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Smoothed wave functions of chaotic quantum systems
Physica D 31 (1988) 169-189 North-Holland, Amsterdam
S M O O T H E D WAVE FUNCTIONS OF CHAOTIC QUANTUM SYSTEMS E.B. BOGOMOLNY The Academy of Sciences of the USSR, LD. Landau Institute for Theoretical Physics, 142432 Chernogolooka, Moscow Region, USSR Received 31 December 1987 Communicated by V.E. Zaldaarov It is shown that wave functions of quantum systems as h ~ 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic
trajectories. The contribution of each trajectoryis expressedthrough the dements of the monodromymatrix of the trajectory. The results are comparedwith the numerical calculations of the wavefunctionsfor the stadium billiard.
1. Introduction
This paper is devoted to the study of wave functions of non-integrable quantum systems in the semiclassical limit h-~ 0. Here non-integrability means that a part of trajectories of the corresponding classical problem does not lie on the invariant toil. Such trajectories are said to be stochastic (or chaotic) and the problem of expressing this stochasticity in quantum language is the subject of the so-called "quantum chaos" or "quantum stochasticity". Conventional semiclassical quantization rules of the Bohr-Zommerfeld type which are strongly based on the assumption that all classical trajectories wind on the invariant tori cannot be applied for similar problems. As a result, the behavior of the eigenvalues and eigenfunctions as h ~ 0 for these systems has not been well studied so far. (See reviews [1, 2] for the problem history and different approach to it.) Consider a quantum problem whose classical motion is ergodic (i.e. almost all classical trajectories spread uniformily over the whole energy surface). The main hypothesis about the behavior of the wave functions for such problems is the assumption by Berry [4] and Voros [5] based on the Shnirelman theorem [3] that the averaged square of the eigenfunctions in the semiclassical limit h--* 0 coincides with the projection of the classical microcanonical distribution to the coordinate space
(l~b(q)12)~po(q) ash~O, fd'p 8(E - H(p, q)) Po(q) = /d"p'd"q'O(E- H(p', q ' ) ) '
(1)
where H ( p , q) is a classical Hamiltonian of the problem. The exact form of the wave functions with such a property is unknown. Usually it is supposed that they are random or at least they are irregular functions of the coordinates and the correlations between their different parts are practically absent [2, 5]. First numeric calculations would seem to confirm a chaotic character of the wave functions of the ergodic systems. In the pioneering paper by McDonald and Kaufman [6] only irregular pictures of the nodal lines (where ~(x) = 0) for high excited eigenfunctions of the quantum stadium billiard were given. 0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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However, in the unpublished McDonald thesis [7] it has been emphasized that certain wave functions manifest regular structures which seem to be connected with the periodic trajectories. The existence of such structures was also mentioned in a short note by Taylor and Brumer [8]. But these results remained practically unnoticed until the papers by Heller [9, 10] where it was emphasized that a large number of high excited wave functions for the stadium billiard have an enhancement in the vicinity of the classical periodic trajectories. This result seems to be unexpected since for ergodic problems all periodic trajectories must be unstable or neutral. Hence, if one constructs a wave packet centered near such a periodic trajectory, it must spread out quickly (per time defined by classical dynamics). It will be shown below* that nevertheless each unstable periodic trajectory gives a definite contribution to the wave function which is maximal either at the very periodic trajectory or near it. If for an ergodic problem one averages the wave function square over small intervals of energy and coordinates the resulting "smoothed" wave function can be represented in the following form:
(l~p(q)12)=po(q)+ht~-l)/2~,,Im
(
I
Ap(x)exp i--h-+i
2h
(2)
where for each periodic trajectory the x axis is chosen along the trajectory and the Ym axes are chosen perpendicularly to it (see fig. 1), n is the number of degrees of freedom (n >_ 2). Sp = ~Pn dqn is a classical action calculated along the trajectory. Ap(x) and Ve'pkm(x)are classical quantities defined through the elements of the monodromy matrix of a given trajectory. The monodromy matrix or the matrix of Poinca:" mapping is the matrix which connects a solution of linearized classical equations of motion in the plane perpendicular to the periodic trajectory per period:
yk(T) ] = Mf Yk(O)),
lye(o)
(3)
where T is the period of the given periodic trajectory. The summation in eq. (2) is performed over the finite number AN of the periodic trajectories which depends on the value AE entering the definition of the energy smoothing. As AE ~ 0, AN---, oo. While changing the energy the contributions of certain periodic trajectories become larger than those of others. One may say that at different energy eigenvalues it is possible to see different periodic trajectories. Eq. (2) allows to express a smoothed wave function as h ---,0 in terms of the quantities defined from classical mechanics and it may serve as the basis of a semiclassical approximation for the ergodic quantum systems. Similar considerations were used in refs. [11-13] to derive the formula connecting the density of energy levels with the sum over unstable periodic trajectories of ergodic problems. However, to the best of author's knowledge they were not used before to get information about the eigenfunctions. The plan of the paper is as follows. In section 2 we discuss a general formalism based on the semiclassical representation of the Green function of the SchriSdinger equation as the sum over all classical trajectories connecting two fixed points. It is shown here that after the averaging over the small intervals of energy and coordinates the main contribution to the Green function as h ~ 0 will be given by the classical trajectories located in the vicinities of short-period trajectories. *A short note about this subject was published in [24].
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In section 3 the expressions for Ap(x) and wpkm(x) in terms of the elements of the monodromy matrix are obtained, the general properties of these formulae are discussed and a physical interpretation of the wave function amplification near the unstable periodic trajectory is given. In section 4 a case of the quantum stadium billiard is considered in detail. The results are compared with the pictures of high excited wave functions published in refs. [9, 10] and it is shown that even small details of the wave function structure are well reproduced analytically. Though the discussed effect is of importance for non-integrable problems only, it also exists in some integrable systems where there exist unstable periodic trajectories. In section 5 it is shown how this effect manifests itself for the integrable elliptic billiard and the similarity and the difference in its manifestations for the integrable and non-integrable systems are discussed. In the Appendix a convenient method for the calculation of the billiard monodromy matrix is presented.
2. General formalism
Let E. be the energy eigenvalues and ft.(q) be the corresponding eigenfunctions. Consider the energy interval of length AE with the center in a point E 0 (AE << E0) and define the averaged square of the wave function modulus by the following relationship: 1 ([~k(q)l 2) = ~ E Iq~.(q)I z, (n}
(4)
where the summation is done over all wave functions whose energy eigenvalues are in the chosen interval
(5)
E o - ½ A E _ < E . < E o+ ½AE
and N is the number of such functions. Thus, from the beginning we decline considering individual wave functions and shall investigate the quantifies averaged over a small energy interval. It will be shown below that such a smoothing gives senseful results from both mathematical and physical points of view. Definition (4) has different meanings for the integrable and non-integrable problems. For multi-dimensional integrable systems degeneracy of high excited levels is inevitable [1, 2] and in the formal limit A E ---, 0 eq. (4) does not give a square of the individual wave functions but an average over a large number of degenerated states. For general non-integrable systems only an accidental degeneracy and a degeneracy of different symmetry states is possible [1, 2] which is not difficult to take into account (see section 3) and eq. (4) as AE ~ 0 and E o ~ E, transforms into the squared modulus of one wave function. Define the Green function of the Schr~Sdinger equation in the energy representation in the usual way:
(1t(p", q") - E)G( q", q', E) Using the series expansion of
=
8(q"- q').
(6)
G(q", q', E) in terms of the eigenfunctions,
~b*(q")~.(q') E+ie-E.
G(q",q',E)=~.. n
'
(7)
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172
it is easy to show that the averaged wave function square defined by eq. (4) is expressed in terms of the Green function (6) as follows: ( I m G ( q , q, E ) )
f(ImG(q', q', E)) d'q"
( l f f ( q ) 12) =
(8)
where
~Eo-~AE
Further calculations are based on the conventional expression for the semiclassical limit of the Green function G(q", q', E) as the sum over all classical trajectories connecting the two fixed points q" and q' (see, for instance, refs. [11, 12] and references therein). For simplicity we shall consider below a two-dimensional case. The generalization for multi-dimensional models is straight-forward. In a twodimensional space one has [11, 12]
G(q,q,E)=
Go(q,E)+ h-~G (q,E)+O
ImG0(q, E) =
fs(e- rap, q))
G°~(q, E) = i(2~ri)1/2 1
Y'. IA 11/2 exp (
,
d2p -~S(q,i
(9) q,E)-
i1,~),
,n
where Go(q, E) is a contribution of the so-called "zero length trajectories" corresponding to the Tomas-Fermi approximation. G°~:(q, E) is a contribution of classical trajectories with a non-zero action starting and ending in a point q. S(q, q, E) is a classical action calculated along one of such trajectories. A is a certain determinant constructed from the second derivatives of the action. J, is a phase equal to a number of conjugated points on the trajectory. For the billiard type systems with wave functions equal to zero on the boundary it is necessary to add a double number of reflections from the boundary to I,. Substituting eq. (9) into eq. (8) and expanding in powers of h, we find 2~fh (l~(q)12)=po(q)---V-((ImG
o~¢
( q , E ) ) - l f(ImG°SC(q',E))d2q'),
(10)
where Po(q) is defined in eq. (1). For two-dimensional systems Po(q) = 1/V, where V is the volume of the available regions in the q space (outside this region Po(q) vanishes). We emphasize that the first term in eq. (10) coincides with expression (1) obtained in refs. [2, 4] from other considerations. Expansions (9) and (10) give a formal representation of the averaged wave function square as the sum over all classical trajectories with fixed energy which start and end at the point q and along which the time of motion may be arbitrary. There are an infinite number of such trajectories and for non-integrable systems the summation over them is practically impossible (see, nevertheless, ref. [13]) or at any rate it is as difficult as a direct solution of the Schriidinger equation. This difficulty is not only of a technical character. All we know about the properties of chaotic systems does not contradict the statement that irregularity of
E.B. Bogomolny/ Wavefunctions of quantum systems
173
the energy levels and the wave functions on small scales is so large that a statistical description becomes the only reasonable way to describe them and the exact calculation of the energy eigenvalues and eigenfunctions seems to be meaningless (similar to the calculation of long trajectories for ergodic systems). However, for the quantities smoothed over small (but fixed) intervals of energy and coordinates one can get quite meaningful formulae. Firstly we consider the result of the averaging of expression (9) for G°~:(q, E) over a small energy interval. Taking into account that AE << E o, one can expand the action in the exponent by the deviation from E 0 and retain the first two terms only,
S(q, q, E o + BE) = S(q, q, Eo) + T(q, Eo)SE,
(11)
where T(q, Eo)= (SS/E)e_eo is the time of motion along the given trajectory. As h ~ 0 the main contribution to sum (9) after the averaging over E will be given by the classical trajectories along which the time of motion is restricted and satisfies the following inequality: h T<~CAE,
(12)
where c is a constant (c -- 2~r). In other words, the summation in eq. (9) over trajectories with a finite time of motion T < TO is equivalent to an averaging of the Green function over an energy interval of width h E - ho~o, where
oao = 2 ~ / T o. Similarly, the averaging of the Green function G(q, q, E) over a small coordinate interval Aq at each point q0 yields that the dominant contribution to sum (9) will be given by the classical trajectories for which the change of momentum during the time of motion,
hp
[Pf-Pi[ -- ~
+
(13)
will be small, h
a g -< c ' -a-p
(14)
(c' is a constant of the order of 1). If the matrix of the second derivatives of the action is not degenerated then it is easy to show that in the vicinity of the given trajectory (the nearness is defined by the smallness of Ap) there will be a real periodic trajectory for which aS/Oq/' + OS/Sq/= 0 at q' = q'. It means that after the above-mentioned averages only the classical trajectories located in small vicinities of short-period trajectories make an essential contribution to the semiclassical Green function. Here it is assumed implicitly that the considered periodic trajectories are isolated (and hyperbolic). For the ergodic systems other variants are impossible (excluding families of neutral trajectories). The problem of a correct calculation of the contributions from stable (elliptical) periodic trajectories existing in general systems is not well studied yet. For more information about this subject see, for instance, refs. [2, 14, 15].
