Quantum dynamics of wave packets on phase space cells in nonlinearly coupled oscillators

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL XXIJI, 47-63 (1983)

Quantum Dynamics of Wave Packets on Phase Space Cells in Nonlinearly Coupled Oscillators

JURGEN BRICKMANN AND PETER C. SCHMIDT Institut fur Physikalische Chemie, Technische Hochschule Darmstadr, Petersentrasse 20,

0-6100 Darmstadt, Federal Republic of Germany

Abstract

The time evolution of harmonic oscillator coherent states (minimum uncertainty wave packet) I W) located on regular and random von Neumann lattice points in phase space is analyzed for nonlinearly coupled anharmonic vibrational systems. The Henon-Heiles system is studied as an example. A quasiprobability measure Fpp, is introduced to investigate the relative probabilities for the transition between phase space cells within a narrow energy range. The probability P,,,,, as well as the information entropy Sfi, is studied as a function of energy. A smooth transition from regular to chaotic motion is found indicated by a change of the fluctuations of these quantities as a function of energy.

1. Introduction

Intramolecular vibrational energy transfer in isolated polyatomic molecules has recently been examined in a large number of papers using classical [ 1-3],* quantum mechanical 14-61, and semiclassical approaches [7]. It was demonstrated that the exchange of energy between different modes can be weakly related to the magnitude of the nonlinear coupling terms between these modes. For low energies the normal mode concept always works well because, in general, the nonlinearities do not lead to a drastic change in the directions of the normal coordinates nor to a breakdown of the regularity of the solutions of the equations of motion. These regular solutions are trajectories on n-dimensional invariant tori in classical systems off degrees of freedom or vibrational wave functions uniquely classified byfvibrational quantum numbers. If the total energy is raised the situation changes according in two respects: (i) The normal coordinates may no longer represent an adequate description of the vibrational motion since for large amplitude intramolecular motion new (mostly curvilinear) coordinates are much more convenient [8]. (ii) The regularity of the solutions of the equations of motion (trajectories or wave functions) no longer holds and a transition to chaotic (ergodic) motion takes place. This transition may occur more or less exactly at a critical energy E,, but there are always some regular solutions in the chaotic do- main and vice versa.

While all measurable quantities related to the trajectory (the wave function) depend on f independent action components Cfquantum numbers) in the regular region, they become a smooth function of the total energy only in the chaotic region. It is obvious that the assumption of fast energy equilibration made in standard theories of uni-

* For an overview, see Ref. 1.

0 1983 John Wiley & Sons, Inc. CCC 0020-7608/83/010047- I7S02.70

48 BRICKMANN A N D SCHMIDT

molecular decomposition (like the R R K M theory) holds in the chaotic region but the opposite statement is not necessarily true. Intramolecular vibrational energy flow in isolated polyatomic molecules is essentially a quantum mechanical phenomenon. This process can be adequately described in terms of initially created nonstationary wave packets by analyzing their time evolution within a time scale of interest. There are in general contributions from the chaotic states as well as the regular states in the expansion of the initial states. Even in the case where one has regular contributions these states may spread over the accessible phase space region in a grained sense as has been demonstrated recently with one dimensional model systems [9- 131. Con- sequently, the analysis of the Hamiltonian of a given vibrational system gives no general answer to the question of whether there is fast energy transfer between the modes or not. One has to take into account the preparation process, i.e., the set of possible initial states. In classical mechanics one has standard initial ensembles like the microcanonical distribution based on the assumption that it is no problem to initially create a particle with given phase space coordinates. This concept can no longer be applied in quantum vibrational systems since the corresponding phase space representations, such as, for example, Glauber's P representation [ 141 and the Wigner distribution function [ 15,161, are quasi probability functions which approach the classical phase space density only in the limit of ti - 0 (which is not valid for most molecular systems).

In this paper we use sets of minimum uncertainty Gaussian wave packets located on a regular and a random von Neumann lattice [ 171 in phase space as initial states. This choice is related to the ideal experimental situation where one can create a particle in each phase space cell with minimum momentum and position uncertainty. In Section 2 some properties of the initial wave packets are reviewed; Section 3 deals with the phase space properties of the set of initial states, while in Section 5 the time evolution and its analysis are described. In Section 6 numerical results are presented concerning a two-dimensional model system, and in Section 7 some conclusions are drawn.

