PH YSICAL REVIE% A VOLUME 23, NUMBER 4 APRIL 1981

Quantum ergodicity for time-dependent wave-packet dynamics

John S. Hutchinson~ and Robert E.WyattDepartment ofChemistry, University of Texas, Austin, Texas 7871Z

(Received 17 November 1980)

The dynamics of nonstationary wave packets in model quantum-mechanical systems are studied for the propertiesof ergodicity and instability. The long-time-average Wigner phase-space distribution is calculated for each wave

packet and analyzed for uniformity on a surface of section. A mild transition from nonergodic to ergodic behavior isobserved with increasing energy. These results are compared to long-time averages of the wave-function probabilitydensity. The "stability" of each packet is also analyzed by several methods, including the "survival" of the initialstate and a new measure of "separation" of similar wave packets. The relationships between the wave-packet

ergodicity and instability and the corresponding classical dynamics are developed. Comparisons to other quantum-

ergodic studies, and implications for intramolecular energy transfer and classical-ergodic studies are discussed.

I. INTRODUCTION

The statistical mechanical property of "ergo-dicity" has become a topic of increasinginterest to chemical dynamicists, and has beenthe subject of extensive experimental' and theore-

, tical' research recently. These studies have im-mediate relevance to a wide range of fields inchemistry and physics, including intramolecularenergy transfer, ' laser excitation of molecules, 4

theories of chemical kinetics, ' semiclassicaltheories for quantum mechanics, ' and of coursestatistical thermodynamics. ' Specifically, theergodic problem poses the question of whetPerthe microcanonical ensemble average of a dynam-ical function over all system states of fixed energyis equal to the long-time average of that functionfor any one of the systems.

Until recently, research on ergodicity has beenconfined to systems obeying classical dynamics.Classically, the state of a system is described byits position in phase space, giving all its positionand momentum coordinates, and a microcanoni-cal ensemble has equal density in all cells on theconstant energy surface in phase space. Clearlythen, ergodicity requires the system trajectoriesto evolve throughout the energy allowed surface inphase space so that the time average and thephase-space average are equivalent. ' Alternately,we may define a density function by the dis-.tribution of ensemble states in phase space, andan ergodic system may be characterized by auniform density in the long-time limit. Thus,classical studies focus on the development in timeof a uniform phase-space density.

Several excellent reviews describe the recentadvances in understanding ergodic behavior inclassical systems. ''. ' However, two of theseresults require noting here. First, classical tra-

jectories, which are of course deterministic, cannonetheless exhibit "chaotic" behavior. The re-sult of this chaotic motion is that a small cell oftrajectories mill eventually spread to cover theentire phase space. A system with this propertyis called a "mixing" system, and it is clear thata mixing system is also ergodic. However, al-though a classical system can thus display ergo-dicity, such behavior is only expected above acertain "critical" energy, in accord with the theo-rem of Kolmogorov, Arnold, and Moser (KAM). 'Thus, the trajectories undergo a transition in be-havior with increasing energy from "regular"periodic or quasiperiodic motion to "irregular"ergodic motion. Second, ergodic trajectories arecharacterized by "unstable" dynamics, in that twoinitially similar trajectories will diverge expo-nentially in time. In contrast, for regular tra-jectories, the distance between similar trajector-ies grows only linearly with time. '

The question remains unanswered, however, asto whether these important classical results arerelevant to molecular systems, for which quantummechanics governs the dynamics. Over the pastdecade, quantum-ergodic studies have appearedincreasingly, but under a wide range of criteriafor ergodicity. The most widely referenced study,by Nordholm and Rice, defined a phase space bya zero-order basis set so that an ergodic statewas one with uniform overlap with the basisstates. ' Percival" and Pomphrey" characterizedergodic eigenstates as unusually unstable under asmall perturbation. McDonald and Kaufman" andStratt, Handy, and Miller" studied the irregular-ity of wave-function nodal patterns. Noid, Kos-zykowski, Tabor, and Marcus" suggested thatergodicity is manifested in wave-function powerspectrum broadening and avoided crossings ofenergy eigenvalues. All of these studies suggest

1567 1981 The American Physical Society

1568 JOHN S. HUTGHINSON AND ROBERT E. WYATT

a transition in the nature of quantum eigenfunc-tions of bound systems from regularity to irregu-larity, similar to the KAM transition of classicaltrajectories

On the other hand, a few recent studies, usingdifferent definitions of ergodicity, have sought toprove the impossibility of quantum ergodicity.Heller" has shown that, for a degenerate quantum

system, a wave packet will spend more time nearits original position than near other positions. Hethus concludes that degenerate systems are nonergo-dic. Kosloff and Rice"developed aquantum analog ofthe classical Kolmogorov entropy. The classicalE entropy is nonzero for a chaotic system, andis thus a predictor of ergodicity. The quantumE entropy is thought to measure the flow of quan-tum density over a set of projections onto operatoreigenvalues. Kosloff and Rice then show that, dueto recurrences in bound quantum motion, thequantum K entropy is identically zero for a systemwith a discrete spectrum. They conclude thatchaotic behavior, and thus ergodicity, is not pos-sible in bound quantum systems. Brumer andShapiro" have reported a computational study ontwo-model potentials, only one of which generatesclassical chaotic behavior. They found the quan-tum dynamics on these surfaces to be similar.This paper has received the attention of Davis,Stechel, and Heller, "as discussed later.

Unfortunately, the relevance of any of thesestudies to the classical ergodic problem is stillunclear. The absence of a uniform phase-spacedistribution as a criterion for ergodicity rendersthese studies irrelevant to the question posedabove: that is, do the results of the classicaldyanmics studies accurately reflect the behaviorof ergodicity in quantum-mechanical systems'P

In order to address this question directly, wehave recently applied the Wigner phase-spacedistribution' to the eigenstates of a model mole-cular system. The results of that study, pre-sented in detail in an earlier paper" and reiter-ated in Sec. III, lead to the conclusion that an ex-cellent correspondence exists between classicaland quantum ergodicity when both are consideredin terms of phase-space densities. In this paper,we will extend these results to time-dependentwave packets in a model two-dimensional system.We will then be able to establish the developmentof ergodicity in initially localized states. In addi-tion, results on the dynamics of these wave pack-ets will be presented, including calculations ofthe "survival" of the initial state and "separation"of similar wave packets. These results are an-alyzed by comparison to the corresponding classi-cal dynamics, from which correlations betweenquantum ergodicity and quantum irregularity may

where (x, p) represents a point in phase space andVis the wave-function dimensionality. F is realfor all points (x, p) but can be negative; hence, theWigner distribution is not strictly a probabilitydensity function. However, F does possess pro-perties we would expect of a phase-space density. "For example, the probability density in positionspace may be recovered by summing F over allpossible p at each x:

P(x) = 4*(x)e(x) =J

I (x, p) dp . (2)

Similarly, the probability density in momentumspace is found by integrating over all positions:

P(p) = C *(p)4 (p) = Jt I' (x, p) d x,

where @(p) is the momentum-space wave functioncorresponding to @(x). The Wigner distributioncan be used to calculate expectation values in thetypical manner of a probability density:

(G) = ~f Jt G(x, p)I"„(x,p)dxdp . (4)

Lastly, since we are primarily interested in aquantum-classical comparison in phase space,we note that the equation of motion

is similar in form to the classical Liouville equa-tion and reduces to it in the limit g -0 (Ref. 22}:

Lq =L, +O(A')+O(h')+ ~ ~ ~,where J, and Lz are the classical and quantumLiouville operators. Because of these properties,the Wigner distribution has been consistently in-terpreted in the literature"'"'" as the quantumanalog of a classical phase-space density. Infact, many other authors have recommended theapplication of F to the ergodic problem. '" Asa final justification of our reliance on F as ameasure of ergodicity, we cite the successful re-sults of our earlier study, as well as the resultsreported here.

be deduced. Comparisons to other studies or er-godicity will be discussed in the conclusion.