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174
3. The representation of the Green function as the sum over periodic trajectories
Let us consider a certain periodic trajectory. In the vicinity of it we introduce the coordinate system choosing the x axis along the trajectory and the y axis perpendicular to it at the point x (see fig. 1). (There is a particular coordinate system for each periodic trajectory.) Expand the action up to the second order in y. The calculations are simple and give the following result [11]: 1
(G°~(q, E ) ) = i(2~ri)1/2 ~
(
(D(x))I/2 exp [~[
~
S "4-
llp-~ y2 _ ._'n
))
,
(15)
where
z (x)
(
I by by
] ]y"-y'-O
[b~s _ b2s 32s) W(x)~ 1~--~"]-2~~- ~-~-~ y"-y'-O'
Iql is the velocity modulus, S is the action along the periodic trajectory and ~ is the maximal number of conjugated points on the periodic trajectory. Let mij(x) (i, j = 1, 2) be the elements of the monodromy matrix in the variables y and )~ at the point x per period, y ( r ) ) = / mix(x) )~(T) /m21(x )
rnx2(/)] ( y ( 0 )
m22(x)]~(O) ).
(16)
Using the relationships p' -- bS/by", p' = OS/by' one can show that the quantities D(x) and W(x) are connected with mij(x ) in the following way: 1
D(x) = m12(x),
W(x) =
mxx(x ) + mz2(x ) - 2
mlz(x)
(17)
Fig. 1. A dosed classical trajectory (solid line) in the vicinity of the periodic trajectory (dashed line). The x axis is chosen along the periodic trajectory, the y axis is perpendicular to it at the point x. p! and Pi are the finite and initial momenta of the trajectory.
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175
Eqs. (17), (15) and (10) permit one to determine the contribution of the periodic trajectory to the Green function and the wave function using the knowledge of the classical parameters of this trajectory. In contrast to eq. (9) the summation in eq. (15) is done over the finite number AN of the periodic trajectories with the period T <<.2~rh/AE, where AE is the interval of the energy averaging. (For ergodic systems A N - exp (c/hE).) Note that multiple passing of each trajectory should be included in sum (15). The parameters corresponding to the n-times repetitions of one primitive trajectory can be obtained as follows:
S,=nS,
T.=nT,
hl -- ~2
~,=nr,,
D , = D x ~1 h~2,
Wn = Dn(X'{ + X~2- 2),
(18)
where hi and ~x 2 = 1/~,1 are the eigenvalues of the monodromy matrix defined in (16). For hyperbolic trajectories in a two-dimensional space, )t i can have one of the two forms X1 = exp ( u ) ,
X2 -- exp ( - u),
X1 -- - e x p (u),
(19a)
h2-- - e x p ( - u ) ,
(19b)
where u > 0. The exact summation over n is possible in the so-called resonance approximation where it is assumed that I~,~1 >> 1~,21. In this case one can consider that D, = D/h~ -1, Wn = W and eq. (15) is reduced to the geometric series. If ~ > 0, then
G~°~(q,E)=
1
2D 1/2
i(2~ri)1/2 E - - - ~
[ i W 2~ - -
(20)
exp [ ~-~y ) F ( E ) ,
where
F(E) =
e(U/2)+i~, _ 1 2 ( c h 2 - cos g})
;u 2
= 2s
(e ("/2)+i*- 1)
+oo E ~=-~
1
2
( ~ _ 2,~rn)2 + ( 2 )
and q} = ( 1 / h ) S - ~ ( ~ r / 2 ) . The summation in eq. (20) is performed over different primitive periodic trajectories. We emphasize that it is not a periodic trajectory itself that gives the contribution to eq. (15) but a closed (non-periodic) trajectory located in its vicinity. To find the coordinates of this trajectory it is necessary to determine )~(0) from the first row of eq. (16),
)~(0) = y
1 - mll(X ) mlz(x)
(21)
It is with this momentum that one must choose the classical trajectory in the point (x, y) which will return to the starting point per time equal to the period. The points of the periodic trajectory where m12(x ) = 0 are the so-called self-focal points and all trajectories going out from them return back through the period. At different points of the periodic trajectory the number of conjugated points can be different. In eq. (15) ~ is the maximal number of conjugated points on the trajectory but in contrast to eq. (9) the quantity D(x) enters without the modulus, i.e. a change of the number of conjugated points can be compensated by
E.B. Bogomolny/ Wavefunctionsof quantumsystems
176
the change of the phase of D(x). It is possible to write [D(x)l in eq. (15), but then in the exponent one must use the quantity v(x), which changes by + 1 when a self-focal point is passed. For the system with time inversion invariance the passing of the same trajectory in the reverse direction should also enter eq. (15) and it really doubles the result. If we integrate both sides of eq. (15) over d2q = dx d y we obtain the expression for the density of the energy eigenvalues as the sum over the periodic trajectories,
f d0(e) dO=(E) d(E) =- Y'.8(E - E,,) = - ~1 d E q l m G ( q , q , E ) = _ _ h 2 + _ _h
(22)
?Z
where
do(E)=f~8(E-H(p,q)), (21r)
1
r
(.g
~)
d°~(E) = ~- )". Re v~.(mxl + m22 _ 2)1/2 exp l ~ - i ~
.
For periodic trajectories which pass twice in the q space, the result is two times smaller. This formula was derived in refs. [2, 11, 13, 15] but eqs. (15) and (10) for the wave functions have not been used before. Qualitatively the contribution of each periodic trajectory is proportional to ~hlD ( x ) l / I q l and can be seen in the narrow band of width ~h/[W(x)l around it. Depending on the parameters values the maximum of the wave function can be either at the very periodic trajectory (see fig. 2a) or at some distance from it proportional to ]h/IW(x)[ (see fig. 2b). Using eq. (15) one should take into account that it holds after averaging over small intervals of energy and coordinates. The averaging over y results in the substitution exp (iWy2/2h) for
/
[iWy2'\
I'Yo+'AY
-~j.
Y
,
exp
(iWyZ~
2h ] d y - - e x p
(iWy2) sinz ~
z
(23)
where
z = WyoAy/h. This equality is of an approximate character but it shows the behavior at Ay --, 0 and Y0 --' ~ correctly. While changing the energy, relative contributions of different trajectories will change rapidly mainly due to the large value of S/h. A considerable enhancement of (IV[ E) will be observed near points where D(x) --, oo. These points which are the analogues of the caustic are the above-mentioned self-focal points where m t 2 ( x ) = 0. In their vicinities a simple semiclassical approximation (9) is not applied. If necessary it is not difficult to obtain formulae for these regions as well. The additional increase of the wave function will take place also at the self-crossing points of one periodic trajectory, at the crossing points of different periodic trajectories and in the vicinities of the cusps of one trajectory. If the problem considered has a discrete symmetry group q'= ~iq then instead of the averaging in eq. (4) over all energy levels it is natural to average over the wave functions transforming by an irreducible representation of this group. One can get it by constructing the corresponding linear combinations of the
E.B. Bogomolny/ Wavefunctions of quantum systems
177
a
b
Fig. 2. An approximate form of the averaged wave function square in the vicinity of the unstable periodic trajectory. The cut across the trajectory is on the left. The relief picture of the wave function along the trajectory is on the right. The periodic trajectory is denoted by the dashed line. A black circle is a self-focal point.
Green function where one of its arguments is subject to the group transformation
GSY~( q, q, E) = ~ , ~ , G ( Y~iq, q, E ).
(24)
i
Constants ~ i depend on the chosen representation of the group. For instance, let the problem be invariant to the coordinate axes inversion. Then the eigenfunctions can be classified by parity and the following combinations contain wave functions of identical parity only: G~++)= ¼ ( a + b + c + d),
G~+_) -- ¼(a + b - c - d ) ,
G(_+) = ¼(a + c - b - d),
Gt__) -- ¼(a + d - b - c),
(25)
where
a = G ( x , y; x, y),
b= G ( - x , y; x, y),
c= G(x, - y ; x, y),
d= G ( - x , - y ; x, y).
The Green function subscript denotes the transformation law of the inversion of x and y respectively.
E.B. Bogomolny/ Wave functions of quantum systems
178
While calculating the smoothed wave function of the factor ¼ is cancelled since trajectories of zero length which give the main contribution to eq. (10) exist for G(x, y; x, y) only. By averaging eq. (24) over small intervals of energy and coordinates one can show that the dominant contribution to G(~q, q; E) will be given by the classical trajectories located near a part of the periodic trajectory passing through the phase-space points (q, 0) and (~iq, ~[lgt) provided such a periodic trajectory exists. For the point with the coordinates (x, y) near the periodic trajectory (see fig. 1) symmetry transformation is reduced to some transformation of the x coordinate and to a possible change of the y sign. For the transformation without a change of the y sign the result coincides with eqs. (15) and (17) with the substitution of the monodromy matrix per half a period instead of the monodromy matrix per period (16). If the considered symmetry transformation changes the y sign then W(x) in eq. (15) should be calculated by the formula
( a2S
a2S
~2S /
W ( x ) = l ~ y a - E a y ' a y " +a ,,---~/ = "Y / y"--y'=O
mll(X) + ~t2z(x) + 2 ml2(X)
(26)
where ~ j ( x ) are elements of the monodromy matrix per half a period. The example of the separation of symmetrical wave function will be given in the next section. Physically the effect of the amplification of the wave function near the unstable periodic trajectory can be interpreted as the result of the scattering on this trajectory. Under scattering the quantum scattering amplitude becomes large near the target mainly due to the slowing of the motion. The eigenfunction can be represented as the superposition of running waves, each of them scatters on the periodic trajectories. Scattering amplitude depends strongly on energy (even by the resonant way as in eq. (20)) and on some trajectories it is larger than on others. It is these trajectories that manifest themselves as the region of the largest enhancement of the wave function.
4. Stadium billiard
While studying quantum stochasticity the problem of quantum stadium billiard is often used as an example. The problem is to define eigenfunctions and eigenvalues of the Laplace operator for the functions vanishing on the boundary of the stadium type region (see fig. 3). It represents a square with the side 2R (or a rectangle with any ratio of the sides) and two semicircles of radius R join it. It is known [16] that a classical problem of the free motion within such a region with elastic reflections from the boundary is
gR Fig. 3. Stadium billiard.