2. Properties of the Initial Wave Packets

The dynamical state of an f-dimensional vibrational system can be uniquely described by the actual values offmass-weighted position coordinates XI, . . . , xJand fmomentum coordinates y l , . . . , y f A minimum uncertainty Gaussian wave packet located at this phase space point may be described as a product

with the one-dimensional Gaussians

which can be interpreted as displaced ground state wave functions of a harmonic mode with angular frequency wj. In the harmonic case the center of the wave packet moves according to the classical equation of motion while its shape remains unchanged. We

WAVE PACKETS 49

designate such a wave packet a harmonic oscillator coherent state (HOCS) and use the short notation la;). The operator ?(a,) which displaces the oscillator ground state 10; ) to the complex point

can be given as [ 161

?(a;) = exp[aj(d, - dj)]

= exp(-l/2la,l 2, exp(a,dj), (4)

with the phonon creation and annihilation operators d,, and if, respectively,

d = (uJ/2h)144, + $,/u,),

d t = (@,/2h)1/2(4, - $,/u,), ( 5 ) where 4, and @, are the position and momentum operator of the j t h mode. It is straightforward to show that the HOCS’S I a, ) are minimum uncertainty wave packets in phase space, i.e., they fulfill the relation

* Apj = ‘/2 h, (6 ) with

Aqj = <a,)q j la j> - (a,14JlaJ)2>

Apj = (ol/1@:Ia,> - ( a ~ l f i j l a ~ ) ~ . (7)

Moreover, the HOCS’S are eigenstates of the annihilation operator d, with eigenvalue a,. HOCS’s can be easily expanded in a basis of eigenstates I n, ) of the harmonic oscillator with eigenvalue equation

hu,d,td, I n, ) = huJn, I n, ) , (8) resulting in expansion coefficients

ck) = exp(--1/21a,I 2)a;(k!)--1/2. (9)

The absolute squares Ick’l represent a Poisson distribution. Two different HOCS’S 1 a, ) and 10, ) are nonorthogonal [ 161. The overlap integral can be given explicitly by

(aJP,) =exp[-’/2(1a,12+ Ib,I2)+ a , P ; 1 7 (10)

and consequently,

I<a,P,>12 = exp(-la, - P,I2). ( 1 1)

The nice analytic properties of the HOCS’S, particularly their relation to the harmonic oscillator eigenstates, will be used in the numerical treatments presented in Section 5.

50 BRICKMANN AND SCHMIDT

3. Phase Space Representation on a Set of HOCS

The set of all coherent states is an overcomplete set [ 14,161. Nevertheless there are several possible expansions of other states and operators with respect to these states. One of them is the Glauber P representation [ 141 which makes use of the identity

and the density operator for thej th state, which takes the form

with the real valued function P(a j ) such that

J P ( a j ) d2aj = 1.

The P representation is a quasiprobability function that becomes a phase space density in the limit h - 0. For finite h, however, definite and perfectly well behaved density operators which include the functions P can freely take on negative values and there is, in general, no way of avoiding that. Nevertheless, the P representation can be used to get some insight into the nature of the eigenstates of a nonlinearity coupled vibrational system. This was recently demonstrated by Weissmann and Jortner [ 181. The authors analyzed the local properties of the two-dimensional representations

pN(a I7a2) = I(wl(a214N)2 (15)

for each eigenstate 1 4 ~ ) of a Henon-Heiles oscillator. Other authors [ 151 used the Wigner distribution function W(x,y) instead of Y ,

where I x) is the eigenstate of the position operator. This distribution also approaches negative values and so cannot be interpreted as a phase space density function for finite h. The major problem with respect to the interpretation of these functions as probability functions are related to the nonorthogonality of the set ( 1 a)) . The overlap integral between two members is strongly related to the Euklidean distance between them in phase space. To minimize this overlap effect one can remove most of the I a ) and define a discrete subset of coherent states as was done first by von Neumann [ 171. It was demonstrated recently [ 19-22] that the set of HOCS’s on a regular von Neumann lattice with lattice points

