Quantum dynamics in classically integrable and non-integrable regions

Volume

76. number

QUANTUM

1

CHEMICAL

DYNAMICS

Michael J. DAVIS.

IN CLASSICALLY

Ellen B. STECHEL

PHYSICS

LETTERS

INTEGRABLE

and Eric J. HELLER

15 Nobember 1980

AND NON-INTEGRABLE

REGIONS *

*;%

Department of Chemrst~~, Unwernty of Cdtfomia. Los Angeles. Cidtforma 9002-1 USA Received

1 July 1980. III fmal form 8 September

The tune evolution

of wavefunctIons@

1980

composed

of superposltlons

of energy elgenstates are compared

when the wave-

functtonsare untlrlted UI (1) the non-mtegrable and (2) the mtegable regunes m the phase space of the Henon-Hedes The behavror oFP(t) = IWlQ(t))l’ IS found to be very difierent III the t\\o regions

Recently, whxzh

ccrtam arguments have been put forward

are trended

as evidence

for significant

qu&tative lfferences between quantum and classIcal dynamics, especially as regards the non-stochastic and stochastic regmes ofclassical motion These arguments by Brumer and Shapiro [ 1J and Kosloff and Rice [2], are rather sweepmg tn their unphcatlons III that they would appear to leave very httle room for the quantum-classical correspondence pnnclple, or for rehance on trajectory data as approxunate measures for molecular stochastlclty. Kosloff and Rice [2] have attempted a quantum generahzation of the Kolmolgorov entropy [3]. which is a well founded measure of the tendency rewards stochasticlty WI classical systems. Their conclusion IS that tlus entropy in a quantum system IS always zero for hanultomans with a &screte spectrum Thus, for any finite fi, there would not be any quahtatlve change in the quantum Kolmolgorov entropy in those bound systems which are known classically to undergo a trans:tlon to stochastic motion as the energy or the anharmoniclty IS Increased. Brumer and Shapiro [l] have exammed the quantum fimchon P(t) = I($l~#~(t))l~ where Q is an uutlal wavefunction (see below) for a separable, classically integrable system and a classically non-mtegrable one. They found no sigruficant lfference m this function in the two cases. Suppose the entropy measure defined by ref. [2] * Work -*

Alfred

supported by NSF Grant CHE77-13305 P Sloan Foundation FeUon,CnmiUe&

Teacher-Scholar

Henry DreyFus

system

and the measure P(r) (with a particular

9) used by

ref.

the known

[ 1] have

physlcal

relevance_

Then.

tendency towards increased chsstcai stochastlcity at higher energies would have little beanng on actual experunents. since the quantum measures of refs. [ 1,2J farled to respond to classIca stochasticlty. Recent researchin thalaboratory [4,5] haspointed to the exact opposite conclusion. We have found very strong correlations between quantum spectra, wavefunctions, energy transfer. and class:& motion. The correspondence is so close we have been able to extract specific spectral features. includmg intensities, bandwidths, and splittings, using trajectory data as mput [4] _ It is possible to do this in both the integrable and non-integrable regimes. Our spectra, which differ quahtatlvely in the two regmes, arise as the Fourier transform of (Qi@((r)) (see below). Stnce the spectra do differ, so must their Fourier transform. in this light, the result of ref. [ 11 that there is not any difference in the function p(t) in the two regimes should seem especially puzAing. Other work has also mdlcated a close connection between trajectones and quantum mechanics. For example, plots of probabihty dennty in coordinate space for specific wavefunctlons and time averages of classical trajectones look strikmgly simdar [4,6]. In phase space, the Wigner transform of specific wavefunctions look very much like their classical counterparts m Poincari surfaces of section [7] _ Furthermore, for come tie, trajectories have been used to generate quantum elgenvalues [6,8], even in the non-integrable

