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VOL. 30, NO. 4


Classical and Quantum Eigenstates of a Kicked Rotor? Shau-Jin Chang and Gabriel Perez* Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. (Received April 8,1992)

We consider the classical kicked rotor as an eigenvalue problem. We formulate the dynamics on a discrete lattice of size N X N and introduce a classical evolution operator. We then modify the classical evolution operator by coarse-graining the delta functions into gaussians and by introducing some phases. These modifications lead to a semiclassical evolution operator, and they reduce the number of participatingeigenstates from N2 to N. The N surviving eigenstates agree exceedingly well with the corresponding quantum eigenstates.

I. INTRODUCTION Usually, the evolution of a classical Hamiltonian system is described by a set of nonlinear Hamilton equations. The quantum Hamiltonian system on the other hand is described by a Schrodinger equation. It has been suggested that since the Schriidinger equation is a linear equation there is some kind of intrinsic linearity to quantum mechanics. This is clearly not a valid argument, as shown by the fact that the linear Liouville equation

(1) contains as much information as the Hamilton equations for a given system, whether they are linear or nonlinear. The Hamilton equations are the characteristic equations of Liouville’s equation. We can express the general solution to the Liouville’s equation in terms of solutions to its characteristic equations and vice versa. In this talk, we wish to establish relations between the wave functions in the Schrodinger equation and the wave functions in the Liouville’s equation. The wave function in the Schrodinger equation can be either a function of 4 or ofp, but not both. On the other hand, the wave function in the Liouville’s equation is a function of bothp and q. To make the comparison meaningful, we need to construct a quantum wave function in both thep and the q space. We can achieve this with the help of coherent-state representation. As we shall see, we also need ’ Refereed version of the invited paper presented at the Symposium on Trends in Particle and Medium Energy Physics, November X-16, 1991, Taipei, Taiwan, R.O.C. 01992 THE PHYSICAL SOCIEI’Y OF THE REPUBLIC OF CHINA



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to coarse-grain the classical evolution operator and to introduce a semi-classical phase.

II. KICKED ROTOR AND CHIRIKOV MAP We shall consider one of the most studied Hamiltonian systems, the kicked rotor Hamil‘~ tonian, 2 H=

$ + V(q) -p(1 - nT), n

where the potential V(q) is periodic with period one. For simplicity, we choose the kick period to be T = 1. The equations of motion for this Hamiltonian can be integrated over one period T and give the following map: Pn+l =

Pn -


qn+l = P,+I + qn

(3) (mod 1).

Herep, and qn are the values of the cannonical variables at the time t = n corresponds to the potential,


+ . The Chirikov map

The Chirikov map is a system that has been studied extensively as a testing ground for diverse issues related to nonintegrable dynamics. The model applies to diverse physical situations, from a single pendulum kicked periodically to the behavior of particles in an accelerator. The system can be quantized by using the commutation rule [P,,, q,,] = -ifi, wherep,, and qn are now the operators at time n, in Heisenberg representation. The evolution of the system can be obtained from the unitary evolution operator 394

lJ = exp

(-&d) exp (--iIf( ,

This evolution operator relates a quantum wave function at time t = n + to that at time t = (n + 1>+,

Nn + 1) = &+J


This equation describes the integrated Schrbdinger equation over a kick period. The quantum eigenvalues and eigenstates obey e-i%+5 = U?J


where w is a pseudo energy and is the analog of an eigenenergy in the usual Hamiltonian system.


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Inp space, < m/p = 2rfim < ml,

the matrix elements of U are

emizxmzfi J0




For Chirikov map, the matrix elements are


e -i2nam~fr(_i)m-m


02) Chang and Shi4 have studied the eigenvalues and eigenstates of this quantum evolution operator. Next we express the kicked rotor as an integrated Liouville’s equation.5’6 To accomplish that we introduce a generalized probability distribution v(p, 4) and an evolution operator K(p’, q’,p, q) such that

$9&J’, q’) =

Jrn dPil W-qP', Q’, P, d?ltt=o(P, d> --oo

K(P’, Q’, PI q> = 6(P’ - P + m2N~z(4’ - Q - P’>l

(13) 04)

where S,(x) = 6(x mod 1). This is a Perron-Frobenius equation that corresponds to an integration over one period of the Liouville equation of the system. The kernel K is orthogonal because the map is area-preserving and invertible. We can introduce the classical eigenvalues and eigenstates of K in a similar way 6-wti(P’, d> = l; dpll WqP’, Q’> P, dvYP9 4.