E.B. Bogomolny/ Wavefunctions of quantum systems
179
ergodic. Numerical calculations of high excited wave functions for this system were performed in refs. [6-10]. Let us apply a general formalism of section 3 to this case. For similar problems the dependence on the energy (or I~1 = 2v r ~ ) and on the scale is defined by the dimensionality. In particular, the quantities entering eq. (15) have the following form: S -- I0[RSo,
R T = ]~- So,
D = [ ~ D o,
W-- [~-~-[Wo,
(27)
where the quantities with the subscript are calculated at R = 1 and I0I = 1. While making calculations by eq. (15) it is not difficult to lind numerically the necessary short-period trajectories and their monodromy matrices. In the appendix a simple method for the calculation of the monodromy matrix for billiard system is presented. For simple trajectories calculations can be made analytically. Certain periodic trajectories, their periods (or actions) and the monodromy matrix traces (at R = 1 and 101 = 1) are given in table I. Trajectories 1 and 13 are singular trajectories and will be discussed below. The problem considered has a discrete symmetry group with respect to the coordinate axes inversion x ~-x, y~-y. As has been mentioned, if one averages over wave functions with a particular symmetry, then (besides full periodic trajectories) the halves of the symmetrical periodic trajectories will make contributions to eq. (15). Monodromy matrices per half a period for the simplest symmetrical trajectories are given in table II. The trajectories corresponding to these monodromy matrices are presented by dashed lines in this table. Knowing the monodromy matrix of the given isolated periodic trajectory one can calculate its contribution to the Green function and to tile wave function using eqs. (15) and (10). As an example consider the periodic trajectory No. 2 from table I. Its monodromy matrix per half a period is given in table II. The classical trajectories, which give the dominant contribution to the Green function G(otx, o2y; x, y) at different values o 1 - + 1 and 02 = + 1 are pictured schematically in fig. 4. During calculations it is necessary to take into account the fact that for the Green function with the changed sign of y, W(x) in eq. (15) should be calculated by eq. (24). As a result we get ( ~ G(olx, o2Y; x, y) = ]~ A,exp i (n)
W-(°2) n
l~'~y •
n+
2
\ ),
(28)
where A,=
2oi. . 2x2,)( 2 ~ l q I ( i ) ~ e x p , i 2 °(2R
Rl! W~ ±~= - 2(2R~q_' x 2) ~ •t -- - ( 3 + ¢cff),
o(x)--0
asx>0,
2 ( 2 R 2 _ x 2)
Ihxl ~
,[h2[" 2,)1J2
~2(X1 + ~2 :~ 2),
X2 = _ (3 - ~fff),
o(x)=-I
asx<0.
At o r = +1 the summation in (28) is done over even n(-~2,4 .... ), at o1= - 1 it is made over odd n ( = 1,3 . . . . ). The coefficient 2 at the be~nning in A, takes into account the existence of two classical trajectories pictured in fig. 4. At x -- 0 the expression for G ( - x , y; x, y) has to be additionally multiplied by 2 since in this case the initial and final points do coincide and each of the trajectories of fig. 4c can be passed in the direct and in the inverse direction. Similar singularities also occur at y -- 0.
180
E.B. Bogomolny/ Wave functions of quantum systems
TableI Certainshort-periodtrajectoriesofthestadiumbilliard No. Trajectory T TrM- 2
No.
1
(I)4.00
--
10
(
3
~
8.60
44.05
12
Q
10.39
8.94
60.00
13
Q
11.32
Trajectory
T )10.00
TrM- 2 672.20
96.00
J
4
@
~
5
Q
8.98
-72.35
14
~
12.94
115.78
6
~
9.24
128.00
15
~
13.20
179.98
7
~
9.66
54.63
16
~
16.65
210.60
8
_
1677
0203
9
Q
16.91
171.98
-D
966
17331
17
9.78
284.64
18
~
181
E.B. Bogomolny/ Wavefunctions of quantum systems Table II The monodromy matrices per half a period for simplest symmetrical periodic trajectories of the stadium billiard No.
Trajectory
Monodromy matrix
Notes
1,2 2
R
5+-~
1--~(L-x)
~(3t2-sx2)
8
2
5
7
1= RI/~
----
24x -7+-- l
-~ (15l 2 + 24x 2 - 24/x) )
/
]
24 1
-4--~-x
(D Q i) 2) 3) 4)
_~R5 ( 3 R 2 _ x 2)
f5-R
-4+-~x
4x
4(x2-312 )
- 7 - - 7-
3
/ 2
I
1= R ~
54- 8~ ~(12--X2+]X) 2 8
7
13- -~
1= RV[2
For trajectories No. 2, 7, 14, 16 from Table I, L = 2R, L = (1 + v/2)R, (1 + VrS-)R, (1 + I ~ ) R , respectively. x is the distance from the centre to the chosen point. x is the distance from the upper right comer of the trajectory. x is the distance from the lower comer of the trajectory.
E.B. Bogomolny/ Wavefunctionsof quantumsystems
182
a
b
c
d
Fig. 4. Classical trajectoriesin the vicinityof the periodic trajectorywhich give the dominant contributions to the differentGreen functions: (a) G(x,y; x,y), (b) G(-x,-y;x,y), (c) G(-x,y; x,y), (d) G(x,-y; x,y). The second classical trajectories are denoted by the dashed lines. For the case (a) the second trajectorycorresponds to the reverse passing of the first one. A schematic relief of the wave function corresponding to eq. (28) is shown in fig. 5. This picture agrees qualitatively well with fig. 2 of ref. [9]. The contributions of other periodic trajectories can be calculated analogously. Singular periodic trajectories passing through the junction points of plane and semicircle sections of the boundary (see fig. 3) are an exception. Two such trajectories are shown as No. 1 and No. 13 in table I. Due to the fact that at their collision points the boundary curvature is discontinuous the monodromy matrix formalism cannot be applied to describe motion in the vicinity of these trajectories. The result of the motion (even in the linear approximation) depends essentially on the boundary point where a collision occurs. Roughly speaking, there exist many different monodromy matrices corresponding to various possibilities of the motion. The usual eq. (9) is not applied in this case. To derive a semiclassical expression it is convenient to use the exact representation of the Green function for the billiard problems with zero boundary conditions in the form of the multiple scattering obtained by Balian and Bloch [18]. According to their paper, oo
G(q",q', E) = Go(q",q' ) + ~_, ( - l i ) " f / o l . . . d o p OG°(q'''eq) OG°(al'a2) p=l
X ...
O G ° ( a ' - l ' or')Go(ap, q'), 0np
~nl
~n2 (29)
Fig. 5. A schematicrelief of the smoothedwave functionin the vicinityof the unstable periodic trajectoryNo. 2 from table I.
E.B. Bogomolny/ Wavefunctions of quantumsystems
183
where Go(q", q') is the free Green function whose asymptotics is as follows:
Go(q",q') = ih(2~rih)l/2(l(llL)-l/2exp
i
, L= Iq"-q'l.
The integral in eq. (29) is taken along the boundary of the given region, alan k is a derivative over the internal normal to the boundary at the point a k. If the boundary is smooth, the integration over do k can be performed by the saddle point method and the classical trajectories for which the incidence angle is equal to the reflection angle give the main contribution. As a result, we get eq. (9). However, if the integration is performed near the singular boundary point, the result is more complex. As an example consider a calculation of the contribution from the simplest singular periodic trajectory No. 1 from table I. Let us consider the trajectories with two reflections from the boundary. In this case there exist four different possibilities corresponding to the collisions on different sides of the singular points (see fig. 6). Their contributions can be determined from the third term of expansion (29). The total length of the trajectory enters the exponent of the obtained expression. Let u, v be the distances along the boundary counted from the collision points as in fig. 6. Taking into account terms up to the second powers of u, v, we get
exp ( i ~ )
G( x, y; x, y) = ih(2,trih )X/2l~la/2(2R( R 2 _ x2))l/2
foOOdufoOOdv ~ exp ( i _ ~ 2 ( u , v)),
(30)
j*l
where qby(U,v) ( j = 1, 2, 3, 4) are contributions from the trajectories of figs. 6a-d respectively.
( y z u ) 2, ,1,1 = 2 ( R - x ) +
( y .--_.0)2,
(u-v) 2
( y = u ) 2, (y+v) 2 ~2 = 2 ( R - x) + 2(R + x )
(y-v) 2
4------T--+ 2 ( R + x ) '
(y + u) 2
~3 = 2 ( R + x) + ~(l~---'"x-)
u2
R +
(u + v) 2
41~
(y + u) 2
,
(y + v) 2
¢h,~= ~(1~2"-Z) + ~(I~+"Z)
02
( . + v) 2
R +
4R
'
u 2+ v2 + (u-
R
o) 2
4R
(31) If lYi >> C~/h- the integrals can be taken by a saddle point method and the main contribution will be given by the real classical trajectories but they do not exist at all positions on the point. For instance, the classical trajectories (b), (c), (d) in fig. 6 are only possible at x > 0, x < 0 and x = 0 respectively. If the classical trajectory is allowed its contribution can be determined by the usual eq. (15). But at finite lYl CV~ a saddle point method is not applied, real classical trajectories are unimportant and the result of the integration in eq. (30) is not expressed in terms of elementary functions (as in the case of the Frenel diffraction). The most prominent difference between contributions from regular and singular periodic trajectories is that the former is symmetric relative to the periodic trajectory (cf. fig. 2) but the latter is asymmetric. For the above-considered singular trajectory the averaged wave function square is larger at y > 0 than at y < 0. Moreover, at small y an additional decrease of the wave function is observed as compared to eq. (15). These properties agree well with fig. 1 of ref. [9].-Similarly one can calculate a contribution from trajectories in a vicinity of the singular trajectory No. 13 in table I. The dominant contribution to G ( - x , y; x, y) will be given by the trajectories (not obviously the classical ones) of the types 5 and 6 in table II. During calculations it is necessary to take into account that if x = 0 the result is amplified due to both self-crossing and additional symmetrical contributions as in eq. (30).
184
E.B. Bogomolny/ Wavefunctions of quantum systems
R
V tt
tt
b
V IJ.
I/
V
Fig. 6. Possibleforms of the trajectories(not obvious are the classicalones) in the vicinityof the singular periodic trajectory(dashed line), u and v are the arc lengths counted from the collisionpoints. The x axis is chosen along the periodic trajectory, the y axis is perpendicular to it. For the stadium billiard there is another class of exceptional trajectories close to the neutral periodic trajectories (which are perpendicular to plane sections of the boundary) for which the above-discussed formalism is not applied since many trajectories are located close to each other. These trajectories are responsible also for a slow decrease of time correlations in the classical problem [19, 20]. Summing over all such trajectories one can calculate their contribution to the wave function. On the whole, all calculations agree well with the results of numerical calculations of high excited wave functions for the stadium billiard performed in refs. [9, 10]. One can explain quite simply all the distinguished details on all pictures of these papers. Unfortunately, in these papers the energy values corresponding to the given pictures of the wave function relief are not mentioned and it is not known on which heights the slices of the wave functions are made. Since the results depend strongly on these quantities it is impossible to make a quantitative comparison with these works.