II. THE V(IGNER DISTRIBUTION

A. Properties

The Wigner phase-space density is defined fora wave function C(x} by the transformation

I'„(x,p) =~. f e'(x+z)e(x —R)e"'*~"dz,

QUANTUM KRGODICITY FOR TINK-DKPKNDKNT %AVK-PACKET. . . 1569

B. Method of calculation

Consider a time-dependent wave function in twodimensions, 4(x, y, t), expressed as an expansionover harmonic-oscillator product basis states

@(x,), t) =p g c„(t)e„(g)@()') .For simplicity, we adopt the convention that thefirst index in c„and the first function +„ listedboth refer to the nth basis function in the x modeharmonic oscillator basis set. Then the Wignerdistribution for 4 can be written in the form

In fact, this might be the most obvious analysis,since P, is not restricted to any one particularvalue of a quantum wave packet. However, Ber-ry" has shown that, for a microcanonical phase-space density uniform throughout the constant en-ergy surface, this function ~ would be decidedlynonuniform on the (x, P,) surface, and in fact,would not be easily discernable from a nonergodicdistribution.

A better technique is to study F on the y =0plane of the constant energy surface in phasespace, so that P, is chosen at each (x, P,) to givea fixed-average wave-packet energy"

W(x, P,) =r.(x, y=O, P„p,(x, P„Z)) .x(x, P,)+ (y, P ),

(8)

where ~" is a Wigner basis function defined by

(x, P, ) =—I 4„"(x+8)c„{x—E)e '~ dz .

For a perfectly ergodic wave function, W wouldappear as a flat "top-hat" distribution for all(x, p, ) on the energy surface. This is in sharpcontrast to the behavior of a nonergodic distribu-tion, for which a localized spire or a wall of den-sity is expected. " Thus, W(x, P,) calculated fromEq. (11) provides a useful criterion for analysisof quantum ergodicity.

The Wigner basis function may be tediously butdirectly integrated analytically (see the Appendix).Notice that these functions depend only on thechoice of the harmonic-oscillator basis and onneither the particular wave function chosen in Eq.(t), nor the time. As such, the necessary 4™may be calculated in advance at all ( x, P,) of in-terest and saved for use in Eq. (8). All that isrequired to calculate F are the expansion coef-ficients c„„(t).

For an eigenvector + of a bound system, thetime-independent coefficients in Eq. ('l) are sim-ply the appropriate elements of the eigenvectortransformation matrix, found by diagonalizing thesystem Hamiltonian in the harmonic-oscillatorbasis. This was done in our earlier study. ' Inaddition, however, Eq. (8) is a particularly con-venient form for transforming time dependent4(x, y, t). We may build any wave packet usingEq. (t), follow its time dependence under theSchrodinger equation, and then insert the c„„(t)into Eq. (8). This will be our approach in Sec. IV,with our choice of Gaussian wave packets.

The phase space for a two-dimensional systemis four dimensional. As usual in ergodic studies,we will restrict our calculations to a two-dimen-sional Poincare surface of section. ' One way toanalyze F would be to calculate the phase-spacedensity on the y=0 surface, summing over allpossible P, at each (x, p, ) point:

C. Model Hamiltonians

Our system of study is the model originally an-alyzed by Henon and Heiles (the HH system)~:

1(P', +P', )+-,'m((d,'x'+(d', y')+Zxy'+(({x'.

(12)They studied classical trajectories for the casem =1, &u, =(d, =1, X=1, and g= ——, (all in atomicunits), for which the escape energy is —,'. TheKAM transition from regular to ergodic behaviorwas first observed in this system, with a criticalenergy of about —' More recent studiess'8 suggest that the critical energy may be as low asThe ergodic HH system will be the primary modelxn th).s study.

For comparison, we will also analyze two var-iations on the HH system for which ergodic behav-ior is not found in classical trajectory studies.The first is the separable HH system, with A. =O

so that x and y motion are uncoupled. The otheris a (classically) nonergodic HH system with m= j., (u, =0.70, (u2=1.30, A. = —0.10, p, =0.01, and 5= 1.0. Since the KAM transition is not observedfor trajectories on either of these surfaces, thenanalysis of the Wigner distributions on each willserve as an acid test for quantum ergodicity.

III. STATIONARY STATE DISTRIBUTIONS (REF. 20)

& (x, P,) =~I I'„(x,y=0, P„P,)dP, . (10) The eigenstates of Hamiltonian (12) were found

by diagonalization in a 324-term harmonic-os-

1570 JOHN S. HUTCHINSON AND ROBERT E. WYATT

cillator basis with k =,. (Reduction of k' has theeffect of making the eigenstates more classical,thus easing the classical-quantum comparison. '4)

The eigenstates of this system may all be classi-fied as either nondegenerate A symmetry states,or degenerate E symmetry states. The eigenvalueswere checked for convergence against a calcula-tion using a basis of 900 functions, " ' and goodagreement was found for all 98 bound eigenstateswith E& —,

' . The Wigner basis functions were cal-culated over a grid of 2500 (x, P,) points. Thedistributions on the surface of section, W(», P,)were then calculated from Eqs. (8) and (11).

Three distinct types of behavior were identifiedfor the different eigenstates. At low energies(E& 0.075), the Wigner distributions appear eitheras spires located near the (», P,} origin, or as a~all of density along the total energy bound on thesurface of section. A spire indicates that thephase-space density is restricted to low-energyregions in the (x, P,), and is generated by a wavefunction dominated by basis functions with small nand large m in the expansion in Eq. (7). Wavefunctions generating a wall of density in phasespace are dominated by basis functions with largen and small m, indicating a large amount of energyin the x'mode. Both types of behavior are corre-lated to a fixed energy in the x mode of the oscilla-tor, and both are highly nonergodic. Classical tra-jectories at this energy are predominantly regular,so the quantum-classical correspondence is clear.

At intermediate energy (E& 0.140), the phase-space distributions are combinations of brokenwalls and spires, implying that constraints stillexist on the energy transfer between the x and ymodes of the oscillator. These distributions arealso conspicuously nonuniform and thus nonergodic,as are the classical, phase-space densities in thisenergy range.

At high energies (Z & 0.140), many of the eigen-state Wigner distributions are uniform in appear-ance. Although these distributions still containoscillations, the surface of section is covered byshort broad peaks of nearly equal height, andthere are no regions of low density as in the low-er-energy distributions. This is consistent witha free transfer of energy between the modes ofthe system. At high energies, then, we expectthe presence of states characterized by quantumergodicity, just as in the classical case.