The evolution operator K has the following classical symmetry: K(p’+ L!?‘,P+ Ld = qP’d,Pd


This means that we can write the eigenfunctions as a Bloch wave multiplying a function periodic inbothp andq +(P, 9) = e-iz*ap dh (p, 4,


&(p+ Lq> = $a(P, Q + 1) = YL(P,d!




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where 0 I a I 1. For the new function q&, 4) we can write an evolution equation

J J dPWG(P’, 1

,-;W(a) $a (p’ ) q’) =



Q’>Pl q)lcla(P, Q)l



with the evolution operator given by

e2nia(+F-N)qp~ -p - N + v’( q))61(*’


N=--03 =

- q - p’)

e.-i2Ta”‘(q)61(pt - p + V’(q))61(q’

- q - p’).

Thus & is a function periodic in bothp and 4 variables.

III. DISCRETIZATION In order to carry out numerical calculation, we need to find a way to approximate the evolution operators by finite matrices. A consistent approximation is to put variablesp, 4 on a discrete lattice with lattice spacing l/N. Let us first work out the discrete classical kicked rotor. The classical map equations become

(21) q’ = q

(mod I),



withp, 4 on lattice obeying - 03 < p,p’ < QJ and 0 < q,q’ I 1. The square brackets in Eq. (21) mean nearest integer, and are included to insure that the map transforms lattice sites into lattice sites. The question of discretizing a classical system on a lattice and its physical consequences have been discussed thoroughly by Chirikov et al in Ref. (7). These authors pointed out that under normal situation there is some transition time below which the behavior of the discrete map should reflect the chaotic behavior of the continuous model.’ Indeed, it has been shown by Ranou’ that even for maps defined in medium size integer lattices (of the order of 400 x 400 points), it is possible to identify in the dynamics most of the phenomenology associated with the continuous real dynamics. The discrete map can now be formulated as an eigenvalue problem



q’> =



c c WP’, Q’IP, [email protected](P, d,

p=-cc q=o

= 6 (P/-P+

~[NV’(dl) 61(q’- P -P’),







In order to make meaningful comparisons between the quantum and the classical eigen states, we need to modify both wave functions. (1) From a quantum eigenstate, we can project it into the coherent state representation. On a doubly periodic lattice, we denote a coherent state with center located atp = m/N and q = n/N by Im,n > where m,n are integers between 1 and N. We choose the coherent states with equal widths inp and q space. The matrix elements of U in these coherent states are < ml,

n1lUlm0, no >

x exp % (m - m&Q + ( ma - m)n - $ - NV(;) i [


II .

where the normalization constant is chosen to make < m,n jm,n > = 1. Even though the matrix elements of U expressed in Eq. (29) form an N2 x N2 matrix, this matrix has only rank N. When we diagonalize this matrix, we obtain N eigenstates with corresponding eigenvalues as phases, exp(-io). These are the true quantum eigenstates. There are also N2 - N eigenstates which have degenerate eigenvalues zeros. These eigenstates do not correspond to any quantum states but their presence is needed for completing the larger space. (2) From the classical evolution operator K&‘, q’, p, q), we can construct a semiclassical evolution operator by including a semiclassical phase factor in & and then coarse-graining this new K, in the phase space over regions of size& This may be viewed as a semiclassical treatment of the path integral representation of U. We expect that only those orbits or paths whose action integrals retain sufficient coherence over the coarse-graining processes can survive. After coarse-graining, the resultant evolution matrix is no longer unitary. In the following, we consider only a = 0 and drop the subscript a in K,. As we shall see that under diagonalization, this resultant matrix reduces approximately to a block diagonalized matrix made of an N x N unitary matrix and an (N2 - N) x (N2 - N) matrix of small eigenvalues. We can then compare these N eigenstates with those in the quantum coherent-state representation. We can achieved the coarse-graining by replacing the product of delta-functions in Eq. (24) by a gaussian,

WA Q’,P, d = N ormalization x


f a c t o r e x p -;- P; + V(q) + 44P’ -4 { I

5 exp[-a(r-I)2-b(r-l)(s-l’)-c(s-1’)2] ’= /.I -00



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s q’ - q - p’,

s z p’ - p + V’(q)



and a, b, and c are complex parameters. The real parts of a, b, and c provide the gaussian coarsegraining, and the imaginary parts an additional phase. In Ref. (9), we obtained a set of u, b, and c from the semiclassical approximation. We also showed in Ref. (9) that the qualitative features of our results do not depend sensitively on the real parts cf these parameters, but depend sensitively on the phase resulting from the imaginary parts of these parameters. V. NUMERICAL RESULTS