21i-h 1 w 1/2 a,, = (s) (mi + i T; n , (m,n> integers,

also forms an overcomplete set which becomes complete if one arbitrary member (for convenience the a00 state) is removed. Since only one-dimensional motion is considered here, the mode index is omitted. In Eq. (1 7), ( is an arbitrary constant. The lattice cell area in the a plane is equal to T which corresponds to a phase space area of h , as

WAVE PACKETS 51

ln W 5 1 I- I I I I I

f z '0 2 L 6 8 10 12 1L

TOTAL ENERGY I hw1

Figure I . Histograms for (a) the number of von Neumann lattice states and (b) the number of eigenstates as a function of energy for a two-dimensional Henon-Heiles- type oscillator with Hamiltonian

can be seen from Eqs. (17) and (3). Consequently, these states appear in a natural way as representatives of the cell structure of phase space. In this work we study the time evolution of minimum uncertainty wave packets located on a particular von Neumann lattice which is isotropical in position and momentum for a two-dimensional nonlinear oscillator. The parameter [ is chosen as (27rh/w)I/* in this case and the lattice points in one dimension become

It can be shown [23,24] that the density of energy expectation values (amfl Ifil amn) agrees with the density of eigenstates p ( E ) of the Hamiltonian A. This can be easily demonstrated in the harmonic oscillator case, where p ( E ) = (hw)-l is given. Since the HOCS'S are eigenstates of the annihilation operator 6 with eigenvalue a, the energy expectation values become

a,, = r'f2(m + in). (18)

Ern, = hw ( a m n I &+a I a m n )

= hwa*,,arn, = h w ~ ( w t 2 + n2) . (19)

Considering m and n as continuously varying variables, the number Z of states up to a given energy E becomes

Z = E / h w , (20)

5 2 B R I C K M A N N A N D S C H M I D T

15 0

0

rn 2:

z o I 0 2 4 6 8 1 0 1 2 1 4

TOTAL ENERGY I h w ~

Figure 2. Histogram for the number of random lattice states as a function of energy for a two-dimensional system as in Figure 1.

which demonstrates that the density of E,, is equal to (hw)- ' as was found for the eigenvalues. The correspondence between the regular von Neumann states and the approximate eigenstates holds also in multidimensional bound systems as is demonstrated with a particular example in Figure 1. This model system is described in Section 4.

4. Two-Dimensional Nonlinear Oscillator

The dynamics of von Neumann states is studied numerically for a two-dimensional nonlinear oscillator represented by a Henon-Heiles Hamiltonian of the form

I2 = l/@: + 0:4: + a ; + w:&) + €(&2 - 1/36;). (21 1 This system has two types of dissociation channels: one in direction of the symmetry axes q 2 and two symmetry equivalent channels for q2 * 0. The eigenstates (4,,, ) were calculated with a linear variational scheme using a nontensor product basis of harmonic oscillator eigenstates I T S ) , where r represents the vibrational quantum number of the ql mode (frequency w1) and s represents that of the q 2 mode. The details of the basis choice will be published elsewhere [25] . The regular von Neumann states for this system are characterized by a pair of complex numbers

for the motion along q1 and 92, respectively. We also used a set of initial wave packets located on a random von Neumann lattice. In this lattice the integers (j,k,rn,n) are replaced by a set of random numbers (r],r~,r3,14) with the same point density in four-dimensional space. In Figure 2 the number of random lattice states as a function of total energy is shown for the example from Figure 1.

For convenience, we introduced generalized coordinates (4, , 4 2 $ 1 9 2 ) for the motion along the two modes

WAVE PACKETS 53

-10 -25 -50 -2.5 0 2.5 5.0 25- 10 X l

Figure 3. Potential energy for the Henon-Heiles-type oscillator as a function of the generalized coordinates 21 $ I and 22 2 q2 and centers of the regular von Neumann lattice states.

In these coordinates the regular von Neumann lattice becomes a four-dimensional primitive lattice with lattice constant ( 2 ~ ) ' / ~ ,

( a l ) = 21/2 Re(ajk) = ( 2 ~ ) I / ~ j ,

0 1 ) = 2'12 Im(a,k) = ( 2 ~ ) I / ~ k ,

For the random von Neumann lattice the vector ('j,k,m,n) is replaced by the random vector (rl 7 2 7 3 ~ 4 ) .