Supporting this evidence for ergodicity are theresults of analysis of the separable HH and thenonseparable but nonergodi. c HH systems. In bothcases, the high- and intermediate-energy Wignerdistributions are predominantly walls and spires,similar to the distributions seen at low energy inthe ergodic system. Classical trajectories on

these systems are regular at all energies. Thus,when the coupling between the x and y modes isturned down or off, ergodicity disappears in boththe quantum and classical systems. We concludethat a uniform Wigner distribution is consistentwith an ergodic state, and that quantum-ergodicstates can be generated by a Hamiltonian whichis classically ergodic.

IV. NONSTATIONARY STATE DISTRIBUTIONS

E rgodicity is, by definition, primarily a dynam-ic property. That is, the ergodic problem askswhether an initially localized wave packet willevolve in time to eventually sample the entirephase space in a uniform manner. The stationary-state Wigner distributions at high energy indicatethat ergodic states can be generated initially.Moreover, the results suggest that the energy ofan ergodic state is uniformly distributed amongthe available modes of the system. However, thedevelopment in time of ergodicity requires cal-culating the time evolution of a nonstationary(localized) wave packet.

(13)

If the element of eigenvector transformation ma-trix T, corresponding to the coefficient of theharmonic-oscillator basis state 4„(»)4„(y), forthe l th eigenvector Q, (x, y), is written as T„,,then

(14)

Equating the expressions for q(», y, f } in Eqs. (7)and (13), inserting Eq. (14) for P, , and projectingonto the n, m basis state yields

c„„(t)=g T„,a, (t), (15)

which provides a convenient means to transformbetween the eigenstate expansion and the harmonic-basis expansion. It is easy to show that, if thewave function in Eq. (13) evolves under the Schro-dinger equation, the time dependence of a, (t) isgiven by

a, (t) =a, (0)e 'si'~",

where E, is the Lth eigenenergy of the system.

A. Time&ependent calculations

The time-dependent wave function defined inEq. (7) may be expressed as a superposition of thebound system eigenstates, P, (x, y), by the follow-mg:

QUANTUM ERGODICITY FOR TIME-DEPENDENT %AVE-PACKET. . . 1.571

From Eqs. (15) and (16), it follows that the timeevolution of c„(t) is given by

I

c (t„(.=g „e„,e-'* '&" g I, »'„,.„CN.. (0((.r

(17)

Therefore, the time dependence of @(x,y, t) de-fined in Eq. ('l), can be calculated easily, providedthat we know the eigenvalues E&, the eigenvectormatrix elements T„r, and the wave-function co-efficients at t =0, c„(0). These coefficients aredetermined by the choice of the initial wave packet,which we have taken as a Gaussian in (x, y) withinitial momentum (P„P,}:

11(x,y, 0) Ae»t" *o~ e»~" "0' e'~"~" e'~2"~" . (18)2 2

Here, (xo, yo} is the initial position of the wave-function peak, y, and y, determine the width of thepacket, and A =(2/s)'~'(y, y,)' '. We have usedy, =y, = &00.0 corresponding to a packet width of0.05 in each coordinate; recall that h =

~p for theergodic HH system. The initial conditions of thepacket (xo, yo, P„P,), were usually determinedrandomly, subject to the constraint that the aver-age energy of the packet equal a chosen fixed val-ue. The average energy of C(x, y, 0), which is ofcourse invariant with time, is given by

k2«& =If(x„y„p„p.) +, (y, + y.)

are restricted) to follow the time evolution of 1on the surface of section.

B. Long-time average calculations

Despite the hindrances of the previous para-graph, the Wigner distribution can still be a usefultool for studying the ergodicity of nonstationarywave functions. Specifically, the ergodic hypothes-is requires only that the long-time average of thephase-space density become uniform in time.Thus, classical trajectories are analyzed for thepattern they generate by intersection with the sur-face of section over a long-time period, and not atany particular time. The detail of the time de-pendence is not as important as the knowledge ofwhere the trajectory has been. Quantum mechan-ically then, we need only generate the long-timeaverage of I' (t), and check to see that all regionsof phase space have been sampled uniformly. Ad-ditionally, the long-time averages will show clear-ly the facility of energy transfer in the system.

The long-time average of F is defined by

(I' (x, y, P„P,)),= lim — I' (x, y, P„P„t)dt.~

~

Z~~ T Jp(21)

When Eq. (8} is inserted, the result may be ex-pressed as

(I' (x, y, p„p,))g

FI1 ((Qd ~(d)~

X x(( 3 l1 «0

y2 I y2 yl

where 8 is given in Eq. (12). The coefficientsc„(0)are given by

(19)

where

=PPgT (:.„., c"."'(», ()e'"(Y, p),n m n m'

(22)

c„(0)=I J @(x,y, 0)(jn(x)tm(y)d «dy (20} f'T

V„~,~, = lim--+ ao p

c+ (t)c„„(t)dt . (23)

which may be integrated analytically.In principle then, we have all the factors needed

to calculate the coefficients c„„(t)as a function oftime. These can be inserted into Eq. (8) to giveF as a function of t. However, we must still con-strain our analysis to a surface of section accord-ing to Eq. (11}, and here a problem arises. If thewave-packet center is remote from the y =0 plane,then 4 is extremely small at y =0, and in fact, the324-term expansion for 4 is insufficient for con-vergence at these remote points of the wave pack-et. Similarly, the calculation of W(«, P,) on they =0 surface does not converge properly with thislimited basis, and the error in the expansion for8'is sufficiently large to wash out the interestingfeatures of F on the surface of section. Thus, itis not possible (with the limited basis to which we

Using Eqs. (15) and (16), it is not difficult to showthat

~&r =+r &

V„„„e e = gg T„„,T„e„e,. ag(0)a, e(0), (24)

where the coefficients a, are defined in Eq. (13},and the sum over l' is restricted to' values forwhich E, =E... i.e., l and l' must be degenerateE states for the ergodic HH system. From this equa-tion, it is possible to calculate (I' ), on the constantenergy surface of section by knowing only the in-itial expansion coefficients, a, (0).

We will also calculate the long-time average ofthe configuration-space probability @*(t)+(t),given by

JOHN S. HUTCHINSON AND ROBERT K. WYATT

1 ~r( 4*4(x, y)},= lim —

I 4*(x, y, t)%(x, y, t)dtP ~00 ~ 0

+nmn'm'+n x +n' xtt tg fl 1Ã

x @„*(y)y ~ (y),

where the V„„.i are defined precisely as before.

C. Results

1. Q

Gaussian wave packets defined in Eq. (18) weregenerated with random initial average conditions(x„yo, P„P,) corresponding to fixed-average en- '

ergies of E= 0.050, 0.075, 0.100, 0.125, 0.150,and 0.165. Typically, 20 or 30 packets were runat each energy. Long-time averages of F werethen calculated for these packets evolving underthe ergodic HH system. Generally, the resultantplots of W(x, P,}can be sorted into three categor-ies.