From the N2 x iV2 coarsely grained classical evolution matrix, we can obtain both the eigenvalues and the corresponding eigenstates. We then compare them with the exactly quantum eigenvalues and the corresponding eigenfunctions in the coherent-state representation. We present here some numerical results for N = 26 (N2 = 676). In Fig. l(a), we choose k = 1 and a, b, c from the semiclassical calculation. The coarsely grained evolution operator K leads indeed to 26 large eigenvalues with magnitudes between IA 1 = 0.994 and 0.996. The remaining 650 eigenvalues are practically zero. Even fork as large as 10 (Fig. l(b)), the separation of large and small eigenvalues is still very clear. The first 26 eigenvalues have magnitudes between 0.862 and 0.926 with the remaining eigenvalues practically zero. In Fig. 2, we change the real parts of a, b and c and consequently modify the shape of the gaussian in the coarse-graining. The separation of large and small eigenvalues still holds. In Fig. 3, we compare the classical eigenfunctions with the exect quantum eigenfunctions in the coherent-state representation. In Figs. 3(a) (b), we have a pair of classical wave functions associated with a coarsely grained K with semiclassical a, b, and c. In Figs. 3(c) (d), we plot the corresponding exact quantum wave functions in the coherent-state representation. In Figs. 4(a) (b), we have the same pair of wave functions in a different coarsely grained K. These three pairs of wave functions are practically indistinguishable. This feature is true for all 26 large-eigenvalue wave functions. In Fig. (5), we show a small-eigenvalue wave function of a coarsely grained K. In general, these small-eigenvalue wave functions are more chaotic in appearance. VI. DISCUSSION We can summarize our findings as follows: (1) Coarse-graining of a classical evolution operator reduces the number of non-trivial eigenstates from N2 to N. This corresponds to the reduction of independent dynamical variables from (p,q) variables in classical mechanics to either p or 4 variable in quantum mechanics. (2) The N eigenstates of coarsely grained K with large eigenvalues (1111 = 1) are indistinguishable to quantum eigenstates. The remaining N2 - N states with small eigenvalues ( IL 1 -c -c 1) have no quantum counter parts.








1. 0 E E fro.6


Ei z E0.6


i i IO.4 6

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ii! 0.2


I 20

0 0

I 40

I 60

I 60

I 100

I 120


: 2 N 0.8 x b ’ 0.6

0 . 2


020 0












(b) FIG. 1.

Magnitude of the eigenvalues (in descending order) for the coarsely grained classical evolution operator K. The lattice parameter is N = 26. The values of k are (a) 1 and (b) 10. Notice that we have plotted only the first 60 eigenvalues. The total number of eigenstates is 262 = 676. All the ignored eigenvalues are practically zero.

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1.2 -

1. 0




L 0














FIG. 2.

Magnitude of the eigenvalues (in descending order) for the coarsely grained Kwith a modified gausSian. The parameters are N = 26, and k = 1. There is a clear separation between the 26 large eigenvalues and the rest of the spectrum. The small eigenvalues are no longer negligible.







FIG. 3.

Magnitude squared of some eigenfunctions of the coarsely grained classical evolution operator K . In (a) and (b) we have used N = 26 and k = 1. The values of the quasienergy w/(m) are (a) 0.2087, and (b) -0.0233. Figures (c) and (d) represent the same states as (a) and (b) obtained with the exact quantum operator. The quasienergies are: (c) 0.2094 and (d) -0.0238. These two sets of graphs are practically indistinguishable.






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FIG. 4. Magnitude squared of two eigenfunctions of the coarsely grained Kwith a modified gaussian. The quasienergies are (a) 0.1745, and (b) -0.0558. Notice that even after the extra approximation introduced by the change in the shape of the gaussian, these states remain practically identical to their counterparts obtained with the exact quantum evolution operator.



FIG. 5. Magnitude squared of one eigenfunction of a coarsely grained classical evolution operator K with small eigenvalue.

(3) For large k, these N2 - N states with no quantum counter parts appear to be more chaotic than those N states with quantum counter parts.

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(4) Both large- and small-eigenvalue states are needed for completeness. A change of coarse-graining affects the small-eigenvalue states but not the large-eigenvalue states. This is reminiscent to a gauge transformation.