The Hamiltonian equation (21) consequently becomes

I? = f i / hwl = */2G? + 4: + K - ~ G ; + a:)] + Z ( i f L j 2 - I/&$), ( 2 5 ) with

In the numerical example treated in this paper, we have chosen a 1 I / 10 high-order resonance system with a classical critical energy for the transition between regular and irregular motion of E , = 12hiwl. The lowest dissociation barrier is E D , = (3 + K ~ ) hW1/(24Z2K) = 14.78hwl, which has a twofold occurence at 21 = f ( 3 / ~ ) ' / ~ / 2 : = f7 .58 and 2 2 = - 1/2Z = -4.59, while the third dissociation barrier is higher: ED^ = 1 / ( 6 ~ % ~ ) = 22.63hw1, X I = 0, x2 = ~ / ( K ~ Z ) = 11.1 1 . In this system (z = 0.1089, K = 1 0 / 1 1 ) there are 107 quasibound states up to the dissociation area. In Figure 3


I I 8 ! I I 1

Figure 4. as in Figure 3.

the location of the centers of the regular von Neumann lattice states in position space is shown with the potential energy lines for energies differing in five quanta of the w1 oscillator. It will be seen that the phase space cell produces a relatively rough graining which is partly in resonance with the symmetry of the system. This is no longer true for the random lattice states as can be seen from Figure 4 where the projections of the lattice points on position space are shown. According to the statement made above there are roughly 100 coherent lattice states up to the dissociation area, i.e., there are 100 phase space cells in the four-dimensional phase space in the area of classical bound motion. For an initial state preparation technique with minimal momentum position uncertainty this means that one can prepare 100 physically different situations at t = 0. In this paper, we observed the time evolution of these initial states in order to decide whether there are different types of dynamical behaviour. In Section 5 the time evolution is evaluated.

5. Time Evolution

The time evolution of the coherent states on a von Neumann lattice can be formally given as

IS@), = exp[-(i/h)At]IW)o, (28)

where H is the total Hamiltonian of the system and p = G,k,m,n) or p = (rl,r*,r3,r4) represents the lattice point of the initial wave packet

(29) l a j k ) I on?,) (regular lattice), [ larIr2) (random lattice).

The time evolved wave packet can be explicitly obtained if the eigenfunctions I $ M )

of H are known. In this treatment we are only interested in the quasibound motion,

WAVE PACKETS 55

i.e., we replace the eigenstates by resonance states and neglect the imaginary part of the energy eigenvalues. Consequently, the time evolution of I ‘k.) only contains that part which is related to the bound motion and we do not consider dissociation processes as a consequence of sub-barrier tunneling processes. The latter effect has recently studied by Miller and co-worker [26] to analyze mode specific tunneling dissociation processes. With the formal expansion of the initial states

I W O = c RbI4M), M

the time evolved wave packets become

where EM are the eigenvalues of fi (which are the real parts of the complex eigenvalues in our model systems). If there are no degeneracies in the energy spectrum of H as in our model system, the probability of finding the system in an eigenstate 1 4 M ) can be easily calculated from the expectation value of the projector 1 4 ~ ) ( I # I ~ I and one obtains the stationary result

(32) P% = t ( * p l 4 M ) ( 4 M l * p ’ ) t = IDI*Ml2.

In the regular regime, i.e., if only regular quantum states 1 4 ~ ) contribute to 1 q p ) ,

there are only a few significant contributions DG, say N,, and the ID$) are the order of Nil. If all contributions to l q p ) ~ come from irregular states, there is a strong mixing of zero order states (which may be related to the most similar integrable Hamiltonian f i 0 relative to I?) and the number of significant contributions becomes roughly equal to N: [ 131. The magnitudeof IDGI is then NL2. It has been demonstrated recently by Brickmann and Russegger [9-113 that in typical one-dimensional cascs the number of contributing eigenstates to a coherent state is of the order of 10, i.e., the absolute values of lD%1* give some insight whether or not a given state’s motion is typically one-dimensional (no energy transfer to another mode). A convenient measure for the distribution of a given initial state with respect to the eigenstates of a system is the information entropy [27]

S@ = - C pG lnp$. (33)

There has been a lot of effort to introduce entropylike quantities as a measure of “chaos” in quantum system for bound motions duriag the last few years [28]. We will not touch the general problem here but we use Eq. (33) to formulate a pragmatic definition:

A nonlinear oscillator of more than one dimension exchanges energy between modes at a given energy E if Sp is only a function of energy for all IS,) with (‘k@lI?I P.) x E.