Many of the phase-space distributions appearas either a sharp spire somewhere on the (x, p,)plane [Fig. 1(a}], or as a complete or brokenwall of density [Fig. 1(b)] roughly paralleling thetotal energy boundary. Both of these distributionsmay be easily interpreted in terms of the dynam-ics of the wave packet. A spire indicates that thepacket is passing through roughly the same valueof x with the same energy in the x mode everytime it passes y =0. The time dependence is nearlyperiodic, and the packet is neither breaking apartnor straying from a regular quasiperiodic orbit,as'is clear from the high localization of the den-sity in one region of the surface of section. This

type of motion is clearly nonergodic. A wall ofdensity is generated by a packet passing throughy =0 with fixed x energy but not necessarily at thesame (x, P,) point. This behavior is quasiperiodicand also nonergodic. Energy transfer out of theinitially localized state is not probable for thesepackets.

The majority of the wave packets generate along-time-average signer distribution such asdisplayed in Fig. 2. In these cases, the entiresurface of section has an appreciable density;there are no regions of near zero density as in theearlier distributions. However, one or more re-gions of the phase space show a sharply higherdensity than the rest, appearing as mountain peaksamong the lesser foothills. Thus, we have theintermediate case where all regions of the phasespace (at least on the surface of section) have beensampled, but not in a uniform manner. The dis-tribution is obviously nonergodic and is generatedby a wave packet which develops a tremendousinterference structure as it flows but for which themotion of the average position is quasiperiodic.The quasiperiodicity can be quite complex (asclassical trajectories can be' ), allowing the packetto wander through most of the phase space, butstill favor certain regions. Energy transfer isprobable for such a state, but is not free, in thatthe transfer is restricted to the quasiperiodicityof the motion.

The la,st type of distribution i.s exemplified bythe plots of W'(x, P,) in Fig. 8. These (I' }, aredominated by short, broad peaks which cover theconstant energy surface of section. Although thesedistributions are very oscillatory and do not re-semble the top hat expected for an ergodic state,it is clear that these densities are more nicely

(0)

FIG. 1. Highly nonergodic (p )&.. Perspective picture of the long-time average of W(x, p~) for Gaussian wave packetson the ergodic HH system, vrith (a) (xp, yp, pg, p2) =(0.11, 0.08, -0.18, 0.08), (E) =0.05; (b) (xp pp&pfyp2) ( 0.19, -'0.08,-o.15, o.21}, (s) =o.o75.

T WAyE-PACKET. .ME-DEPENDENT WqUANTT&& ERgOD FOR TI

la)

CP-0)

) for Gauss j wavetjme average o '

) =(p 26,jcture of the "g '

~ p 063 (b) (xp 30pf P2

): Perspe«i p'p p pp, p. pp), &&

e nonergodic) =(0.20, 0 20

IG. 2. Intermedjate «) +0 zopf P2he ergodic HH system, wj) =(0.02, 0.00, 0.05,

packets on

odlcWe thus con-hose discussed above.uniform than those id' at least in ates to be ergo xc,

sense. In fac, w-gect a flat-top en

'xp

zs not a true p robability dens& y,re-the uncertainty prin 'p . 8 ry

' ci le. Berry

f erfect uniformity for ergth tth'4 It is worth s ress'

tofd t b tot b far of this sor o

o l 6d' l d

wave pac e swith E =0.165, on y

signer distribution.an ergodic i

(a)

G ' ackets on theicture o e average of W(x, pf) for GGaussian wave p( e icture of the long-time averagHH system, with (a) (xo gp fergodic HH sys em,

0.30), (E) =0.166.

1574 JOHN S. HUTCHINSON AND ROBERT E. WYATT

smaller at lower energies. Nonetheless, it isclear that quantum ergodicity can develop frominitially localized states and that facile energytransfer is possible.

The three types of( I' ), obey loosely the follow-ing energy trends: The highly nonergodic dis-tributions are predominant at low energy (E& 0.075) where the coupling in the potential has asmall effect. About 10%%up of the packets at theseenergies do display the intermediate behavior.At moderate energies (0.075& Z &0.150), theWigner distributions are nearly all (roughly 3 outof 4) the intermediate type. There are a lessernumber of the highly nonergodic densities. Er-godicity is still not seen in this energy range.At the high energies above the quantum "criticalenergy" (E) 0.150), some ergodic distributionsdo appear, as noted before. Again, however, theintermediate behavior is far more predominant,even at the very high energy E =0.165. Less than10% of the distributions generated at high energyare the highly nonergodic type. Thus, there ex-ists a noticeable but slight trend towards ergod-icity with increasing energy. At least quantumergodicity is restricted to high energy, in ac--cordance with the KAM theory and the classicalresults, and the extent of nonergodic behaviordiminishes with increased energy. However, itis clear that ergodicity is far less significantfor the quantum-mechanical system, and thatquantum ergodicity requires a higher criticalenergy than the classical E, .

Wave packets were generated and long-time-average Wigner distributions calculated, for theseparable HH system, for which no ergodicity isexpected. At all energies studied on this surface,the phase-space densities are walls or spires,

as seen in Fig. 4, corresponding to the low-ener-gy behavior of the ergodic system. The packetsare thus quasiperiodic and nonergodic, even athigh energy. We conclude that uniform distribu-tions are not possible without the mode couplingterm in Hamiltonian (12), just as ergodic traject-ories are not possible in this case.

Wave packets were finally generated on the non-ergodic HH system w ith fixed-average energiesof 8 =4.0, 8.0, 11.0, and (I'„), was calculatedfor each. (Recall that ff is taken to be 1.0 for thissystem. ) Once again, the plots of W(x, p, ) in Fig.5 are dominated by walls or spires at all energies.Since no uniform Wigner densities and no ergod-ic-classical trajectories are found, we concludethat the classical-quantum correspondence isupheld. On the systems for which there is littleor no mode coupling and thus no irregular classi-cal motion and no ergodic trajectories, uniformphase-space distributions are not generatedquantum mechanically. When the system can pro-duce classical ergodicity at high energy, ergodicWigner distributions are also possible in thelong-time average.

2. &4*4&,

The majority of authors previously studyingquantum ergodicity have restricted their analysis,to the configuration space of the system. ' " Theargument is then that the wave-function density4*(x,y)4(g, y) should become uniform in the long-time limit for an ergodic state, or at least thata trend toward uniformity may create irregular-ity in the structure of the density. The ease ofcalculation of 0 ~4, plus its familiar interpreta-tion as the system probability dens'ity at position

FIG. 4. Nonergodic separable (P~)&. Perspective picture of the long-time average of W(x, p~) for Gaussian wavepackets on the separable HH system with (a) (xo, yo, pt, pz) =(-0.16, -0.01, 0.16, 0.33), (E) =0.100; (b) (xo, yo pt pt)= (-0.03, 0.29, 0.20, 0.37), (E) =0.150.

QUANTUM ERGODICITY FOR, TIME-DEPENDENT %AVE-PACKET. . . 1575

(b

FIG. 5. Nonergodic (P~}&. Perspective picture of the long-time average of W(x, pt) for Gaussian wave packets on thenonergodic HH system with (a) (xo, yo, PI iP&) =(-0.82, 2.58, 0 47, 0.72), (E}=8.00 (5) (xp yp Pt Pt)= (2.52, —0.76 3.56,1.39), (E}= 11.00.