ACKNOWLEDGMENTS One of us (SJC) wishes to thank Professor Pauchy Hwang and the Physics Department of National Taiwan University for their hospitality during his visit. This work was supported in part by the U. S. National Science Foundation under grant No. NSF-PHY-87-01775.

REFERENCES * Present Address: International Centre for Theoretical Physics, P.O.B. 586,341OO Trieste ITALY. 1. V. B. Chirikov, Phys. Rep. 52,263 (1979). 2. J. Greene, J. Math. Phys. 20,1183 (1979). 3. G. Casati, B. V. Chirikov, F. M. Iwailev, and J. Ford in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Vol. 93 of Lecture Notes in Physics, edited by G. Casati and J. Ford (Springer, New York, 1979). 4. S. J. Chang and K. J. Shi, Phys. Rev. Lett. 55,269 (1985); and Phys. Rev. A 34,7 (1986). 5. H. J. Korsch and M. V. Berry, Physica 3D, 627 (1981). 6. S. J. Chang and J. Wright, Phys. Rev. A 23,1419 (1981). 7. B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, Soviet Scientific Reviews 2C, 209 (1981). 8. F. Ranou, Astron. Astrophys. 31, 289 (1974); see also M. Henon in Chaotic Behavior of Deterministic Systems, Lectures at Les Houches 1981, Session XXXVI, edited by G. Iooss, R. H. G. Helleman, and R. Stora (North Holland, Amsterdam, 1983). 9. S. J. Chang, G. Perez, Phys. Rev. A 42,5898 (1990).


VOL. 30, NO. 4


On q-Deformed Bose-Einstein Condensationt Chin-Rong Lee Institute of Physics, National Chung Cheng University, Chia-Yi, Taiwan 62117, R.O.C. (Received January 9,1992)

We study the Bose-Einstein condensation of the q-bosons (particles which obey the q-deformed commutation relations) in a box and find that the condensation temperature will rise as the deformation parameter moves away from q = 1. We note that this kind of condensation can occur even in two space dimensions.



Bose and Fermi statistics are characterized by commutation and anticommutation relations, respectively: [Ui,

af ]

= Sij ,

(0 {bi,bj} =


where the indices i, j denote the whole set of quantum numbers (momentum, spin,...) and [A,B] = AB - BA, {A,B} = AB + BA. Recently, Macfarlane’ and Biendenharn2 have considered a new operator relation which may be used to provide a realization of the quantum group3 SU(2)q in terms of the q-analogues of the quantum harmonic oscillator. This bosonic q-oscillator is defined by three elements a, at and N which satisfy the relations

[v] = [&A] = 0,

uut - q&a = q-N, [N, A] = at, [N,


u] = --a,

where N is the number operator of the bosonic type oscillator and the deformation parameter q is taken to be real for simplicity. The Hilbert space with basis In > is constructed such that ’ Refereed version of the invited paper presented at the Symposium on Trends in Particle and Medium Energy Physics, November 15-16, 1991, Taipei, Taiwan, R.O.C. 491








Jln >= [n+ 1p1n-t 1 >,

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a]n >= [&I - 1 >, where the notation [ ] is defined in terms of the deformation parameter q as [z] = q= - q-= q-q-1 .

p] has the following properties: (i) When 4 + 1, [x] + X; (ii) [l] = 1 and [0] = 0; (iii) [-xl = -[xl; (iv) k] is invariant under the transformation q * 4-l. It follows from Eqs. (2) and (3) that in the Hilbert space there exist the identities4 - N

ata = [N] = QNq-q-1 -q

aat = [N + 11, and the operator N is such that Nln >= nllz >


In the limit q + 1, one reproduces the familiar bosonic results, in particular

[d] = 1, (7)

a t a=N.

II. STATISTICAL DISTRIBUTION OF q-GASES It has been pointed out that,’ by using the law of detailed balance, one can derive the statistical distribution function for the q-gases which obey the q-deformed commutation relations given by Eq. (2). We briefly recapitulate the essential point of the derivation here for the sake of completeness. Let fk be the average number of particles in the k state, which k denotes the momentum (energy) and spin of the particles. Because of collisions, each particle continuously change its state. For example, considering the reaction 1+2++3+4,


the states 1 and 2 interact to change into states 3 and 4 and vice versa. The direct reaction rate