5. Elliptic bmiard For ergodic systems the hyperbolic (and neutral) periodic trajectories are the only invariant manifolds which in the semiclassical limit can serve as the carders of the wave function structures [21]. However, unstable periodic trajectories can be present in other systems as well. For instance, they exist in some integrable problems. It is rather instructive to observe how the general formulae can be applied for the
E.B. Bogomolny/ Wave functions of quantum systems
185
integrable models where, in principle, everything can be calculated exactly. In this section we consider as an example the problem of the calculation of the eigenfunctions and eigenvalues of the Laplace operator with the zero conditions on the ellipse boundary: x2 y2 A--¢ + ~ = 1,
(32)
where A = ch u o, B = sh u 0. It is well known that this so-called elliptic billiard is integrable and in the elliptic coordinates,
x=chucosv,
y = s h u s i n p,
(33)
the variables are separated in both the classical and the quantum equations and the eigenfunctions of the Laplace operator can be written as the product of two functions depending only on u and v:
vt,(x, y) =f(u)g(v). The functions h 2d2f du 2
(34)
f(u) and g(v) satisfy the
+p2(u)f--O,
2d2g
h ~
+
Mathieu equations
k2(v)g=O,
(35)
8
b Fig. 7. Elliptical billiard. (a) The motion of the "whispering gallery" type. (b) The motion of the "bouncing ball" type. The caustics of quantum wave functions are depictured by the solid line. The dashed line corresponds to the unstable periodic trajectory.
186
E.B. Bogomolny/ Wavefunctions of quantum systems
where p2(u) = 2E shE u - a, k2(l ,) = a - 2E sin2 I, and a is a parameter of the separation of variables (an additional integral of motion). At different a the motion is of a different character. The case 0 < a < 2 E s h 2 u 0 corresponds to the motion of the "whispering gallery" type (see fig. 7a) and the case -2E < a < 0 corresponds to the motion of the "bouncing ball" type (see fig. 7b). We shall be interested in the unstable periodic trajectory which is the large ellipse axis (see fig. 7). Locally it is not distinguished from the stadium periodic trajectory No. 2 in table II. In the problem considered there exist invariant tori and the rules of semiclassical quantization are of the usual form, n x=
n2=
u)du+~,
k(~)dv+-~--,
(36)
where n x and n 2 are integers. At a > 0, a x = 1, ot2 ----0; at a < 0, a t = 0, ot2 = 2. At first glance it seems that these formulae do not contain any information about the considered periodic trajectory. Especially, the existence of the contribution of this trajectory to energy level density which follows from eq. (22) seems strange. Let us consider this subject in detail. Denote by nl(E , a) and n2(E, a) the exact expressions which should be equal to integers from the quantization conditions (as in eq. (36)). Using the Poisson summation formula (see, for instance, refs. [2, 22]) we get
d(E)= Y'~ I~(E-E(?/1,?I2))= ~ fdnldn2exp(2~riMln 1 + 2~riM2n2)8(E-E(nl, n2) ) nl, n2
Mz, M2
= Y'. fD(E,a)daexp(2~riMln 1 + 2¢riM2n2),
(37)
M1,M~ where D(E, a) is the Jacobian of the transformation from the variables (nl, n2) to the variables (E, a),
D(E, a ) = 0nl
On2 0nl
0n2
0E Oa
~E ~ a "
The functions n~(E, a) contain the terms proportional to 1/h. Therefore, the usual way to calculate the integral in eq. (37) as h---} 0 is to use a saddle point method [2, 22] according to which a dominant contribution to eq. (37) will be given by the resonance tori for which the frequencies are commensurable, Ml~O1 + M2a~2 = 0,
(38)
where
In this case
d(E)=
do(e) ~
+ ~
1
]~
ImAvex p i--~- ,
(39)
M1,M2 where Sp is a classical action calculated over resonance toms, Ap is a certain easily calculable amplitude and the summation is done over all Mr, M 2. Here it is implicitly assumed that at all M1, M 2 a saddle point method can be applied. But one can check that for elliptic billiard at M x = 2 M 2 a saddle point method is
E.B. Bogomolny/ Wavefunctions of quantum systems
187
not applicable and a region of small a gives the dominant contribution to the integral. However, at small a quantization conditions (36) are incorrect since this case corresponds to the motion near a double zero of the potentials in eq. (35) and the conventional eqs. (36) hold only if the momenta p2(u) and k20,) have a first order zero on the boundary. It is straightforward (but formidable) to fred the corresponding formulae for the quantities hi(E, a) matching the semiclassical wave function with the parabolic cylinder functions on the boundary,
~n2(E,a)=2
+~ln
+ ½argF(½- ia) + ~ + X q: ½arctg(-a,~
,
(40) 2~m~(E,a)--2
chuo- ahith
-~n2(E,a)+,~+~,
where a = a/(2h2¢2-E). The signs + correspond to odd (even) functions with respect to the transformation u - - - , - u , v - - * - v . Substituting these values into eq. (37) one can show that at MI=2M 2 the result coincides with the contribution to the level density from the considered periodic trajectory and its multiple passings as in eq. (22). Similarly, one can be convinced that besides contributions from the resonance tori the averaged wave function square contains a contribution from the unstable periodic trajectory (a large ellipse axis) equal to eq. (15). Special wave functions for which the parameter a in eq. (35) is small and a usual semiclassical approximation is not applicable [23] give this contribution. It is worthwhile to emphasize that eq. (15) is formed only after the summation over all degenerated states with a > 0 and a < 0. The individual wave functions with small a have another form. Roughly speaking they have maxima near the caustics of the type pictured in fig. 7, but the corresponding ellipses and hyperboles at small a are compressed to a part of the large ellipse axis, immitating the increase of the wave function near the periodic trajectory. From this example one can clearly see that the discussed effect is related to the motion near the periodic trajectory, i.e. to the scattering on it (see the end of section 3). In the integrable systems such as the elliptic billiard, a trajectory passing in the phase space near the unstable periodic trajectory cannot go far from it. Thus only a small number of wave functions have a marked enhancement near the unstable periodic trajectory, but the magnitude of this effect in the individual wave function is large. (Of course, the value averaged over all wave functions coincides with eq. (15).) For ergodic systems almost any trajectory passes near all periodic trajectories and it is natural to assume that almost any eigenfunction contains contributions from all periodic trajectories as in eq. (15) but their contributions are small ( - qr~-). The discussions above show that in integrable problems there may exist non-saddle point contribution corresponding to the account of periodic trajectories. In the limit h ---, 0 this contribution is smaller than the ones from the resonance tori but larger than semiclassical corrections to them. Such a contribution may occur both from unstable periodic trajectories (as in the elliptic billiard) and from families of neutral trajectories. The contribution to the level density of the spherical billiard from the sphere diameter which can be extracted from ref. [18] can serve as an example of the latter one. The existence of such contributions for integrable systems is rarely discussed in the literature.
6. Conclusions
The main result of this paper consists in the fact that hyperbolic periodic trajectories make a calculable contribution to wave functions both for non-integrable and for integrable systems. In all cases the
E.B. Bogomolny/ Wavefunctions of quantum systems
188
averaged wave function square can be calculated by eqs. (15) and (10). For integrable systems such a contribution exists only in a small number of wave functions and for averaged quantities it is small comparable to the resonance tori contribution. The exact behavior of individual wave functions for ergodic systems is not known so far. One can assume that each wave function can be represented as the sum over all periodic trajectories. The coefficients of this expansion are rapid and irregular functions of energy. At a particular energy eigenvalue the contributions of certain trajectories become larger than those of others and the wave function shows a noticeable enhancement in the vicinity of these trajectories. Physically this effect can be interpreted as the result of the scattering on the periodic trajectories. Taking some freedom, one can say that quantum stochasticity develops hidden regular structures of the classical chaotic motion. In classical mechanics these structures have a zero measure and they can be only observed while choosing the initial conditions directly on the periodic trajectories (and their incoming separatrices). The described method is useful for the calculation of wave functions smoothed over the finite energy interval. In this case the dominant contribution to all quantities of interest will be given by short-period trajectories for which the monodromy matrix elements are the least. For the exact calculation of the individual wave function of the ergodic systems this method cannot be practically applied since the number of necessary periodic trajectories is exponentially (of the smoothing papemeter) large. For physical problems where small-scale irregular fluctuations are not essential, the semiclassical formalism of the periodic trajectories allows one to calculate rather simply both a smoothed level density [2, 11, 13] and a smoothed wave function in the limit h ~ 0. An accurate comparison of the obtained semiclassical formulae with the numerically calculated wave functions for ergodic quantum problems is of considerable interest.
Acknowledgements The author is grateful to D.L. Shepelyansky for pointing out the paper [9] and to L.N. Shur and D.E. Khmelnitsky for numerous useful discussions.
Appendix Calculation of the monodromy matrix for billiard problems Here we describe a convenient method the calculation of the monodromy matrix for the billiard problems based on refs. [17, 16]. Let a periodic trajectory consists of n straight lines. Fix a certain point on this trajectory. Denote by t~ (i = 1,2 . . . . . n) the times of sequentional collisions with the boundary and by tn+ 1 the time of motion from the last collision to the chosen point. Let us construct two sequences K2(a), K j a ) , . . . , "Xn+/t'(a)l(tit = 1, 2) satisfying the recurrent relationship k'(~) = + d,. --,.+t 1 + t , . K ( = ~) '
(A.1)
where d,. = - 2 / ( R , . c o s y m ) and R= is the curvature radius at the collision point, 7m is the incidence angle. (For a plane section of the boundary dm= 0.)
E.B. Bogomolny/ Wavefunctions of quantum systems
189
The first terms of these sequences are as follows: 1 K(1) ----dl' K2(2)= ~'1 + d1"
(A.2)
For each sequence we construct a product
G,(a)= f i (1 + m--2
t,.K(a)).
(A.3)
Then the monodromy matrix defined by eq. (16) is equal to
mll
c 1 (1 + t n + l rcln + l ] ,
m21 = G n(1)Kn(1) +l,
m12 ~
txc (2 (1 + tn+ 1 r
n+l],
(2) m 2 2 -- t 1G n(2)Kn+l"
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
I.C. Percival, Adv. Chem. Phys. 36 (1977) 1. M.V. Berry, Les Houches Summer School Prec. 36 (1981) 177. A.L. Shnirelman, Uspekhi Matem Nauk 29 (1979) 181. A. Voros, Lecture Notes in Physics 93 (1979) 326. M. Shapiro and G. Geclman, Phys. Rev. Lett. 53 (1984) 1714. S.W. McDonald and A.N. Kaufman, Phys. Rev. Lett. 42 (1979) 1189. S.W. McDonald, Lawrence Berkley Lab. Report LBL-14837 (1983). R.D. Taylor and P. Brumer, Faraday Discuss. Chem. Soc. 75 (1983) 170. E.I. Heller, Phys. Rev. Lett. 53 (1984) 1515. F.J. Heller, Lecture Notes in Physics 263 (1986) 162. M.C. Gutzwiller, I. Math. Phys. 12 (1971) 343. M.V. Berry and M.E. Mount, Rep. Progr. Phys. 35 (1972) 315. M.C. Gutzwiller, Physica 5D (1982) 183. V.I. Amol'd, Funct. Anal. Appl. 6-2 (1972) 12. M. Tabor, Physica 6D (1983) 195. L.A. Bunimovich, Funct. Anal. Appl. 8 (1974) 73. Ya.G. Sinai, Uspekhi Matem. Nauk 25 (1970) 141. R. Balian and C. Blech, Ann. Phys. 60 (1970) 401; 69 (1972) 76. F. Vivaldi, G. Casati and I. Guameri, Phys. Rev. Lett. 51 (1983) 727. L.A. Bunimovich, ZhETF 89 (1985) 1452. I.W. Helton and M. Tabor, Physica 14D (1985) 409. M.V. Berry and M. Tabor, Prec. R. Soc. London A 356 (1977) 375. V.G. Osmolovsky, J. Comp. Math. and Math. Phys. 14 (1974) 365. E.B. Bogomolny, Pis'ma v ZhETF 44 (1986) 436.