It is clear that the energy exchange in a given system is not automatically only a function of the Hamiltonian alone. In the case of the minimum uncertainty wave packets on the von Neumann lattice the critical energy for a switch between “ergodic”

M

56 BRICKMANN AND SCHMIDT

and “nonergodic” behavior may be different for other preparation techniques with other complete sets of initial states like local mode excitations [29,30].

The order of magnitude of S , can be estimated with a simplifying example. Let there be N , contributions of the eigenstates to a given initial state and let all these contributions be equal. Then one has 1 D P l 2 = N;’ and

N P

S% = -Nil C InN;’ = InN,. M = l

If there are N: equal contributions with 1 0 ~ 1 2 = N i 2 (irregular case), one has N$

Sg = -N i2 C lnN;* = InN: = 2S%, M = l

(34)

(35)

i.e., the fluctuation of S , is expected to be of the order of magnitude of S” itself in the case of regular motion and should decrease if the contributions from the irregular states increase.

There are some other properties which can be used as an indication of a transition from regularly moving coherent states to irregular behavior. The initial state population probability

p,, = l t ( * ~ ( W o 1 2 , (36)

which can be directly related to the Franck-Condon structure of absorption spectra, shows different behaviour for very regular moving wave packets and for more com- plicated motion. This quantity as well as its time average was studied in a series of one- and two-dimensional treatments [9-12,31-331, but to our best knowledge no quantitative criterion exists from finite time analysis for deciding whether regular or nonregular motion is dominant, since even in the regular case, the time evolution of

I . P,, may appear to be very “chaotic” [9-1 We have calculated the time average

K L , = c M

as well as the time average probability

D ~ A 4, (37)

M

for a transition between a lattice state I!€’,) to an energetically adjacent state I!€’,’). The results of our calculations are discussed in Section 6 .

6. Results and Discussion

We have investigated the time evolution of harmonic oscillator coherent states (HOCS) initially located on lattice points in the four-dimensional phase space of a two-dimensional nonlinear oscillator with the Hamiltonian

fi = 1/2@:j? + S t + “/10@1 + s t ) ] + 0.1089 (9:q2 - “/I, - I&:), (39)

WAVE PACKETS

1 p , , , , , , , ' ' , ,

OQ 5 10 TOTAL ENERGY I hw,

Figure 5 . Entropy S p for different initial states on a regular von Neumann lattice as a function of the energy expectation value (W( H I Be).

where 41 and 4 2 are generalized coordinates, and fi is measured in units of h q . The lattice points were chosen in such a way that the density of initial states corresponds to the density of phase space cells, i.e., each initial state may be interpreted as a representative of one phase space cell. A regular lattice of wave packets of this type was first introduced by von Neumann [ 171 and it was demonstrated recently [ 19-22] that such a set of states forms a complete set for the Hilbert space of the systems as well as the set of eigenstates of the Hamiltonian. Unfortunately, these states are not or- thogonal, so that one cannot easily represent density operators with respect to this set. We therefore observed the time evolution of each state separately to find out whether there is a characteristic change in its time evolution as a function of energy and whether such a change can be related to the set of chaotic trajectories in the corresponding classical system. The critical energy for the latter transition was found to be E, = 1 2hwl from classical trajectory calculations, while the lowest dissociation saddle point is at E D ] = 14.78hw1, but there are several periodic trajectories in the energy range

We have extended the original regular von Neumann lattice idea by replacing the lattice points by a random lattice in which the density of points was conserved. While, in the original von Neumann lattice, some wave packets are located on symmetry centers of the Hamiltonian, this can no longer be observed in the random lattice. The density of expectation values (*PI@] \ k p ) for both the regular and the random lattice as functions of the energy roughly agree with the density of eigenstates of the system. This is demonstrated in Figures 1 and 2. The projections of the lattice points for both cases on the position space are shown in Figures 3 and 4. The interference of the regular lattice with the symmetry of the potential function in the regular lattice case is im- mediately obvious.