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is given by [fil[f2lP + f31P +



where, according to Eq. (3), vi] and Lf2] are the probabilities of annihilating one particle in the states 1 and 2 respectively and [l + f3] and [l + f4] are the probabilities of creating one particle in the states 3 and 4. R denotes the reaction strength, which depends on the structure of the particles and the details of its interacting mechanism. This reaction rate must be equal to that in the reverse direction (the law of detailed balance), i.e.

vmfll~ + f3][1 + f‘I]R =

V3lVd1 + fr][l+ f#’


By time reversal symmetry, we have then R = R’ and hence [fll [f21 [f31 [f41 [l+=[Imfil


We also have to consider the conserved quantities of the reaction. Before and after the reaction (Eq. (S)), the energy and the particle number are conserved, i.e. particle number = 2,


e n e r g y = 61 + e2 = f3 + e4,

Hence, a reasonable solution to Eqs. (11) and (12) is given by

[fkl = exp(-a - ,bk), -

[1 + fk]


Where a and /I are constants, the distribution functionfk is then determined by solving Eq. (13). In the dilute gas approximation, j&] < < 1, we have (in the limit q + 1) [fk] = fk = exP(-a - bk).


This is the familiar result of Boltzmann distribution, and the constants a and /I are known as the chemical potentialp (and a = -&L) and the inverse of the temperature, (k~7’)-~, respectively. As q --, 1, the Bose-Einstein distribution, 1 exp(CX + /?Ek) - 1’


is recovered from Eq. (13). Finally, we should mention that all results remain the same for multiparticle scattering reactions.



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III. THE BOSE-EINSTEIN CONDENSATION As a simple application of the previous discussion of the q-deformed statistical distribution6 we consider how the Bose-Einstein condensation temperature (Tc) would be modified, if the particles were to satisfy the deformed commutators, Eq. (2). The mean occupation number of particles, f, which in the lowest state with energy E = 0 is given by the relation (in accordance with Eq. (13)) if1 - =

[1 + fl


Since the average number f must be positive or zero, they cannot be negative. It follows that the chemical potentialp of a q-bosonic gases system must be non-positive (as ordinary bosons7): P I 0. .


The energy density of states (for a given volume V), is 2xV/h3 (2n1)~‘~~“~ E CE”~ (m being the mass of a particle of the gas). Hence the total number of particles N is

where the statistical distribution function of the q-bosons,fc ,is determined by = If4

I1 + fcl



for a given temperature. Let us vary the temperature of the gas, keeping the particle density NIV constant. The condensation temperature T, is such that for T = T, the chemical potential vanishes: p = 0. The temperature T, is then given from Eq. (18) by (~3~s = 7 and PC = l/k~T,) N


M aaf J0


-dv h 00


$(_ qea - 1


1 q-le’l _ lJdq


-;ca;~&(;)(F+) -F;(-a)),

where Bose-Einstein integral F,(a) E & SOW &dz(I’( n)

being the Gamma function)

and q = ea. This implies (Eq. (20)), with the particle density N/V being kept constant, that the condensation temperature T, is a function of deformation parameter q (or a).

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Some numerical values of Bose-Einstein integral, F,,(a), have been tabulated by London (1954).8 We list here some of the results:








Tc (a) TC(a = 0)







We note that for smaller values of q, the condensation temperature T, is higher. Finally, we should point out that this kind of condensation can occur even in two space dimensions.

ACKNOWLEDGMENTS The author would like to thank Prof. Choon-Lin Ho and Dr. Wai-Ping Lam for numerous useful discussions and reading the manuscript. This work is supported in part by R.O.C. Grant NSC 81-0208-M-194-07.

REFERENCES A. J. Macfarlane, J. Phys. A 22,4581(1989). L. C. Biedenharn, J. Phys. A 22, L873 (1989). M. Jimbo, Lett. Math. Phys. lo,63 (1985). P. P. Kulish and E. V. Damaskinsky, J. Phys. A 23, LA15 (1990). C. R. Lee and J-P Yu, Phys. Lett. A 150,63 (1990). Zhe Chang and Hong Yan, Phys. Lett. A 154,254 (1991); C. R. Lee, Chinese J. of Phys. 29,91 (1991); 28,381(1990); C. R. Lee and J-P. Yu, Phys. Lett. A 164,164 (1992). 7. R. Kubo, M. Toda, and N. Hashifume, in “Statistical Physics I. ZZ” (Springer-Verlag Rerlin, Heidelberg 1983). 8. F. London, in “Super&ids Vol II” (John Wiley and Sons, New York, 1954).

1. 2. 3. 4. 5. 6.