(A.4)
S M O O T H E D WAVE FUNCTIONS OF CHAOTIC QUANTUM SYSTEMS E.B. BOGOMOLNY The Academy of Sciences of the USSR, LD. Landau Institute for Theoretical Physics, 142432 Chernogolooka, Moscow Region, USSR Received 31 December 1987 Communicated by V.E. Zaldaarov It is shown that wave functions of quantum systems as h ~ 0 have an extra density near unstable periodic trajectories of the classical problem. The averaged wave function square is represented as the sum over a finite number of periodic
trajectories. The contribution of each trajectoryis expressedthrough the dements of the monodromymatrix of the trajectory. The results are comparedwith the numerical calculations of the wavefunctionsfor the stadium billiard.
1. Introduction
This paper is devoted to the study of wave functions of non-integrable quantum systems in the semiclassical limit h-~ 0. Here non-integrability means that a part of trajectories of the corresponding classical problem does not lie on the invariant toil. Such trajectories are said to be stochastic (or chaotic) and the problem of expressing this stochasticity in quantum language is the subject of the so-called "quantum chaos" or "quantum stochasticity". Conventional semiclassical quantization rules of the Bohr-Zommerfeld type which are strongly based on the assumption that all classical trajectories wind on the invariant tori cannot be applied for similar problems. As a result, the behavior of the eigenvalues and eigenfunctions as h ~ 0 for these systems has not been well studied so far. (See reviews [1, 2] for the problem history and different approach to it.) Consider a quantum problem whose classical motion is ergodic (i.e. almost all classical trajectories spread uniformily over the whole energy surface). The main hypothesis about the behavior of the wave functions for such problems is the assumption by Berry [4] and Voros [5] based on the Shnirelman theorem [3] that the averaged square of the eigenfunctions in the semiclassical limit h--* 0 coincides with the projection of the classical microcanonical distribution to the coordinate space
(l~b(q)12)~po(q) ash~O, fd'p 8(E - H(p, q)) Po(q) = /d"p'd"q'O(E- H(p', q ' ) ) '
(1)
where H ( p , q) is a classical Hamiltonian of the problem. The exact form of the wave functions with such a property is unknown. Usually it is supposed that they are random or at least they are irregular functions of the coordinates and the correlations between their different parts are practically absent [2, 5]. First numeric calculations would seem to confirm a chaotic character of the wave functions of the ergodic systems. In the pioneering paper by McDonald and Kaufman [6] only irregular pictures of the nodal lines (where ~(x) = 0) for high excited eigenfunctions of the quantum stadium billiard were given. 0167-2789/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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However, in the unpublished McDonald thesis [7] it has been emphasized that certain wave functions manifest regular structures which seem to be connected with the periodic trajectories. The existence of such structures was also mentioned in a short note by Taylor and Brumer [8]. But these results remained practically unnoticed until the papers by Heller [9, 10] where it was emphasized that a large number of high excited wave functions for the stadium billiard have an enhancement in the vicinity of the classical periodic trajectories. This result seems to be unexpected since for ergodic problems all periodic trajectories must be unstable or neutral. Hence, if one constructs a wave packet centered near such a periodic trajectory, it must spread out quickly (per time defined by classical dynamics). It will be shown below* that nevertheless each unstable periodic trajectory gives a definite contribution to the wave function which is maximal either at the very periodic trajectory or near it. If for an ergodic problem one averages the wave function square over small intervals of energy and coordinates the resulting "smoothed" wave function can be represented in the following form:
(l~p(q)12)=po(q)+ht~-l)/2~,,Im
(
I
Ap(x)exp i--h-+i
2h
(2)
where for each periodic trajectory the x axis is chosen along the trajectory and the Ym axes are chosen perpendicularly to it (see fig. 1), n is the number of degrees of freedom (n >_ 2). Sp = ~Pn dqn is a classical action calculated along the trajectory. Ap(x) and Ve'pkm(x)are classical quantities defined through the elements of the monodromy matrix of a given trajectory. The monodromy matrix or the matrix of Poinca:" mapping is the matrix which connects a solution of linearized classical equations of motion in the plane perpendicular to the periodic trajectory per period:
yk(T) ] = Mf Yk(O)),
lye(o)
(3)
where T is the period of the given periodic trajectory. The summation in eq. (2) is performed over the finite number AN of the periodic trajectories which depends on the value AE entering the definition of the energy smoothing. As AE ~ 0, AN---, oo. While changing the energy the contributions of certain periodic trajectories become larger than those of others. One may say that at different energy eigenvalues it is possible to see different periodic trajectories. Eq. (2) allows to express a smoothed wave function as h ---,0 in terms of the quantities defined from classical mechanics and it may serve as the basis of a semiclassical approximation for the ergodic quantum systems. Similar considerations were used in refs. [11-13] to derive the formula connecting the density of energy levels with the sum over unstable periodic trajectories of ergodic problems. However, to the best of author's knowledge they were not used before to get information about the eigenfunctions. The plan of the paper is as follows. In section 2 we discuss a general formalism based on the semiclassical representation of the Green function of the SchriSdinger equation as the sum over all classical trajectories connecting two fixed points. It is shown here that after the averaging over the small intervals of energy and coordinates the main contribution to the Green function as h ~ 0 will be given by the classical trajectories located in the vicinities of short-period trajectories. *A short note about this subject was published in [24].
E.B. Bogomolny/ Wavefunctions of quantum systems
171
In section 3 the expressions for Ap(x) and wpkm(x) in terms of the elements of the monodromy matrix are obtained, the general properties of these formulae are discussed and a physical interpretation of the wave function amplification near the unstable periodic trajectory is given. In section 4 a case of the quantum stadium billiard is considered in detail. The results are compared with the pictures of high excited wave functions published in refs. [9, 10] and it is shown that even small details of the wave function structure are well reproduced analytically. Though the discussed effect is of importance for non-integrable problems only, it also exists in some integrable systems where there exist unstable periodic trajectories. In section 5 it is shown how this effect manifests itself for the integrable elliptic billiard and the similarity and the difference in its manifestations for the integrable and non-integrable systems are discussed. In the Appendix a convenient method for the calculation of the billiard monodromy matrix is presented.
2. General formalism
Let E. be the energy eigenvalues and ft.(q) be the corresponding eigenfunctions. Consider the energy interval of length AE with the center in a point E 0 (AE << E0) and define the averaged square of the wave function modulus by the following relationship: 1 ([~k(q)l 2) = ~ E Iq~.(q)I z, (n}
(4)
where the summation is done over all wave functions whose energy eigenvalues are in the chosen interval
(5)
E o - ½ A E _ < E . < E o+ ½AE
and N is the number of such functions. Thus, from the beginning we decline considering individual wave functions and shall investigate the quantifies averaged over a small energy interval. It will be shown below that such a smoothing gives senseful results from both mathematical and physical points of view. Definition (4) has different meanings for the integrable and non-integrable problems. For multi-dimensional integrable systems degeneracy of high excited levels is inevitable [1, 2] and in the formal limit A E ---, 0 eq. (4) does not give a square of the individual wave functions but an average over a large number of degenerated states. For general non-integrable systems only an accidental degeneracy and a degeneracy of different symmetry states is possible [1, 2] which is not difficult to take into account (see section 3) and eq. (4) as AE ~ 0 and E o ~ E, transforms into the squared modulus of one wave function. Define the Green function of the Schr~Sdinger equation in the energy representation in the usual way:
(1t(p", q") - E)G( q", q', E) Using the series expansion of
=
8(q"- q').
(6)
G(q", q', E) in terms of the eigenfunctions,
~b*(q")~.(q') E+ie-E.
G(q",q',E)=~.. n
'
(7)
E.B. Bogomolny/ Wavefunctions of quantum systems
172
it is easy to show that the averaged wave function square defined by eq. (4) is expressed in terms of the Green function (6) as follows: ( I m G ( q , q, E ) )
f(ImG(q', q', E)) d'q"
( l f f ( q ) 12) =
(8)
where
~Eo-~AE
Further calculations are based on the conventional expression for the semiclassical limit of the Green function G(q", q', E) as the sum over all classical trajectories connecting the two fixed points q" and q' (see, for instance, refs. [11, 12] and references therein). For simplicity we shall consider below a two-dimensional case. The generalization for multi-dimensional models is straight-forward. In a twodimensional space one has [11, 12]
G(q,q,E)=
Go(q,E)+ h-~G (q,E)+O
ImG0(q, E) =
fs(e- rap, q))
G°~(q, E) = i(2~ri)1/2 1
Y'. IA 11/2 exp (
,
d2p -~S(q,i
(9) q,E)-
i1,~),
,n
where Go(q, E) is a contribution of the so-called "zero length trajectories" corresponding to the Tomas-Fermi approximation. G°~:(q, E) is a contribution of classical trajectories with a non-zero action starting and ending in a point q. S(q, q, E) is a classical action calculated along one of such trajectories. A is a certain determinant constructed from the second derivatives of the action. J, is a phase equal to a number of conjugated points on the trajectory. For the billiard type systems with wave functions equal to zero on the boundary it is necessary to add a double number of reflections from the boundary to I,. Substituting eq. (9) into eq. (8) and expanding in powers of h, we find 2~fh (l~(q)12)=po(q)---V-((ImG
o~¢
( q , E ) ) - l f(ImG°SC(q',E))d2q'),
(10)
where Po(q) is defined in eq. (1). For two-dimensional systems Po(q) = 1/V, where V is the volume of the available regions in the q space (outside this region Po(q) vanishes). We emphasize that the first term in eq. (10) coincides with expression (1) obtained in refs. [2, 4] from other considerations. Expansions (9) and (10) give a formal representation of the averaged wave function square as the sum over all classical trajectories with fixed energy which start and end at the point q and along which the time of motion may be arbitrary. There are an infinite number of such trajectories and for non-integrable systems the summation over them is practically impossible (see, nevertheless, ref. [13]) or at any rate it is as difficult as a direct solution of the Schriidinger equation. This difficulty is not only of a technical character. All we know about the properties of chaotic systems does not contradict the statement that irregularity of
E.B. Bogomolny/ Wavefunctions of quantum systems
173
the energy levels and the wave functions on small scales is so large that a statistical description becomes the only reasonable way to describe them and the exact calculation of the energy eigenvalues and eigenfunctions seems to be meaningless (similar to the calculation of long trajectories for ergodic systems). However, for the quantities smoothed over small (but fixed) intervals of energy and coordinates one can get quite meaningful formulae. Firstly we consider the result of the averaging of expression (9) for G°~:(q, E) over a small energy interval. Taking into account that AE << E o, one can expand the action in the exponent by the deviation from E 0 and retain the first two terms only,
S(q, q, E o + BE) = S(q, q, Eo) + T(q, Eo)SE,
(11)
where T(q, Eo)= (SS/E)e_eo is the time of motion along the given trajectory. As h ~ 0 the main contribution to sum (9) after the averaging over E will be given by the classical trajectories along which the time of motion is restricted and satisfies the following inequality: h T<~CAE,
(12)
where c is a constant (c -- 2~r). In other words, the summation in eq. (9) over trajectories with a finite time of motion T < TO is equivalent to an averaging of the Green function over an energy interval of width h E - ho~o, where
oao = 2 ~ / T o. Similarly, the averaging of the Green function G(q, q, E) over a small coordinate interval Aq at each point q0 yields that the dominant contribution to sum (9) will be given by the classical trajectories for which the change of momentum during the time of motion,
hp
[Pf-Pi[ -- ~
+
(13)
will be small, h
a g -< c ' -a-p
(14)
(c' is a constant of the order of 1). If the matrix of the second derivatives of the action is not degenerated then it is easy to show that in the vicinity of the given trajectory (the nearness is defined by the smallness of Ap) there will be a real periodic trajectory for which aS/Oq/' + OS/Sq/= 0 at q' = q'. It means that after the above-mentioned averages only the classical trajectories located in small vicinities of short-period trajectories make an essential contribution to the semiclassical Green function. Here it is assumed implicitly that the considered periodic trajectories are isolated (and hyperbolic). For the ergodic systems other variants are impossible (excluding families of neutral trajectories). The problem of a correct calculation of the contributions from stable (elliptical) periodic trajectories existing in general systems is not well studied yet. For more information about this subject see, for instance, refs. [2, 14, 15].