(x, y) make such an analysis highly attractive.This property has been analyzed by wave-functionnodal patterns" "as well as other techniques.We have studied the eigenstate nodal patterns forthe ergodic HH system with little informativesuccess. We were unable to identify consistentpatterns of irregularity in the wave-functionnodal lines, nor could we see any relation to ourearlier phase-space analysis. In addition, it hasbeen pointed out that it is possible to devise sep-arable (hence, nonergodic) systems with ii regu-lar nodal patterns. "

The relevance of a uniform configuration-spacedensity to the classical-ergodic problem issomewhat unclear. For example, it is wellknown that it'is difficult to label a trajectory asergodic or nonergodic by following its path inposition space, but a look at the surface of sectionin phase space reveals its nature clearly. Moresignificantly, it is not hard to show that a uniformdensity 4'~4 will not produce a uniform Wignerdistribution F„, nor is the reverse implicationcorrect. " If 4(x) is defined to be constant overthe range ~x~ ~ L, then the Wigner distribution isgiven by the wildly oscillatory function

(,p)-— 2P/(L-x)/~

p

Inversion of a constant I'„gives a similar (sinx)/g function for 4*%.

Thus, in order to understand more fully therelationship between 4*4 and I', we have calcu-lated the long-time average of the wave-functiondensity (4*4'), for the wave packets generatedpreviously. In general, the results may be quali-tativeI. y sorted into two types. In the first type,the density is peaked along a knife edge and falls

off either side slowly, as depicted in Fig. 6(a).This may be interpreted by thinking of a wavepacket "sloshing" across the potential surface ina periodic manner. The packet spreads to eitherside of the main line of motion, but the densitypeak remains along the l.ine, building an edge.This type of packet, obviously, generates a sharp-ly peaked ( I' ), and is thus always nonergodic.

The second' type of (4*4'), plot is exemplified byFig. 6(b). Here, the wave packet has developed atriangular density with three peaks. Interestingly,this type of density is produced by a large numberof different wave packets, particularly those atmoderate and high energy, and cannot be inter-preted in terms of one simple mode of dynamicbehavior. In fact, this pattern of (4*4), is notassociated with any one of the previously discussedpatterns of (I' ), . The corresponding Wignerdistributions may be uniform, sharply spiked, orof the intermediate type; there is no evident re-lationship between the two densities. Therefore(4 ~4), is not a sensitive indicator for quantumergodicity. Moreover, we @ave found no indicationof uniformity in the suave function dens-ity, evenfor Packets with uniform (I' ), . [ The plots of(I' ), in Fig. 3(a) and (4*%), in Fig. 6(b) are forthe same ergodic wave packet. ] The closest touniformity we could find were for those whoseknife edge was not sharply peaked, and these wavepackets were already seen to be nonergodic as de-termined by (I' ), . Thus, uniformity of (4~4 ),is also not an accurate or relevant test for ergod-icity.

We conclude that it is insufficient to study thenature of the wave-packet dynamics in configura-tion space to determine its ergodic properties.Uniformity of (4*4), doke not imply ergodicity

1576 JOHN S. HUTCHINSON AND ROBERT K. WYATT

(a)

FIG. 6. Perspective picture of (4 0)&, the long-time average of the configuration-space wave-function density, forwave packets on the ergodic HH system, and (a) (xo, yo, pq, pz) =(0.50, 0.00, 0.10, 0.10), (E) =0.111; (b) (xo, yo, P, ,p, )=(-0.27, 0.22, -0.15, 0.37), (E) =0.150 [same initial conditions as in Fig. 3(a)i.

or vice versa. Our results indicate that only bygoing beyond configuration-space analysis (e.g. ,by analyzing Wigner distributions) can one predictergodicity.

Finally, we would like to point out that we havenumerically tested the argument by Heller" on.the impossibility of ergodieity for degeneratequantum systems. Using arguments based ongroup theory, he proved that a wave packet willspend more of its time in the vicinity of its orig-inal position than in the vicinity of a symmetrical-ly equivalent position. He has contended that thisdifference could be a factor of 2. We have cal-culated the transition probabilities suggested byHeller and have found his predictions to be quiteaccurate. It might be expected that this effectwould also produce a very nonuniform (4*4),.Since the ergodie HH potential is symmetric in y,then a packet started with y, o 0 might be expectedto produce a (4*%), considerably greater at y thanat -y. However, this is not observed in our cal-culations; in fact our (4*4'), are symmetric (towithin 10%%uc) about the y axis. Thus, the influenceof degeneracy on quantum ergodicity might notbe as significant as previously believed.

D. Stability of wave-packet dynamics

One extremely important feature of ergodic-classical trajectories is the instability of the dy-namics under a small perturbation. ' This proper-ty has been. fundamental to proving and quantifyingergodicity in model physical systems, and hasbeen used to predict the critical energy of theKAM transition. "'"'" Specifically, if one re-

moves an initial trajectory point from itself by asmall distance, then for an ergodic trajectory,the distance from the perturbed path to the actualtrajectory increases exponentially in time, atleast for relatively short times. "Exponentiatingtrajectories" have been studied in great detail fora wide range of systems as an indication of theproperty of "mixing, " which in turn implies er-godicity. '

Our hope would be to establish a similar proper-ty of quantum wave packets; that is, we might ex-pect wave packets which generate uniform ( I'„),to exhibit some instability in their motion. Themost immediately obvious notion is to considertwo narrow wave packets, alike in size and shapebut slightly separated, and to follow the time de-pendence of the overlap I(4', (t)I 4, (t)& I'. How-

ever, both functions may be expanded in the sys-tem eigenfunction basis

4', (t)=pa, (t)Q, =Pa, (0)e 'st ' "p

so that the overlap between the two becomes timeinvaxian~:

(27)

This surprising result renders the overlap oftwo packets useless as a measure of stability.

Another technique, obviously analogous to the

. QUANTUM KRCODICITY FOR TIME-DEPENDENT %AVE-PACKET. . .

study of classical trajectory. exponentiation, is tofollow the time dependence of the average phase-space position of the wave packet. It is ratherconvenient to calculate the average position inphase space for the wave packet expressed as asuperposition of harmonic-oscillator states. Wecan find ((x), (y), (p, ), (p, ) ) at each time fortwo wave packets, initially separated slightly,but with the same average energy, so it shouldbe possible to follow the distance in phase spacebetween the averages as a function of time. Un-fortunately, for wave packets at sufficiently highenergy that short-time exponentiation is expected,the phase-space density rapidly spreads and dis-sipates. The average position in phase spacethen spirals toward the origin for both wavepackets, and the distance between the averagesshrinks rather than grows. This result does notimply ergodicity or nonergodicity of the wavepackets; it is simply an indication of the spread-ing of the wave packets at high energy. Clearly,the most intuitively obvious methods to study in-stability in the quantum dynamics are not viable.We have, however, studied two other dynamicproperties of interest.