regime [9,10]. 21

Volume 76. number 1 WMe we certamly

CHEMICAL PHYSICS LE’I-rERS do not deny that m terestmg &f-

ferences exist between classical and quantum dynamics (indeed we have searched for and found such differences [ II], and continue to search for others), we beheve the results of refs. [1.2] are quite maleadmg. This paper presents additional evidence to support the quanrum-classical correspondence and to questlon the relevance of the conclusions of refs [ I,?]. We begin with a simple argument based on Ehrenfest.s theorem [ 121. Imagine that we start wlrh a wavepacket Q on a smooth but non-separable potential Furface Ehrenfest’s theorem ensures us that for a time the average posItion and momentum of the packet are very close to the classical values. and that the dapersion about the classical mean IS small. By making fi sufficiently small. we can keep a wavepacket pmpomt locahzed about the exact classical path for thousands or brlhons of vibrational periods If we hke. What, then. IS the relevance of measures whch do not sense whether the pinpoint packet IS undergomg non-mtegrable or integrable motion? If the measures provide no hint of the stochasticity m thus physically clear htmg case, can they be used for the more subtle problems mvolvmg reahstx parameters? Given the above argument concernmg the necessary classica&like dynamics of a wavepzcket 9(f) for small h. why XI rzf. [I] 15 ther.z no essential Afference behveen the stochastic and non-stochasuc hamdtomans III the quantity P(t)? There 1s a sunple answer to tlus question havmg to do \wth an unfortunate choice for the state Q Denotmg the eigenstates of the hamrltoruan by {$J~) and the eigenenerges by {E,] we have

9=

Fantin,

O(t) = C an exp(-lE, n

f)G, .

(1)

Rather than speclfymg @ on the basis of some physlcalIy relevant imtial con&tion and determming the a,, the a, of ref [2] were instead constramed to follow a gausslan Istnbuuonla,12 0: exp[-ar(E,

- Ec)?/A2

]

(3)

Such smooth &srnbuhons of the an, encompassing many states, were explicitly shown in ref. [ 131 to correspond to a priori stochastic irzrtiai condrhons That IS, if one wants a wavefunctlon which is as spread out as possible over the phase space of the system, and wluch ~IU, indeperldentof the dynamics, tune-average to a uniform phase space distnbution, then a simple 22

argument

15 November 1980

shows that eq. (2) 1s a good choice

[ 131

This IS true provided that many states are encompassed under the bandlvldth of the gausslan envelope, as was the case for ref. [2] The classical analog of this situation 1s to have a mlcrocanomcal cbstrlbutlon as an initial test density to determme classIcal stochasticlty. However, being spread out uniformly, a microcanorucal density does not provide any information whatever on flow between regons of phase space. and would also faA to show any quahtatne difference m the mtnnsic dynamics To check this Idea, we have taken the gaussian dlstnbutlon of am lrtudes slown m fig. la (peak heights G correlate \lth a;). \mvlth energies and wavefunctions gn’en by the Henon-Helles hamlltoman [ 14,151. The wavefunctlon

$J~(.u,_I~)was generated

by eq (1) with

the gaussian a, USLII~the Q,,(x,_L~) determmed by the ab uutio methods of ref. [ 161 The result IS fig. 2. where the outer contour line IS determmed by I/(x. y) = 12 5. wluch IS the equal to an energy of EC + A [defined in eq. (2)] The highly delocahzed Q’(x,y) tend to confirm the inference that Q is a stochastic uutlal state. The quantity P(t) IS plotted In fig 1b. confirmmg the general features seen in ref [2] for a slmrlar gausslan dlstribuuon of amphtudes. The Henon-Helle: hamiltonian 1s non-separable, and IS substannally lrregular III the energy re@on chosen here. However two uncoupled harmomc oscillators with incommensurate frequencies will defiitely show sumlarly delocahzed #(.Y, y) and qualitatively sunllar P(t) If we rmpose a smooth distnbution of the an The observation that P(r) is sumlar m the regular and Irregular cases for an unposed smooth set of amphtudes should not be interpreted as any f&ure of quantum mechamcs to respond to the two different types of classical motion Instead. the amdarity is due to delocabzed mitial states which are illappropriate for testmg the nature of the dJ’namics This 1s also understandable from a purely mathematical point of view as follows: The value of P(t) =

C [ n

(fzn)’ e\p(--If&

2

1

t)2

(3)