E.B.Bogomolny/ Wavefunctionsofquantumsystems
174
3. The representation of the Green function as the sum over periodic trajectories
Let us consider a certain periodic trajectory. In the vicinity of it we introduce the coordinate system choosing the x axis along the trajectory and the y axis perpendicular to it at the point x (see fig. 1). (There is a particular coordinate system for each periodic trajectory.) Expand the action up to the second order in y. The calculations are simple and give the following result [11]: 1
(G°~(q, E ) ) = i(2~ri)1/2 ~
(
(D(x))I/2 exp [~[
~
S "4-
llp-~ y2 _ ._'n
))
,
(15)
where
z (x)
(
I by by
] ]y"-y'-O
[b~s _ b2s 32s) W(x)~ 1~--~"]-2~~- ~-~-~ y"-y'-O'
Iql is the velocity modulus, S is the action along the periodic trajectory and ~ is the maximal number of conjugated points on the periodic trajectory. Let mij(x) (i, j = 1, 2) be the elements of the monodromy matrix in the variables y and )~ at the point x per period, y ( r ) ) = / mix(x) )~(T) /m21(x )
rnx2(/)] ( y ( 0 )
m22(x)]~(O) ).
(16)
Using the relationships p' -- bS/by", p' = OS/by' one can show that the quantities D(x) and W(x) are connected with mij(x ) in the following way: 1
D(x) = m12(x),
W(x) =
mxx(x ) + mz2(x ) - 2
mlz(x)
(17)
Fig. 1. A dosed classical trajectory (solid line) in the vicinity of the periodic trajectory (dashed line). The x axis is chosen along the periodic trajectory, the y axis is perpendicular to it at the point x. p! and Pi are the finite and initial momenta of the trajectory.
E.B. Bogomolny/ Wavefunctions of quantum systems
175
Eqs. (17), (15) and (10) permit one to determine the contribution of the periodic trajectory to the Green function and the wave function using the knowledge of the classical parameters of this trajectory. In contrast to eq. (9) the summation in eq. (15) is done over the finite number AN of the periodic trajectories with the period T <<.2~rh/AE, where AE is the interval of the energy averaging. (For ergodic systems A N - exp (c/hE).) Note that multiple passing of each trajectory should be included in sum (15). The parameters corresponding to the n-times repetitions of one primitive trajectory can be obtained as follows:
S,=nS,
T.=nT,
hl -- ~2
~,=nr,,
D , = D x ~1 h~2,
Wn = Dn(X'{ + X~2- 2),
(18)
where hi and ~x 2 = 1/~,1 are the eigenvalues of the monodromy matrix defined in (16). For hyperbolic trajectories in a two-dimensional space, )t i can have one of the two forms X1 = exp ( u ) ,
X2 -- exp ( - u),
X1 -- - e x p (u),
(19a)
h2-- - e x p ( - u ) ,
(19b)
where u > 0. The exact summation over n is possible in the so-called resonance approximation where it is assumed that I~,~1 >> 1~,21. In this case one can consider that D, = D/h~ -1, Wn = W and eq. (15) is reduced to the geometric series. If ~ > 0, then
G~°~(q,E)=
1
2D 1/2
i(2~ri)1/2 E - - - ~
[ i W 2~ - -
(20)
exp [ ~-~y ) F ( E ) ,
where
F(E) =
e(U/2)+i~, _ 1 2 ( c h 2 - cos g})
;u 2
= 2s
(e ("/2)+i*- 1)
+oo E ~=-~
1
2
( ~ _ 2,~rn)2 + ( 2 )
and q} = ( 1 / h ) S - ~ ( ~ r / 2 ) . The summation in eq. (20) is performed over different primitive periodic trajectories. We emphasize that it is not a periodic trajectory itself that gives the contribution to eq. (15) but a closed (non-periodic) trajectory located in its vicinity. To find the coordinates of this trajectory it is necessary to determine )~(0) from the first row of eq. (16),
)~(0) = y
1 - mll(X ) mlz(x)
(21)
It is with this momentum that one must choose the classical trajectory in the point (x, y) which will return to the starting point per time equal to the period. The points of the periodic trajectory where m12(x ) = 0 are the so-called self-focal points and all trajectories going out from them return back through the period. At different points of the periodic trajectory the number of conjugated points can be different. In eq. (15) ~ is the maximal number of conjugated points on the trajectory but in contrast to eq. (9) the quantity D(x) enters without the modulus, i.e. a change of the number of conjugated points can be compensated by
E.B. Bogomolny/ Wavefunctionsof quantumsystems
176
the change of the phase of D(x). It is possible to write [D(x)l in eq. (15), but then in the exponent one must use the quantity v(x), which changes by + 1 when a self-focal point is passed. For the system with time inversion invariance the passing of the same trajectory in the reverse direction should also enter eq. (15) and it really doubles the result. If we integrate both sides of eq. (15) over d2q = dx d y we obtain the expression for the density of the energy eigenvalues as the sum over the periodic trajectories,
f d0(e) dO=(E) d(E) =- Y'.8(E - E,,) = - ~1 d E q l m G ( q , q , E ) = _ _ h 2 + _ _h
(22)
?Z
where
do(E)=f~8(E-H(p,q)), (21r)
1
r
(.g
~)
d°~(E) = ~- )". Re v~.(mxl + m22 _ 2)1/2 exp l ~ - i ~
.
For periodic trajectories which pass twice in the q space, the result is two times smaller. This formula was derived in refs. [2, 11, 13, 15] but eqs. (15) and (10) for the wave functions have not been used before. Qualitatively the contribution of each periodic trajectory is proportional to ~hlD ( x ) l / I q l and can be seen in the narrow band of width ~h/[W(x)l around it. Depending on the parameters values the maximum of the wave function can be either at the very periodic trajectory (see fig. 2a) or at some distance from it proportional to ]h/IW(x)[ (see fig. 2b). Using eq. (15) one should take into account that it holds after averaging over small intervals of energy and coordinates. The averaging over y results in the substitution exp (iWy2/2h) for
/
[iWy2'\
I'Yo+'AY
-~j.
Y
,
exp
(iWyZ~
2h ] d y - - e x p
(iWy2) sinz ~
z
(23)
where
z = WyoAy/h. This equality is of an approximate character but it shows the behavior at Ay --, 0 and Y0 --' ~ correctly. While changing the energy, relative contributions of different trajectories will change rapidly mainly due to the large value of S/h. A considerable enhancement of (IV[ E) will be observed near points where D(x) --, oo. These points which are the analogues of the caustic are the above-mentioned self-focal points where m t 2 ( x ) = 0. In their vicinities a simple semiclassical approximation (9) is not applied. If necessary it is not difficult to obtain formulae for these regions as well. The additional increase of the wave function will take place also at the self-crossing points of one periodic trajectory, at the crossing points of different periodic trajectories and in the vicinities of the cusps of one trajectory. If the problem considered has a discrete symmetry group q'= ~iq then instead of the averaging in eq. (4) over all energy levels it is natural to average over the wave functions transforming by an irreducible representation of this group. One can get it by constructing the corresponding linear combinations of the
E.B. Bogomolny/ Wavefunctions of quantum systems
177
a
b
Fig. 2. An approximate form of the averaged wave function square in the vicinity of the unstable periodic trajectory. The cut across the trajectory is on the left. The relief picture of the wave function along the trajectory is on the right. The periodic trajectory is denoted by the dashed line. A black circle is a self-focal point.
Green function where one of its arguments is subject to the group transformation
GSY~( q, q, E) = ~ , ~ , G ( Y~iq, q, E ).
(24)
i
Constants ~ i depend on the chosen representation of the group. For instance, let the problem be invariant to the coordinate axes inversion. Then the eigenfunctions can be classified by parity and the following combinations contain wave functions of identical parity only: G~++)= ¼ ( a + b + c + d),
G~+_) -- ¼(a + b - c - d ) ,
G(_+) = ¼(a + c - b - d),
Gt__) -- ¼(a + d - b - c),
(25)
where
a = G ( x , y; x, y),
b= G ( - x , y; x, y),
c= G(x, - y ; x, y),
d= G ( - x , - y ; x, y).
The Green function subscript denotes the transformation law of the inversion of x and y respectively.
E.B. Bogomolny/ Wave functions of quantum systems
178
While calculating the smoothed wave function of the factor ¼ is cancelled since trajectories of zero length which give the main contribution to eq. (10) exist for G(x, y; x, y) only. By averaging eq. (24) over small intervals of energy and coordinates one can show that the dominant contribution to G(~q, q; E) will be given by the classical trajectories located near a part of the periodic trajectory passing through the phase-space points (q, 0) and (~iq, ~[lgt) provided such a periodic trajectory exists. For the point with the coordinates (x, y) near the periodic trajectory (see fig. 1) symmetry transformation is reduced to some transformation of the x coordinate and to a possible change of the y sign. For the transformation without a change of the y sign the result coincides with eqs. (15) and (17) with the substitution of the monodromy matrix per half a period instead of the monodromy matrix per period (16). If the considered symmetry transformation changes the y sign then W(x) in eq. (15) should be calculated by the formula
( a2S
a2S
~2S /
W ( x ) = l ~ y a - E a y ' a y " +a ,,---~/ = "Y / y"--y'=O
mll(X) + ~t2z(x) + 2 ml2(X)
(26)
where ~ j ( x ) are elements of the monodromy matrix per half a period. The example of the separation of symmetrical wave function will be given in the next section. Physically the effect of the amplification of the wave function near the unstable periodic trajectory can be interpreted as the result of the scattering on this trajectory. Under scattering the quantum scattering amplitude becomes large near the target mainly due to the slowing of the motion. The eigenfunction can be represented as the superposition of running waves, each of them scatters on the periodic trajectories. Scattering amplitude depends strongly on energy (even by the resonant way as in eq. (20)) and on some trajectories it is larger than on others. It is these trajectories that manifest themselves as the region of the largest enhancement of the wave function.