1. Separation of wave packets

The dynamic instability of ergodic-classicaltrajectories is usually studied by slightly dis-placing two initial states from one another andfollowing the separation with time. An equivalenttechnique of study is to follow the time dependenceof one trajectory, and then, taking the sameinitial conditions (with P, adjusted to fix the totalenergy), follow. the time dependence of the tra-jectory evolving under a slightly perturbed poten-tial. The distance between these two trajectorieswill also initially exponentiate in time for ergodictrajectories. This sensitivity to the potentialperturbation is evident even at very small per-turbations, although the time at which exponen-tiation first begins is shorter for larger per-turbations.

For quantum wave packets, the correspondinganalysis consists of evolving the same initialpacket on two potential surfaces differing by asmall perturbation, and following the overlapwith time. Thus, if the packet on the unperturbedsurface is given by

(28)m

and the same wave packet evolving on the per-turbed surface is given by

4"(t) = QQc' (t)4'„(x)@ (y),

then the overlap is given by

(e(t) ~e (t)) = gg c+„(t)c„(t), (30)

where the time dependence of c (t) is given byEg. (1V). Of course, in Eq. (1V), the elementsof the transformation matrix T and the eigen-values E, are different for the two wave packets.

Actually, we have calculated a separation mea-sure D(t) defined by

so that D is initially zero and remains at zerofor zero perturbation. D can grow to a maximumlimit of 1.0 in the extreme case that 4(t) and@'(t) become orthogonal. Thus, increasing D(t)is an indication of separation of the two packets.Our choice of perturbation is to multiply bothA. and p in Eg, (12) by a number (1+a), wheren is a number taken to be less than 0.01. Thefirst result that is easily found (e.g. , by usingperturbation theory) is that reducing a increasesproportionally the time scale over which D(t)grows; i.e., if + is reduced from 0.01 to 0.001,then the time scale of the growth of D(t) increasesby tenfold, but the shape of the D(t) versus t plotis unchanged. Therefore, we can focus on onevalue of e only, taken to be +=0.01.

D(t) versus time was calculated for each of thewave packets generated earlier on the HH system.The results as a function of energy are shownfor three different wave packets in Fig. V(a).Rather evidently, the final extent of separationof the perturbed packets increases with energy,as does the rate of growth toward that final sep-aration. At E =0.165, D(t) rises rapidly to a valuenear 1.0 and remains nearly that large for theremainder of the time.

However, this behavior does not appear to bean indication of unusual instability at high en-ergies. We reach this conclusion for severalreasons. First, the trend with increasing en-ergy is slow and steady. Classically, the tran-sition from regular to ergodic behavior occursover a short energy range and begins abruptly.There is no such sudden leap in the D(t) pl'otswhen energy is increased. Second, as mentionedearlier, the rate of growth of D(t) is very sensitiveto the strength of the perturbation, unlike theclassical result, in which exponential growth isfound even for o. =10 '. Moreover, the time scaleover which separation occurs is a factor of tengreater in the quantum case. Third, and mostimportantly, the qualitative features of the D(t)versus t plots do not depend strongly on the choiceof initial conditions. In Fig. V(b), we show D(t)versus time for three wave packets at E=0.165

1578 JOHN S. HUTCHINSON AND ROBERT E. WYATT

" I 00D(t)

I 000.TIME (a.u3

(a)

I

2000.

we are calculating. A final possible conclusionis that-all of the wave packets are displayingstability, yet some are still ergodic. Typically,instability of trajectories implies the propertyof mixing, which in turn indicates ergodicity.However, trajectories can be ergodic and not

mixing and may therefore be ergodic and stable. 'Similarly, it is possible that the wave packetsstudied here are dynamically stable, but prop-agate and spread to cover the entire phase spaceuniformly over a long-time period. These pos-sibilities are discussed further in the conclusion.

-. I.OO2. Survival of wave packets

1000.T I ME (a.u3

2000.

but with different initial conditions. It is im-mediately seen that the short-time growth andthe long-time-average separation are not sig-nificantly affected by the change in initial con-ditions. However, the surprising result is thatthese three choices of initial conditions generateone each of the three distinct type of (1'g, plots,as labeled in Fig. 7(b). Thus, the complete rangefrom nonergodicity to ergodicity is present inFig. 7(b), despite the qualitative similarities ofthe curves.

Several different conclusions may be deducedfrom this result. The first possibility is thatnone of the wave packets are unstable and there-fore none are ergodic. %e reject this conclusionas inconsistent with the %igner distribution re-sults. A second conclusion could be that D(f) doesnot really measure instability. This remains avalid possibility. In fact, our earlier results for(4"%), indicate that restricting the analysis toconfiguration space is insufficient. The stabilityof the nonergodic wave packets may be moreevident in a full phase-space density, and theinstability may be an artifact of the space inwhich

FIG. 7. D(t) versus time for Gaussian wave packetson the ergodic HH system: (a) as a function of increasingaverage energy; (b) as a function of initial conditions andcharacter at fixed energy, (E) = 0.165; N, nonergodic(I' )t; I, intermediate (I' )&, E, ergodic (I' ),.

A frequently studied property is the survivalof an initial state defined by

S(f) = ~(@(o)~+(~)& ~'.

The survival thus measures how much of the wavepacket remains in the vicinity of the initial posi-tion. By insertion of the expansion for 4(t) inEqs. (13) and (16), S(f) is given by

S(f) = gg (n, (0) ('~n, (0) ~'e-'" ""'". (88)

Therefore, S(t) is completely determined by thespectrum of frequencies generated by the systempotential and by the "power spectrum" of theinitial wave packet. Brumer and Shapiro'~ studiedS(t) for wave functions on two different surfaces,only one of which is classically ergodic, and didnot find any difference in the qualitative behaviorof the two systems.

S(t) was calculated as a function of time for thewave packets generated previously on the ergodicHH system. Two types of dramatically differingresults were found. Many of the wave packetsshow a very regular, highly periodic structurein S(f), as exemplified by Fig. 8(a). These plotsshow clearly a fundamental frequency independentof energy, and several envelope structures. Theremainder of the wave packets show a very ir-regular, nonperiodic, patternless structure,lacking any clear "signature, " just as found byBrumer and Shapiro [see Fig. 8(b)). Generally,the regular survivals can be found exclusivelyat low energies, decreasing in predominance asenergy is increased, until the high-energy packetsare dominated by irregular behavior.

One is highly inclined initially to label thisas an ergodic transition, with the irregular sur-vival plots indicating ergodic wave packets. How-ever, further analysis shows that the situationis more complex. First, the wave packets withirregular survival plots do not necessarily show

QUANTUM KRGOOICITY FOR TIME-DEPENDENT %AVE-PACKET. . . 1579

"0.50S(t) ..

(0)

.L., L t. I L L. II ll 6 L I. .L {II ~ A Jl Ia IJ J 6)II.GII 4 Ulled

IOO. 200.TIME (a.u.)

"030

IOO. 200TIMF (a.u.)