IS of course dependent only on the smooth properties of the ni for times short enough not to resolve the energy spacmg between the adJacent states. It matters little whether the Et1 come from a regular or irregular harrultonian

ifsuch a smooth set ofa,, islmposed

on the

Volume

76, number

1

CHEMICAL

PHYSICS

LE-ITERS

15 November

1980

ENERGY

TIME

Fig

1 (a) A ~auss~~~ dlstrlbutlon

of Fran&-Condon

factors

were brondencdw~rha_gnuss-Lneshape\*hich*ves

ti,,

IS

the nrh eIge.nvalue

(b) 1!~1q(r)>l’

for fis

X(w)=

X,r,[T/(%r)

~~~]e~p[-~T~~,-,,$]

so that &lo,i’ T=57

= 1. The spectra forourclsem,d

la

Fig 2 Plot of 10 1’ for the gaus_uan dtstributton of 10 I2 shown tn fig la The ~7,~ were taken to be real so that nn = (~,lW The contour IS drawn at E = E, + A = 12 5 to show the delocalIzatlonofI~I*

with u = In 2. A = 1 5 normahzed

system. As noted previously [4,5], the overall breadth of the distnbution in ener,T determines the rapidity of the first fall off in P(t). and any organized structure of the an other than the broad envelope will cause concomitant structure m P(r). The small amplitude oscillatron of this quantity seen at later times in fig 1b 15:not organized recurrence but “noise” coming from the inabrhty of the unequally spaced En to completely Interfere at all times even with smooth a,. Furthermore, any arbltt-ary choice of the a,l, even If fluctuating, would result m sumlar time dependence for two &fferent sets of En, provided that botha, were chosen on the basis of the same criteria. We are then led to consider a, and P(t) which result when physically WIterestitzg Initial states are taken. This question has already been examined [4,5,13 J, but we give further evidence here, directly addressed to the question of irregular motion. In classical mechanics, the fundamental measure for energy or phase space flow is the isolated trajectory_ 23

Volume 76, number 1

15 November 1980

CHEhIICAL PHYSICS LEMERS

The quantum analog of this IS the phase-space locahzed wavepacket hioreover, Iocallzed wavepackets are actually created in Fran&-Condon absorption and emIsslon [ 17]_ For the Henon-Heties system. we thus examine the amphtudes

and time-dependence

resultmg

from a wavepacket mitialiy placed in (1) an Integrable regon rn phase space and (2) a non-integrable regon. Fig 3 shows the Henon-Hedes (v, pY) surface of section map for an energy of 10 0. whxh corresponds to the energy of the center of the wavepacket. The hamlltoman is g;ven by

+ 0.1 11803(x2~

q3/3),

Ed==

-I

13.3333

_

The circles labeled 1 and 2 show the approumate (fwhm) and position of ql and ~2, Gi = n-1/Texp[-$x2

+ 3 762 ix - $(r

-1

(4)

x~+39761x-5(y+

8

G

i

4

6

sue l-1~. 3.The

tknon-HHed~.(_t

, P,

) surfaceof

Sectionat

anener-

0 Shown also are the two wavepackets (fwhm) w hose dynamics are consldered \Va\epacher 1 hes m an mtegrable region while aavepacket 2 hes m a non-mtegrable regon The wavepackets have been chosen so that E,,((q).

) = 10 0 and &jqrn = 110

gy

- 2 707)‘], 1.914)’

+ 0.96 1 iy]

0 Y

Ga) @-, = 7r-l12 exp[-:

-2

of

10

(sb)

l-l 8

‘1

“II,

9.

ENERGY

TItlE

Fig. 4.(a) The spectra for wavepacket

24

1 m fig. 3. The broadening

1s the same as that of fig. la

(b) I(@l@(t))l* for fig 4a.

Volume 76, number 1

CHEhlICAL PHYSICS LETTERS

Obvtously. these are well locahzed m~hal states, m contrast to the 42 obtained rn fig. 2 for the gausstan distnbutlon of a,, . In tig. 4, the converged quantum

intereshng

15 November 1980 &stnbutlon.

By itnposittg

amplitudes

on

the system,

amphtudes and tune dependence are plotted for the wavepacket I_ In fig. 5, the same is done for wave-

