3). Theyarry
infor-trajetorieslyingin dierentpositionsonphasespae.
Figure 3. The numerial umulative density of states for
therst2500eigenvaluesusingthesetofparameters: ~
0:4583;r=0:40andÆ=0:19. Theinsertpitureshowsthe
numerial umulativedensityof statesand thesoft
umu-lativedensityofstatesforbilliards.
Figure4. Thedeviation(N =(N)numerial (N)soft)
betweenthenumerialumulativedensityofstatesandthe
softumulative density ofstatesusingthe samesetof
pa-rametersofFig.(3). Bothumulativedensitiesdieraround
be-Figure 5. The levels quantum dynamis. A sequene of
eigenvaluesis shownfor dierent valuesoftheeentriity
using thefollowing setof parameters~ 0:2731;r=0:20
andÆ=[0:0;0:75℄.
Finallythesequantum analytialand
numer-ialresults tompletely with the known results
on-erningthisbilliardasestablishedintheliteratureand
essentially they assert that the QSM is an eÆient
quantizationmethod.
Aknowledgments
We thank the Brazilian sienti agenies
Funda~ao de Amparo a Pesquisa do Estado de S~ao
Paulo (FAPESP) and Funda~ao para o
Desenvolvi-mento da Unesp (Fundunesp) for nanial support.
JSEO is partiularly aknowledged to the LPTMS at
Orsay Cedex, Frane for kind hospitality during last
visitandtoDr. D. Ullmoforfruitfuldisussions.
Appendix - The Radial Green Funtions
Hereitisonsideredthepartialregion2where
thepartial radialGreen's funtion isonstrutedwith
the following boundary onditions: it is nullat =r
and itis arbitraryat = 0
. These onditionsleadto
thedenitions
g
2 (
0
;) =AfY
n (r)J
n
() J
n (r)Y
n
()g ;r<<0
~ g
2 (
0
;) =BfJ
n
()+ iY
n
()g ;
0
<;
(21)
whereAandBareonstantsto bedeterminedfrom theadditionalrestritionsat = 0
g
2 (
0
;) ~g
2 (
0
;)j
= 0
=0;
1
d
d fg
2 (
0
;) ~g
2 (
0
;)gj
= 0
= 1:
(22)
Solvingeq.(A-2)yelds,
g
2 (
0
;) =H (1)
n (
0
)H (2)
n
(r)fY
n (r)J
n
() J
n (r)Y
n
()g ; r < < 0:
(23)
Bysimilar argumentstheGreen'sfuntion inthepartialregion(1)isgivenby
g
1 (
0
;) =H (1)
n
(R )H (2)
n (
0
)fY
n (R )J
n
() J
n (R )Y
n
()g; 0
< R < R:
(24)
d
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Figure 6. Sarred eigenfuntions using the same set of parameters of Fig.(3). The even solutions are loated on the
left of the gure and the odd solutions are loated on the right. The eigenvalues used in the plots are the following:
a) = 16:8961439227900 ; b) = 16:91942974911557; ) = 27:3224162938150 ; d) = 27:3035930847495; e) =
27:4114121918047; f) =27:3452473498958 ; g)=31:6387754925558; h)=31:6383421881693; i)=31:7006255714796;