4. Stadium billiard
While studying quantum stochasticity the problem of quantum stadium billiard is often used as an example. The problem is to define eigenfunctions and eigenvalues of the Laplace operator for the functions vanishing on the boundary of the stadium type region (see fig. 3). It represents a square with the side 2R (or a rectangle with any ratio of the sides) and two semicircles of radius R join it. It is known [16] that a classical problem of the free motion within such a region with elastic reflections from the boundary is
gR Fig. 3. Stadium billiard.
E.B. Bogomolny/ Wavefunctions of quantum systems
179
ergodic. Numerical calculations of high excited wave functions for this system were performed in refs. [6-10]. Let us apply a general formalism of section 3 to this case. For similar problems the dependence on the energy (or I~1 = 2v r ~ ) and on the scale is defined by the dimensionality. In particular, the quantities entering eq. (15) have the following form: S -- I0[RSo,
R T = ]~- So,
D = [ ~ D o,
W-- [~-~-[Wo,
(27)
where the quantities with the subscript are calculated at R = 1 and I0I = 1. While making calculations by eq. (15) it is not difficult to lind numerically the necessary short-period trajectories and their monodromy matrices. In the appendix a simple method for the calculation of the monodromy matrix for billiard system is presented. For simple trajectories calculations can be made analytically. Certain periodic trajectories, their periods (or actions) and the monodromy matrix traces (at R = 1 and 101 = 1) are given in table I. Trajectories 1 and 13 are singular trajectories and will be discussed below. The problem considered has a discrete symmetry group with respect to the coordinate axes inversion x ~-x, y~-y. As has been mentioned, if one averages over wave functions with a particular symmetry, then (besides full periodic trajectories) the halves of the symmetrical periodic trajectories will make contributions to eq. (15). Monodromy matrices per half a period for the simplest symmetrical trajectories are given in table II. The trajectories corresponding to these monodromy matrices are presented by dashed lines in this table. Knowing the monodromy matrix of the given isolated periodic trajectory one can calculate its contribution to the Green function and to tile wave function using eqs. (15) and (10). As an example consider the periodic trajectory No. 2 from table I. Its monodromy matrix per half a period is given in table II. The classical trajectories, which give the dominant contribution to the Green function G(otx, o2y; x, y) at different values o 1 - + 1 and 02 = + 1 are pictured schematically in fig. 4. During calculations it is necessary to take into account the fact that for the Green function with the changed sign of y, W(x) in eq. (15) should be calculated by eq. (24). As a result we get ( ~ G(olx, o2Y; x, y) = ]~ A,exp i (n)
W-(°2) n
l~'~y •
n+
2
\ ),
(28)
where A,=
2oi. . 2x2,)( 2 ~ l q I ( i ) ~ e x p , i 2 °(2R
Rl! W~ ±~= - 2(2R~q_' x 2) ~ •t -- - ( 3 + ¢cff),
o(x)--0
asx>0,
2 ( 2 R 2 _ x 2)
Ihxl ~
,[h2[" 2,)1J2
~2(X1 + ~2 :~ 2),
X2 = _ (3 - ~fff),
o(x)=-I
asx<0.
At o r = +1 the summation in (28) is done over even n(-~2,4 .... ), at o1= - 1 it is made over odd n ( = 1,3 . . . . ). The coefficient 2 at the be~nning in A, takes into account the existence of two classical trajectories pictured in fig. 4. At x -- 0 the expression for G ( - x , y; x, y) has to be additionally multiplied by 2 since in this case the initial and final points do coincide and each of the trajectories of fig. 4c can be passed in the direct and in the inverse direction. Similar singularities also occur at y -- 0.
180
E.B. Bogomolny/ Wave functions of quantum systems
TableI Certainshort-periodtrajectoriesofthestadiumbilliard No. Trajectory T TrM- 2
No.
1
(I)4.00
--
10
(
3
~
8.60
44.05
12
Q
10.39
8.94
60.00
13
Q
11.32
Trajectory
T )10.00
TrM- 2 672.20
96.00
J
4
@
~
5
Q
8.98
-72.35
14
~
12.94
115.78
6
~
9.24
128.00
15
~
13.20
179.98
7
~
9.66
54.63
16
~
16.65
210.60
8
_
1677
0203
9
Q
16.91
171.98
-D
966
17331
17
9.78
284.64
18
~
181
E.B. Bogomolny/ Wavefunctions of quantum systems Table II The monodromy matrices per half a period for simplest symmetrical periodic trajectories of the stadium billiard No.
Trajectory
Monodromy matrix
Notes
1,2 2
R
5+-~
1--~(L-x)
~(3t2-sx2)
8
2
5
7
1= RI/~
----
24x -7+-- l
-~ (15l 2 + 24x 2 - 24/x) )
/
]
24 1
-4--~-x
(D Q i) 2) 3) 4)
_~R5 ( 3 R 2 _ x 2)
f5-R
-4+-~x
4x
4(x2-312 )
- 7 - - 7-
3
/ 2
I
1= R ~
54- 8~ ~(12--X2+]X) 2 8
7
13- -~
1= RV[2
For trajectories No. 2, 7, 14, 16 from Table I, L = 2R, L = (1 + v/2)R, (1 + VrS-)R, (1 + I ~ ) R , respectively. x is the distance from the centre to the chosen point. x is the distance from the upper right comer of the trajectory. x is the distance from the lower comer of the trajectory.
E.B. Bogomolny/ Wavefunctionsof quantumsystems
182
a
b
c
d
Fig. 4. Classical trajectoriesin the vicinityof the periodic trajectorywhich give the dominant contributions to the differentGreen functions: (a) G(x,y; x,y), (b) G(-x,-y;x,y), (c) G(-x,y; x,y), (d) G(x,-y; x,y). The second classical trajectories are denoted by the dashed lines. For the case (a) the second trajectorycorresponds to the reverse passing of the first one. A schematic relief of the wave function corresponding to eq. (28) is shown in fig. 5. This picture agrees qualitatively well with fig. 2 of ref. [9]. The contributions of other periodic trajectories can be calculated analogously. Singular periodic trajectories passing through the junction points of plane and semicircle sections of the boundary (see fig. 3) are an exception. Two such trajectories are shown as No. 1 and No. 13 in table I. Due to the fact that at their collision points the boundary curvature is discontinuous the monodromy matrix formalism cannot be applied to describe motion in the vicinity of these trajectories. The result of the motion (even in the linear approximation) depends essentially on the boundary point where a collision occurs. Roughly speaking, there exist many different monodromy matrices corresponding to various possibilities of the motion. The usual eq. (9) is not applied in this case. To derive a semiclassical expression it is convenient to use the exact representation of the Green function for the billiard problems with zero boundary conditions in the form of the multiple scattering obtained by Balian and Bloch [18]. According to their paper, oo
G(q",q', E) = Go(q",q' ) + ~_, ( - l i ) " f / o l . . . d o p OG°(q'''eq) OG°(al'a2) p=l
X ...
O G ° ( a ' - l ' or')Go(ap, q'), 0np
~nl
~n2 (29)
Fig. 5. A schematicrelief of the smoothedwave functionin the vicinityof the unstable periodic trajectoryNo. 2 from table I.
E.B. Bogomolny/ Wavefunctions of quantumsystems
183
where Go(q", q') is the free Green function whose asymptotics is as follows:
Go(q",q') = ih(2~rih)l/2(l(llL)-l/2exp
i
, L= Iq"-q'l.
The integral in eq. (29) is taken along the boundary of the given region, alan k is a derivative over the internal normal to the boundary at the point a k. If the boundary is smooth, the integration over do k can be performed by the saddle point method and the classical trajectories for which the incidence angle is equal to the reflection angle give the main contribution. As a result, we get eq. (9). However, if the integration is performed near the singular boundary point, the result is more complex. As an example consider a calculation of the contribution from the simplest singular periodic trajectory No. 1 from table I. Let us consider the trajectories with two reflections from the boundary. In this case there exist four different possibilities corresponding to the collisions on different sides of the singular points (see fig. 6). Their contributions can be determined from the third term of expansion (29). The total length of the trajectory enters the exponent of the obtained expression. Let u, v be the distances along the boundary counted from the collision points as in fig. 6. Taking into account terms up to the second powers of u, v, we get
exp ( i ~ )
G( x, y; x, y) = ih(2,trih )X/2l~la/2(2R( R 2 _ x2))l/2
foOOdufoOOdv ~ exp ( i _ ~ 2 ( u , v)),
(30)
j*l
where qby(U,v) ( j = 1, 2, 3, 4) are contributions from the trajectories of figs. 6a-d respectively.
( y z u ) 2, ,1,1 = 2 ( R - x ) +
( y .--_.0)2,
(u-v) 2
( y = u ) 2, (y+v) 2 ~2 = 2 ( R - x) + 2(R + x )
(y-v) 2
4------T--+ 2 ( R + x ) '
(y + u) 2
~3 = 2 ( R + x) + ~(l~---'"x-)
u2
R +
(u + v) 2
41~
(y + u) 2
,
(y + v) 2
¢h,~= ~(1~2"-Z) + ~(I~+"Z)
02
( . + v) 2
R +
4R
'
u 2+ v2 + (u-
R
o) 2
4R
(31) If lYi >> C~/h- the integrals can be taken by a saddle point method and the main contribution will be given by the real classical trajectories but they do not exist at all positions on the point. For instance, the classical trajectories (b), (c), (d) in fig. 6 are only possible at x > 0, x < 0 and x = 0 respectively. If the classical trajectory is allowed its contribution can be determined by the usual eq. (15). But at finite lYl CV~ a saddle point method is not applied, real classical trajectories are unimportant and the result of the integration in eq. (30) is not expressed in terms of elementary functions (as in the case of the Frenel diffraction). The most prominent difference between contributions from regular and singular periodic trajectories is that the former is symmetric relative to the periodic trajectory (cf. fig. 2) but the latter is asymmetric. For the above-considered singular trajectory the averaged wave function square is larger at y > 0 than at y < 0. Moreover, at small y an additional decrease of the wave function is observed as compared to eq. (15). These properties agree well with fig. 1 of ref. [9].-Similarly one can calculate a contribution from trajectories in a vicinity of the singular trajectory No. 13 in table I. The dominant contribution to G ( - x , y; x, y) will be given by the trajectories (not obviously the classical ones) of the types 5 and 6 in table II. During calculations it is necessary to take into account that if x = 0 the result is amplified due to both self-crossing and additional symmetrical contributions as in eq. (30).
184
E.B. Bogomolny/ Wavefunctions of quantum systems
R
V tt
tt
b
V IJ.