packet 2 The drfferertce betweett the itttegrahle (case I. fig. 4J and the non-wregrable (case 2, fig 5J regtons LS profound. In case 1. we have a highly structured spectrum wrth many an having neghgrble mtensity: the tune dependence IS hkewise structured. In case 2 (mtegrable), the an are much more democratrcally Istrtbuted (m conformatton of the cntena put forth m ref [ 13 J)- The trme dependence 1svery drfferent from case 1 and IS much more hke that m fig. 1, which recall was for the unposed set of gaussran amphtudes. Thus, our results are dtametrtcally opposed to the imphcatrons of refs [ 1.21. We see very significant dtfferences in the quantum mechamcs of classically integrable and non-mtegrable regions. However, in classical and quantum systems, one must have control over the mural states so that rt represents a phystcally

one has very httle control over the initial t%avefunctlon Ref. [I] contains an apparent paradox caused by usmg an imposed set of amphtudes. It attempts to resolve thts by stating that drfferent systems may respond differently to external tntluences and therefore the intttal wavefunction Q and thus P(f), will doffer. Whrle true, this statement is not very relevant because it holds equally well for the comparison of two dtfferent separable systems as rt does for the companson of an integrable and non-integrabie one. The dtrect unplication of our numerical results which are summanzed m figs. 4 and 5. is: The time-dependence as measured by P(t) of the satttc locahzed initial state, Q WIU doffer srgmficantly tf placed m a classically Integrable as opposed to a non-Integrable system. Srmtiarly, t\vo different preparations on the same system one creating a state m a classically integrable region, the other in a non-Integrable regron wi!l also result m very different time dependences.

ENERGY

TIME

Fig S.(a) The spectra for wavepacket 2 m fig. 3. The broadening IS the same as that of fig_ la. (b) ](~?1]0(f))]*for f&_ 5a.

25

Volume 76, number 1

CHEMICAL PHYSICS LETTERS

Refemwes [ 11P. Brumer and M. Shapiro, Chem. Phys. Letters 72 (1980) 528. 121 R. Kosioffand S.A. Rice, preprint (1980). 131 B.V. Chirikov,‘Phys. Rept. 52 (19791263. [4] E.J. Heller, E.B. Stecheland M.J. Davis, J. Chem. Phys., to be published. [5] E.J. Hellcr, E.B. Stecheland M.J. Davis, J. Chem. Phys. 7: (1979) 47.59. [6] D.\li. Noid,M.L. Koszykowskiand R.A. Marcus, J.Chem. Phys. 67 (1979) 2804. [7] J.S. Hutchinson and RX. Syatt, Chem. Phys. Letters 72 (1980) 378; M.V. Berry, N.L. Balazs, M. Tabor and A. Voros, Ann. Phys. 122 (1979) 26. [S] W. Eastesand R.A. Marcus, J. Chem. Phys. 61 (1974) 4031; D.W. Noid and R.A. Marcus, J. Chem. Phys. 62 (1975) 2119:67 (19771559; S. Chapman, B.C. Garrett and W.H. Miller. J. Chem. Phys, 64 (1970) 502;

26

15 November 1980

K.S.SorbieandN.C. Handy,MoI. Phys. 32 (19763 1322; 1.C. Percival, J. Phys. A7 (1974) 794. (91 R.T. Swimm and J.B. Delos, J. Chem. Phys. 71 (1979) 1706. [ ld] C. Jaffe and W.P. Reinhardt, J. Chem. Phys., to be pubIished. (111 E.J. HeUer,Chem. Phys. Letters60 (1979) 338. [ 12) A. Messiah, Quantum mechanics, Vol. 1 (North-Holland, Amsterdam, 1962) ch. 6. j13J E.J. He&r, J. Chem. Phys. 72 (1980) 1337. 1141 M. Henon and C. Heiles, Astronom. J. 69 (1964) 73. [ 1.51 J. Ford, Advan. Chem. Phys. 24 (1973) 15S;in: Fundamental problems in statistical mechanics, Vol. 3,ed. E.D.G. Cohen (North-Hound, Amsterdam, 1975) p. 215; M.V. Berry, in: Topics in nonlinear dynamics, AIP ConferenceProceedings No. 46, ed. S. Jorna, (AIP, New York, 1978) p. 16. 1161 M.J. Davisand E.J. Heller, J. Chem. Phys. 71 (1979) 3383. [ 171 E.J. HeUer, J. Chem. Phys. 68 (1978) 2066.