The entropy S P , as defined in Eq. (33) as a function of the energy, is shown in Fig- ures 5 and 6 for the regular case and the random case, respectively. In the regular case the calculated points are roughly located on two branches, a high and a low entropy branch. The points near the low entropy branch are related to initial wave packets near simple periodic orbits (fixed points in the PoincarrC surface of section [IS]) and the

E, < E ED^.

58

0.16

BRICKMANN A N D SCHMIDT

-

2l O o 0

1

TOTAL ENERGY I hw,

Figure 6 . Entropy S@ for different initial states on a random von Neurnann lattice (as in Fig. 4) as a function of the energy expectation value (Wlfi( !P*).

occurence of such points is a consequence of the interference of the lattice with the symmetry of the system. Consequently, the approach of the two branches may possibly not be a generic property of the system which appears for an arbitrary initial distribution of phase space cells. The situation here is similar to that of Waite and Miller [ 2 6 ] . These authors found two branches for the tunneling dissociation rate from resonance states of different symmetry. The situation changes up on turning to the random lattice (see Fig. 6 ) . Now the single results more or less uniformly cover a stripe in the diagram. The relative fluctuation Asye[= ASp/Sp decreases monotonically with increasing energy. For E = 3hwl one has ASYel = 0.4. For E = 12hwl, = 0.25 re-

0.10

0.081

0.06

o.o*}

0

@ O

0

0

0

0 0

0

0

01 ' t , 0 2 C 6 8 1 0 1 2 1 4

TOTAL ENERGY l a w ,

Figure 7 . Time averaged initial state population probability P,, for the wave packets initially located on a regular von Neumann lattice.

WAVE PACKETS 59

0 0 0.08 ““1 OO0

0 2 L 6 8 10 1 2 1 . 4

TOTAL ENERGY1 hw,

Figure 8. Time averaged initial state population probability cated initially on a random von Neumann lattice.

for wave packets lo-

sults. Near the dissociation energy ED^ a drastic drop to AS;el = 0.1 can be observed, indicating an increased contribution from irregular states to the initial wave packets.

The quantitative differences found for the initial states on a regular and on a random lattice are also to be seen in the analysis of the initial state population probabilities F,, (see Figs. 7 and 8). For energies near the dissociation energy there are states on the regular lattice with p,, = 0.05 and also with F,, = 0.01. The former correspond to roughly 20 contributing states while for the latter 100 are necessary following the simplifying arguments of Section 5. Since there are approximately 20 quanta in one degree of freedom up to the dissociation energy, the wave packets with the maximal value of FPp can only move very regularly. For the initial states located on a random lattice, F,, does not exceed 0.02 for energies larger than 1 lhwl indicating a contribution of roughly 50 eigenstates. Again, near the dissociation energy there is only a small fluctuation for F,, as a function of energy.

The time averaged probabilities P, ,--] for a transition from one initial state I*.) to another state I*,-]) adjacent in the energy scale ((WlZ?I*.) ( WL- I 1fiIW-l)) is shown in Figures 9 and 10 for the wave packets on the regular and the random lattice as functions of the energy. For the regular lattice these quantities show large fluctuations within the whole energy range considered here. This is partly due to the fact that there are several degeneracies in the expectation values, i.e., 1 W - I ) may have the same energy as the initial state I*,), while for other energetically adjacent states a relatively high energy difference occurs. In the random lattice case F, p - l is much more focused with increasing energy and tends to a minimum value approaching the dissociation area.

60 BRICKMANN AND S C H M l D l

2 L 6 8 10 12 li, TOTAL ENERGY OF STATE U l h w ,

Figure 9. Time averaged probability Pw,,-i for a transition of an initial wave packet I W) to an energetically adjacent one I W-' ) (both on a regular von Neumann lattice).

The numerical results were obtained with the aid of a linear variational scheme using 500 products of harmonic oscillator functions selected according to the symmetry of the system. The details will be published elsewhere [ 2 5 ] . The quantities shown in Figures 5-10 do not change within the accuracy of the drawings obtained from test calculations with a larger basis set. Moreover, we checked the norm of the initial wave packets after expanding them relative to the approximate eigenstates

This quantity was equal to one within a maximal relative error of 1% for all wave packets considered here, indicating that our basis set is extended sufficiently also to calculate time averaged overlap effects between different states.