I/
V
Fig. 6. Possibleforms of the trajectories(not obvious are the classicalones) in the vicinityof the singular periodic trajectory(dashed line), u and v are the arc lengths counted from the collisionpoints. The x axis is chosen along the periodic trajectory, the y axis is perpendicular to it. For the stadium billiard there is another class of exceptional trajectories close to the neutral periodic trajectories (which are perpendicular to plane sections of the boundary) for which the above-discussed formalism is not applied since many trajectories are located close to each other. These trajectories are responsible also for a slow decrease of time correlations in the classical problem [19, 20]. Summing over all such trajectories one can calculate their contribution to the wave function. On the whole, all calculations agree well with the results of numerical calculations of high excited wave functions for the stadium billiard performed in refs. [9, 10]. One can explain quite simply all the distinguished details on all pictures of these papers. Unfortunately, in these papers the energy values corresponding to the given pictures of the wave function relief are not mentioned and it is not known on which heights the slices of the wave functions are made. Since the results depend strongly on these quantities it is impossible to make a quantitative comparison with these works.
5. Elliptic bmiard For ergodic systems the hyperbolic (and neutral) periodic trajectories are the only invariant manifolds which in the semiclassical limit can serve as the carders of the wave function structures [21]. However, unstable periodic trajectories can be present in other systems as well. For instance, they exist in some integrable problems. It is rather instructive to observe how the general formulae can be applied for the
E.B. Bogomolny/ Wave functions of quantum systems
185
integrable models where, in principle, everything can be calculated exactly. In this section we consider as an example the problem of the calculation of the eigenfunctions and eigenvalues of the Laplace operator with the zero conditions on the ellipse boundary: x2 y2 A--¢ + ~ = 1,
(32)
where A = ch u o, B = sh u 0. It is well known that this so-called elliptic billiard is integrable and in the elliptic coordinates,
x=chucosv,
y = s h u s i n p,
(33)
the variables are separated in both the classical and the quantum equations and the eigenfunctions of the Laplace operator can be written as the product of two functions depending only on u and v:
vt,(x, y) =f(u)g(v). The functions h 2d2f du 2
(34)
f(u) and g(v) satisfy the
+p2(u)f--O,
2d2g
h ~
+
Mathieu equations
k2(v)g=O,
(35)
8
b Fig. 7. Elliptical billiard. (a) The motion of the "whispering gallery" type. (b) The motion of the "bouncing ball" type. The caustics of quantum wave functions are depictured by the solid line. The dashed line corresponds to the unstable periodic trajectory.
186
E.B. Bogomolny/ Wavefunctions of quantum systems
where p2(u) = 2E shE u - a, k2(l ,) = a - 2E sin2 I, and a is a parameter of the separation of variables (an additional integral of motion). At different a the motion is of a different character. The case 0 < a < 2 E s h 2 u 0 corresponds to the motion of the "whispering gallery" type (see fig. 7a) and the case -2E < a < 0 corresponds to the motion of the "bouncing ball" type (see fig. 7b). We shall be interested in the unstable periodic trajectory which is the large ellipse axis (see fig. 7). Locally it is not distinguished from the stadium periodic trajectory No. 2 in table II. In the problem considered there exist invariant tori and the rules of semiclassical quantization are of the usual form, n x=
n2=
u)du+~,
k(~)dv+-~--,
(36)
where n x and n 2 are integers. At a > 0, a x = 1, ot2 ----0; at a < 0, a t = 0, ot2 = 2. At first glance it seems that these formulae do not contain any information about the considered periodic trajectory. Especially, the existence of the contribution of this trajectory to energy level density which follows from eq. (22) seems strange. Let us consider this subject in detail. Denote by nl(E , a) and n2(E, a) the exact expressions which should be equal to integers from the quantization conditions (as in eq. (36)). Using the Poisson summation formula (see, for instance, refs. [2, 22]) we get
d(E)= Y'~ I~(E-E(?/1,?I2))= ~ fdnldn2exp(2~riMln 1 + 2~riM2n2)8(E-E(nl, n2) ) nl, n2
Mz, M2
= Y'. fD(E,a)daexp(2~riMln 1 + 2¢riM2n2),
(37)
M1,M~ where D(E, a) is the Jacobian of the transformation from the variables (nl, n2) to the variables (E, a),
D(E, a ) = 0nl
On2 0nl
0n2
0E Oa
~E ~ a "
The functions n~(E, a) contain the terms proportional to 1/h. Therefore, the usual way to calculate the integral in eq. (37) as h---} 0 is to use a saddle point method [2, 22] according to which a dominant contribution to eq. (37) will be given by the resonance tori for which the frequencies are commensurable, Ml~O1 + M2a~2 = 0,
(38)
where
In this case
d(E)=
do(e) ~
+ ~
1
]~
ImAvex p i--~- ,
(39)
M1,M2 where Sp is a classical action calculated over resonance toms, Ap is a certain easily calculable amplitude and the summation is done over all Mr, M 2. Here it is implicitly assumed that at all M1, M 2 a saddle point method can be applied. But one can check that for elliptic billiard at M x = 2 M 2 a saddle point method is
E.B. Bogomolny/ Wavefunctions of quantum systems
187
not applicable and a region of small a gives the dominant contribution to the integral. However, at small a quantization conditions (36) are incorrect since this case corresponds to the motion near a double zero of the potentials in eq. (35) and the conventional eqs. (36) hold only if the momenta p2(u) and k20,) have a first order zero on the boundary. It is straightforward (but formidable) to fred the corresponding formulae for the quantities hi(E, a) matching the semiclassical wave function with the parabolic cylinder functions on the boundary,
~n2(E,a)=2
+~ln
+ ½argF(½- ia) + ~ + X q: ½arctg(-a,~
,
(40) 2~m~(E,a)--2
chuo- ahith
-~n2(E,a)+,~+~,
where a = a/(2h2¢2-E). The signs + correspond to odd (even) functions with respect to the transformation u - - - , - u , v - - * - v . Substituting these values into eq. (37) one can show that at MI=2M 2 the result coincides with the contribution to the level density from the considered periodic trajectory and its multiple passings as in eq. (22). Similarly, one can be convinced that besides contributions from the resonance tori the averaged wave function square contains a contribution from the unstable periodic trajectory (a large ellipse axis) equal to eq. (15). Special wave functions for which the parameter a in eq. (35) is small and a usual semiclassical approximation is not applicable [23] give this contribution. It is worthwhile to emphasize that eq. (15) is formed only after the summation over all degenerated states with a > 0 and a < 0. The individual wave functions with small a have another form. Roughly speaking they have maxima near the caustics of the type pictured in fig. 7, but the corresponding ellipses and hyperboles at small a are compressed to a part of the large ellipse axis, immitating the increase of the wave function near the periodic trajectory. From this example one can clearly see that the discussed effect is related to the motion near the periodic trajectory, i.e. to the scattering on it (see the end of section 3). In the integrable systems such as the elliptic billiard, a trajectory passing in the phase space near the unstable periodic trajectory cannot go far from it. Thus only a small number of wave functions have a marked enhancement near the unstable periodic trajectory, but the magnitude of this effect in the individual wave function is large. (Of course, the value averaged over all wave functions coincides with eq. (15).) For ergodic systems almost any trajectory passes near all periodic trajectories and it is natural to assume that almost any eigenfunction contains contributions from all periodic trajectories as in eq. (15) but their contributions are small ( - qr~-). The discussions above show that in integrable problems there may exist non-saddle point contribution corresponding to the account of periodic trajectories. In the limit h ---, 0 this contribution is smaller than the ones from the resonance tori but larger than semiclassical corrections to them. Such a contribution may occur both from unstable periodic trajectories (as in the elliptic billiard) and from families of neutral trajectories. The contribution to the level density of the spherical billiard from the sphere diameter which can be extracted from ref. [18] can serve as an example of the latter one. The existence of such contributions for integrable systems is rarely discussed in the literature.
6. Conclusions
The main result of this paper consists in the fact that hyperbolic periodic trajectories make a calculable contribution to wave functions both for non-integrable and for integrable systems. In all cases the
E.B. Bogomolny/ Wavefunctions of quantum systems
188
averaged wave function square can be calculated by eqs. (15) and (10). For integrable systems such a contribution exists only in a small number of wave functions and for averaged quantities it is small comparable to the resonance tori contribution. The exact behavior of individual wave functions for ergodic systems is not known so far. One can assume that each wave function can be represented as the sum over all periodic trajectories. The coefficients of this expansion are rapid and irregular functions of energy. At a particular energy eigenvalue the contributions of certain trajectories become larger than those of others and the wave function shows a noticeable enhancement in the vicinity of these trajectories. Physically this effect can be interpreted as the result of the scattering on the periodic trajectories. Taking some freedom, one can say that quantum stochasticity develops hidden regular structures of the classical chaotic motion. In classical mechanics these structures have a zero measure and they can be only observed while choosing the initial conditions directly on the periodic trajectories (and their incoming separatrices). The described method is useful for the calculation of wave functions smoothed over the finite energy interval. In this case the dominant contribution to all quantities of interest will be given by short-period trajectories for which the monodromy matrix elements are the least. For the exact calculation of the individual wave function of the ergodic systems this method cannot be practically applied since the number of necessary periodic trajectories is exponentially (of the smoothing papemeter) large. For physical problems where small-scale irregular fluctuations are not essential, the semiclassical formalism of the periodic trajectories allows one to calculate rather simply both a smoothed level density [2, 11, 13] and a smoothed wave function in the limit h ~ 0. An accurate comparison of the obtained semiclassical formulae with the numerically calculated wave functions for ergodic quantum problems is of considerable interest.
Acknowledgements The author is grateful to D.L. Shepelyansky for pointing out the paper [9] and to L.N. Shur and D.E. Khmelnitsky for numerous useful discussions.
Appendix Calculation of the monodromy matrix for billiard problems Here we describe a convenient method the calculation of the monodromy matrix for the billiard problems based on refs. [17, 16]. Let a periodic trajectory consists of n straight lines. Fix a certain point on this trajectory. Denote by t~ (i = 1,2 . . . . . n) the times of sequentional collisions with the boundary and by tn+ 1 the time of motion from the last collision to the chosen point. Let us construct two sequences K2(a), K j a ) , . . . , "Xn+/t'(a)l(tit = 1, 2) satisfying the recurrent relationship k'(~) = + d,. --,.+t 1 + t , . K ( = ~) '
(A.1)
where d,. = - 2 / ( R , . c o s y m ) and R= is the curvature radius at the collision point, 7m is the incidence angle. (For a plane section of the boundary dm= 0.)
E.B. Bogomolny/ Wavefunctions of quantum systems
189
The first terms of these sequences are as follows: 1 K(1) ----dl' K2(2)= ~'1 + d1"
(A.2)
For each sequence we construct a product
G,(a)= f i (1 + m--2
t,.K(a)).
(A.3)
Then the monodromy matrix defined by eq. (16) is equal to
mll
c 1 (1 + t n + l rcln + l ] ,
m21 = G n(1)Kn(1) +l,
m12 ~
txc (2 (1 + tn+ 1 r
n+l],
(2) m 2 2 -- t 1G n(2)Kn+l"
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(A.4)