FIG. 8. S{t)versus time for Gaussian wave packets onthe ergodic HH system: {a) (xp, yp, pg, p2) = {-0.35, -0.03,0 12 0 26) {E)=0 165' {b}{Xp go pg p2) ={ 0 06 0 26-0.12, 0.46), (E) =0.165.

uniform (I'„),densities. In fact, it is quite pos-sible to find wave packets showing the totallynonergodic types of distribution, yet having irre-gular S(t). Second, we have calculated S(t) forour wave packets on the separable HH system.Recall that all (I'„),plots indicate nonergodicityof all wave packets for this system. Surprisingly,however, irregular survival plots are still foundfor a few high-energy packets, even without themode coupling in the potential. Third, we haveanalyzed S(f) for the wave packets on the non-ergodic HH system, and have again found someirregularity, even though all wave packets onthis system are nonergodic in the%igner analysis.

To assist the interpretation of these results,we have calculated the dynamics for "swarms"of classical trajectories initially centered atthe average initial position of each wave packet.For a set of classically unstable initial conditions,the trajectory swarm may be seen in Fig. 9 tospread throughout the energy allowed configura-tion space in a short time. Over the same period,a stable trajectory swarm will remain coalesced,as in Fig. 10.

We have found that all wave packets with regularstructure in the survival plot begin with averageinitial conditions within the classically stableregions of phase space. In addition, most (butdefinitely not all) wave packets which produceirregular S(t) begin with classically unstableinitial conditions. The exceptions are those

O,O

FIG. 9. Time dependence of an unstable trajectory swarm. Eighty-one trajectories with average energy E = 0.150 areshown in the (x, y) plane. Limits on the x axis are -1.00» x» 1.00. Limits of the y axis are -0.75» y» 0.75. Solidcurve in each plot is the total energy boundary in the (x,y) plane.

15801

JOHN S. HUTCHINSON AND R'OBERT E. WYATT

t =0.0 t = &0.0 t = 20.0

t = 30.0 = 40.0 = 50.0

FIG. 10. Time dependence of a stable trajectory swarm with average energy E=0.100. Axes defined as in Fig 9~ ~

wave packets initially near the border betweenthe stable and unstable regions of phase space.The transition across this border is seen byanalyzing a set of initial conditions in a stableregion and then smoothly varying the initial con-ditions into an unstable region. At the fringe ofthe stable region, unstable trajectories beginto appear. Irregular wave packets, however,actually appear with initial conditions just insidethe classical stability boundary. There are thuswave packets with irregular S(t) plots which haveclassically stable initial conditions; this is alsoevident in our stated results on the separableHH and nonergodic HH systems. However, any-time we choose conditions which are classicallyunstable, the survival plot of the correspondingwave packet is always irregular.

We can also correlate these results with ourearlier results on the long-time average-Wignerdistributions (I' ),. As mentioned above, irre-gular survival structure is not unique to eitherergodic or nonergodic wave packets. However, itis definite that a wave packet generating an ergo-dic (I ), will show an irregular survival plot.Moreover, such a packet always has initial con-ditions which are classically unstable. Conver-sely, a wave packet with a regular survival willalways generate a nonergodic (I'g, . Virtuallyall nonergodic wave packets are initially centeredin a classically stable region.

All of these relationships can be summarizedconcisely as showt) in the chart in Fig. 11. Hereme have displayed the correlations among the

three types of Wigner distributions, the twotypes of survival plots, and the two types of,trajectory dynamics. These relationships arebased on a study of 20 random wave packets ateach of three average energies (E=0.10, 0.14,0.15) for which (I' ), and S(t) were plotted. A

swarm of 81 trajectories clustered around the

Quantum

~~w&V/

Classical

AGO-e stable

~0~~

qi 6, ~i regular

~64 ~~ /~i'8

irlter~ediateir r egular

4'

ergodic ~——————————————. unstable

: requir ed

————-+ allowed, not required

FIG. 11. Correlation char& between classical and quan-tum behavior. Relationships are given between ( I'~)&and S(t) for quantum wave packets and the classical tra-jectory swarms with the same initial conditions.

QUANTUM ERGODICITY FOR TIME-DEPENDENT WAVE-PACKET. . . 1581

wave-packet initial conditions (with width 0.02in each phase-space coordinate and with P, ad-justed to match the quantum and classical ener-gies) was also calculated for each set of initialconditions. Correlations are shown as follows:If the initial conditions of the wave packet ortrajectory swarm possess a particular property(i.e., an ergodic (I' ),), then a solid arrow indi-cates that such conditions must necessarily leadto the property at the end of the arrow i.e., foran ergodic (I' }„anirregular S(t) and unstabletrajectory swarms are a1so required . A dashedarrow indicates that the property at the head of thearrow is allowed but not required by the propertyat the tail of the arrow. These correlations areweaker than those indicated by solid arrows. Ina few cases, these arrows are annotated as tothe extent of the correlation. For example, nearlyall (but not all) classically stable initial conditionsproduce a wave packet with a regular S(t). Veryfew of these conditions produce an irregular sur-vival. If a single set of initial conditions nevershare two properties, these properties are not.joined by any arrow. Thus, ergodic wave packetsnever origniate in a classically stable region.

The chart in Fig. 11 is a rather comp1ete sum-mary of the relationships between classical andquantum ergodicity; the influence of the corre-spondence principle on the quantum-ergodic prob-lem is clearly seen. Quantum ergodicity onlyappears for wave packets initiated in regions ofphase space for which classical ergodicity isfound. Moreover, both quantum ergodicity andclassica1 instability require quantum stochasticityin wave packets, as evidenced in irregular wave-packet survival. Quantum nonergodicity is al-most completely restricted to classically non-ergodic conditions. A lack of quantum stochastic-ity in a wave packet is a guarantee of both quantumand classical nonergodicity. Thus, we concludethat the KAM behavior does appear to be relevantto quantum-mechanical dynamics.

Note from the chart that our results are en-tirely in accord with those found by Brumer andShapiro: It is quite possible to find structure-less, irregular survival plots for both ergodic andnonergodic systems. We conclude, however, thatthis is not an indication of nonergodicity of wavepackets on both systems. The impact of the Brum-er and Shapiro study on our results will be dis-cussed further in the Conclusions.

V. CONCLUSIONS

We have been primarily concerned with therelevance of KAM theory to intramolecular energy

transfer. The ergodic transition predicted by theKAM theorem for classical-mechanical systemsis a rigorous result, supported by the great bulkof classical trajectory studies. However, we

also know that molecules obey quantum mechanics,so that an ergodic transition in molecular sys-tems is neither required nor prohibited by theKAM theory. Moreover, in order to study KAM

behavior in a quantum system, we believe it isnecessary to study the quantum states in the samemanner as the classical trajectories, i.e. , in aphase-space analysis. Our calculations on theWigner phase-space density for both stationaryand nonstationary states indicate that a gradualtransition with increasing energy does appear forsystems which are classically ergodic. Thus, itappears that a correspondence does exist betweenclassical and quantum ergodicity and that theKAM theorem is thus indeed relevant to the prob-Lem of intramolecular energy transfer. However,our results also point to two other conclusions:First, the critical energy of the transition ishigher for the quantum systems than for the cor-responding classical systems. Second, the extentof ergodicity is not nearly as great for the quantum

system.These conclusions are in good agreement with

the study of Kay, ' who has analyzed transitionprobabilities in a coupled anharmonic-. oscillatorsystem. He has shown the existence of quantum

ergodicity in this system, and has also discoveredthe lesser extent of such ergodicity compared tothe classical results. It appears, then, that clas-sical trajectory studies may be overly optimisticin their predictions of high-energy ergodicity.