'.i O o 0 o j 0 0

1 0 - s o s 2 L 6 8 10 12 1.4

TOTAL ENERGY OF STATE JJ I h w ,

Figure 10. Time averaged transition probability P,,- , as in Figure 9 for random lat tice states.

WAVE PACKETS 61

7. Discussion and Conclusions

In this study the time evolution of two dimensional minimum uncertainty Gaussian wave packets [harmonic oscillator coherent states (HOCS)] moving in a nonlineary coupled oscillator system (Henon-Heiles model) was analyzed by expanding these states according to a truncated set of eigenfunctions of the total Hamiltonian. The eigenfunctions were calculated using a variational procedure. Two different sets of wave packets were chosen to represent the relevant part of the phase space (bound area):

(i) initial states located on a four dimensional cubic von Neumann lattice wherein the volume Vcel] of the unit cell covers just one phase space cell, i.e., Vccl, = h2, and

(ii) initial states located on a random von Neumann lattice for which the single locations of the wave packets were chosen as random but with an average cell volume Vce1l as above.

The restriction to a numerable set of discrete phase space points as starting positions instead of continuous distribution functions (like the Wigner distribution) enables us to interpret these states as representative of the possible phase space cells and to study their time evolution independently.

The aim of this investigation was to find out whether there is a characteristic difference for the wave packets dynamics turning from the classically regular to the irregular region in phase space. We calculated three quantities for each wave packet W, the entropy SM, the time averaged initial state population probability Prr, and the time averaged transition probability Prp. for a “hopping” from state \kr to an energetically adjacent one W’ as functions of the energy expectation value E = (Wlfil q r ) . In classical systems there is a transition from regularly moving trajectories at low energy (motion on invariant tori) to irregular motion at a critical energy Ec. The quantum correspondences to these trajectories are regular and irregular quantum states, respectively [24]. The amounts of irregularity in the motion of a particular initial wave packet depends on the portion of irregular states contributing to its expansion. We can formally write

I*’) = p R l q p ) + P [ I * ’ ) , (41)

with the projection operators PR and I?] projecting on the sets of regular and irregular states, respectively. Obviously the expectations

R W% = ( ? W l B R I W ) = 2 ID$#,

M

I wr = 1 - W$ = ( \ k p * [ P $ P p ) = c ID”M1’ (42) M

can be taken as measures for regular and irregular contributions. The indices R and I indicate a summation only over regular and irregular states, respectively. I f only irregular states contribute to the initial states in a particular energy range (corresponding to the ergodic situation in classical dynamics) it is be expected that the quantities S p , ppr, and pppl become smoothly varying functions of energy alone. An increase of the regular contribution leads to an increasing dispersion. In our model


system the classical transition energy E, for regular-irregular motion is roughly 4/5

of the lowest dissociation barrier. Consequently, for most of the initial wave packets the regular contributions predominate over the irregular ones, i.e., W, > W,. Even for ('kfllal 'kf l ) z E, there is a non-negligible portion of regular states in the energetically equivalent wave packets. This regular part leads to a dispersion in the quantities described above up to the dissociation energy (see Figs. 5-10). The dispersion is much larger for initial states on a regular von Neumann lattice (Figs. 5,7, and 9) than for those on a random lattice (Figs. 6,8, and lo), but for both there is a dispersion decrease with increasing energy above E,. This decrease, however, is much more smooth than in the classical case, i.e., even if almost all trajectories move ir- regularly in a particular energy range, the quantum motion of the corresponding wave packets is already sensitive to the particular initial conditions. We therefore conclude from our present results (a) that for a classical-like preparation of initial wave packets (local in phase space) the transition from regular motion to irregular motion as a function of energy is much smoother than in the classical case, and (b) that, consequently, even for wave packets with energy expectation values perceptibly larger than the critical energy E, the dynamics remains sensitive to the initial conditions. The latter result may become important in connection with mode selective chemistry as was recently discussed [ 2 6 ] .

Acknowledgments

One of us (J.B.) thanks Professor J. Jortner and Professor M. Bixon, both from Tel Aviv, for many helpful discussions. We also thank Professor M. Sage and Professor J . Stone for making available unpublished material, and Dr. Robert Pfeiffer for carefully proofreading the manuscript. This research was supported by the Deutsche Forschungsgemeinschaft, Bonn, the Fonds der Chemischen Industrie, Frankfurt, and the Bundesminister fur Forschung und Technologie, Bonn. The numerical calculations were carried out in the Rechenzentrum der Technischen Hochschule Darmstadt on an IBM 370/168.

Bibliography

[ I ] U. L. Hase, in Potential Energy Surfaces and Dynamics Calculations, D. G. Truhlar, Ed. (Plenum, New York-London, 1980). pp. 1-36.

[2] N . De Leon and B. J . Berne, J . Chem. Phys. 75,3495 (1981). [3] P. Brunner, Adv. Chern. Phys. 47,201 (1981). [4] S. A. Rice, Adv. Chem. Phys. 47, 117 (1981). [5] D. W. Noid, M. L. Koszykowski, and R. A. Marcus, Ann. Rev. Phys. Chem. 32,267 (1981). [6] S. A. Rice, in Quantum Dynamics of Molecules, R. G. Woolley, Ed. (Plenum, New York-London,

1980), pp. 257-356. [7] M. Tabor, Adv. Chem. Phys. 46,73 (1981). [8] L. A. Careira, R. C. Lord, and T. B. Malloy Jr., in Topics Current Chern. 82,3 (1979). [9] J . Brickmann and P. Russegger, J . Chem. Phys. 75, 5744 (1981).

[ lo] J. Brickmann and P. Russegger, Chem. Phys. (to appear). [ 1 1 ] J. Brickmann and P. Russegger, Chem. Phys. Lett. (to appear). [ 121 J . Brickmann, J. Chem. Phys. (to appear).

WAVE PACKETS 63

[ 131 J. Brickmann, in Intramolecular Dynamics, J . Jortner, Ed., (Reidel, Dordrecht, to appear). [I41 R. J . Glauber, in Quantum Optics, S. M. Kay and A. Maitland, Eds., (Academic, London-New York,

[15] J. S. Hutchinson and R. A. Wyatt, Phys. Rev. A 23, 1567 (1981). 1161 W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973). [ 171 J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton U.P., Princeton,

[ I 81 Y. Weissmann and J. Jortner, J. Chem. Phys. (to appear). [I91 V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, Rept. Math. Phys. 2, 221 (1981). [20] A. M. Perelemow,Teor. Math. Fiz. 6, 213 (1971) [Theor. Math. Phys. 6, 156 (1971)l. [21] H. Bakry, A. Grossmann, and J. Zak, Phys. Rev. B 12, 1178 (1975). 1221 M. Boon and J. Zak, Phys. Rev. B 18,6744 (1978). 1231 J . Brickmann (unpublished). [24] I . C. Percival, J . Phys. B 6, L299 (1973); Adv. Chem. Phys. 36,l (1977). [25] J. Briekmann and P. C. Schmidt (unpublished). [26] B. W. Waiteand W. H. Miller, J. Chem. Phys. 74,3910 (1981). [27] R. D. Levine, Adv. Chem. Phys. 47,239 (1 98 1). [28] J. Stone, E. Thiele, and M. F. Goodman, in Intramolecular Dynamics, J. Jortner, Ed. (Reidel, Dor-

[29] M. Sage and J. Jortner, Adv. Chem. Phys. 47,293 (1981), [30] M. Sage, in Intramolecular Dynamics, J. Jortner, Ed. (Reidel, City, to appear). [31] Y . Weissmann and J. Jortner, J. Chem. Phys. (toappear). [32] E. Heller, J. Chem. Phys. 67, 3339 (1977); Chem. Phys. Lett. 60, 338 (1979); J. Chem. Phys. 72,

[33] M. Bixon and J. Jortner (unpublished); M. Bixon, in Intramolecular Dynamics, J . Jortner, Ed. (Reidel,

1970). p. 53.

NJ, 1955).

drecht, to appear); E. A. Thiele (unpublished).

1337 (1980); E. Heller, E. B. Stechel, and M. J . Davies, ibid. 73, 4720 (1980).

Dordrecht, to appear).

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Quantum dynamics of wave packets on phase space cells in nonlinearly coupled oscillators