Our conclusions are not in accord with the re-sults of Kosloff and Rice." ' The prohibitionof quantum chaotic behavior as proven by Kos-loff and Rice for bound systems is equally rig-orous, at least for their formulation of theK entropy. It is apparently undeniable that

quantum states do recur periodically. However,the same recurrence was shown for classicalsystems by Poincare, "yet chaotic behavior isallowed nonetheless for classical trajectory en-sembles. This is because the recurrence timesdiffer for each trajectory, so that at a particulartime only a few trajectories are near their initialconditions. Similarly, Kosloff and Rice defineK entropy in terms of the partitioning of a singlequantum state, rather than of an ensemble ofstates. Moreover, the recurrence of a quantumstate says nothing about its behavior over its per-iod or the facility of energy transfer. Indeed,Heller and co-workers have pointed out that theKosloff and Rice formulation of the K entropy isnot at all sensitive to the stochasticity of the

1582 JOHN S. HIj TCHINSON AND ROBERT E. WYATT

wave-packet dynamics. Our results show that thetransition in (1 ), for quantum states is unique toclassically ergodic systems, and that a uniform(1' ), is only possible for classically ergodic initialconditions. These facts are hard to overlook.Finally, Kosloff and Rice really seek only toprove the lack of chaotic behavior in discretebound systems. While the lack of chaos in phase-space dynamics denies the property of mixing,ergodicity is allowed nonetheless. Recall thatour D(t) calculations gave no indication of any-thing chaotic in the configuration-space wave-packet dynamics. It is possible that we are ob-serving ergodic but nonmixing evolution. Westress that this possibility is tentative: A betteranalysis of the stability properties of the wavepackets must await a time-dependent calculationof 1'„(t), so that the stability may be analyzed inthe full phase space. As discussed earlier, thiscalculation will be time consuming and diff'. cult.

It remains to reconcile our conclusions with thenumerica1 results and conclusions of Brumer andShapiro. Specifically, how can we fit their calcu-lations into the correlation chart in Fig. 11& Wehave already seen that the irregularity they ob-served for wave packets on both ergodic and non-ergodic systems is allowed but not required byFig. 11. The question is why they did not observeany regularity in the nonergodic system. Hellerand co-workers' have very observantly notedthat this result of Brumer and Shapiro is due en-tirely to the choice of the initial wave packet.Specifically, Brumer and Shapiro define the initialstate as an expansion over the system eigenfunc-tions, as in Eq. (13), but with expansion coeffi-cients given by a Gaussian in the energy

ergodicity from initially nonergodic packets onthe separable HH system, the conclusion must bethat the Brumer and Shapiro wave packets areergodic initially. Thus, it is not surprising thatthey did not observe regular survival plots on thenonergodic surface. We conclude that, thesestudies notwithstanding, quantum ergodicity isindeed possible. In fact, it is our belief that,when one analyzes quantum states in terms ofphase-space densities, the close correspondencebetween classical and quantum ergodicity is quiteclear.

ACKNOWLEDGMENTS

This work was supported in part by grants fromthe National Science Foundation and the Robert A.Welch Foundation.

4„(x)=Z„e ~"21f„(~"x) (Al)

where N„ is a normalization constant given by

(3n f )-1/2(&/&)i/4 (A2)

APPENDIX: CALCULATION OF 0"„"(~~, )

The Wigner basis functions defined in Eq. (9)may be calculated analytically for harmonic-oscil-lator basis states. We write the nth harmonic-oscillator state as

n(E, E,)2'l

Iei(» I'= —„2 expII-

I

where E, is approximately the average energy ofthe packet &=ln2, and 6 measures the energybandwidth of the packet. However, such a packethas been shown to be initially delocalized in con-figuration space. ' More importantly, Hellerhas argued that any wave packet with smoothlyvarying coefficients as in Eq. (34) will have auniform phase-space distribution indePendent ofthe dynamics.

We have calculated (1 ), for wave packets, de-fined by Eqs. (13) and (34), on both the ergodicand separable HH systems. Our results showthat, exactly as predicted, the wave packetschosen by Brumer and Shapiro are ergodic evenon the separable HH system. Since we have al-ready shown that it is not possible to develop

Q = IB(0~/8. (A3)

The choice of m and v, is determined by thezeroth-order potential. H„(x) is the nth Hermitepolynomial, given by

a„(x)= Q a!"'x' (A4)

f& oo

+"" (x p ) = 4~(x+e)4 „,(x —e)e"»'~" de .d~oo

(A5)

We insert Eqs. (A1) and (A4) directly, and applythe binomial expansion for the (x+z)' terms toarrive at

and the a,.'"' are the known Hermite coefficients. "According to Eq. (9), the Wigner basis func-

tion 4"" is given by

QUANTUM ERGODICITY FOR TIME-DEPENDENT %AVE-PACKET. . . 1583

(A6)

I1t

y+(~+a@ (& s) N N e-nx e-o& &(n&u(n')n&i+J)/a ( 1))&i+i a-)-su+i'? n' n n'

i=o i=o I-0=o &=o s %IP y gC

'The necessary integral is finished by multiplying by e"t}-2' " and completing the square in the argument ofthe exponent:

Qnn'(g p) N N e- x e-'/ ~ g&n)g&n')n(i+9)/2tO f 1

i=o i=o.

~+oo

x! Qg (-1}'x'"'' ' s"'exp —'n' 's —,/,' dz!.

(AV)

To evaluate this integral, let u= n' 's —ip, /n' 'k:

s'+)e~ in e p.~

ds —n(a++ &/ap . &&' pl)) . ua+) )eu-du (A8)

This is a known integral, which vanishes if (k+1 —t) is odd, but which has the value for even (k+1 —t}:

1 8 5 (k+f-t-1)2(0+2- t) /2 (A9)

We summarize the resultant expression for +"" as follows:

& p) —1N N e "e' ~ a'"'a'"'&"(x p 1

i=o i=o

r ~ ~rjqp ) n((+~)/2~(+~QQ ( 1)) f/k+)(~ p)

y=o l=o

( ~PU'(~ p )=~-'n-"""'g~„, 1(e-t),

t=o

(A10)

0, m odd

I(m)= 1 8 ~ 5 ~ ~ ~ (m-1)!

. v )&', m even

*Present address: Department of Chemistry, Universityof Denver, Denver, . Colorado 80208.

I. Oref and B. S. Rabinovitch, Acc. Chem. Res. 12, 166(1979).

M. Tabor, Adv. Chem. Phys. (in press).P. Brumer, Adv. Chem. Phys. (in press).S. A. Rice,

'in Advances in Laser Chemistry, edited by

A. Zewail (Springer, New York, 1978), p. 2.W. L. Hase, in Dynamics of Molecular Collisions B,edited by W. H. Miller (Plenum, New York, 1976), p.121.

M. V. Berry, in ToPics in Non-Linear Dynamics, AIPConference Proceedings No. 46', edited by S. Jorna(American Institute of Physics, New York, 1978), p.

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