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On the application of the adiabatic invariance method for the identification of “quantum chaos”
Volume 155, number 6
17 March 1989
CHEMICAL PHYSICS LETTERS
ON THE APPLICATION OF THE ADIABATIC INVARIANCE METHOD FOR THE IDENTIFICATION OF “QUANTUM CHAOS” Gert Due BILLING Chemistry Laboratory III, H. C. Brstedlnsfitute. University of Copenhagen, DIG2100 Copenhagen 0, Denmark
and Georges JOLICARD Laboraroire de Physique MolPnrlaire, WA 722 du CNRS, Fact& des Sciences et des Techniques, La Bouloie. 25030 BesancoanCedex. France
Received 11 October 1988; in final form 14 January 1989
The quantum-mechanical adiabatic invariance method is applied to the H&non-Heiles Hamiltonian and used to discuss the criteria for a quantum analog to classical chaotic motion.
1. Introduction There has been much interest in exploring the quantum-mechanical analog to classical chaos, and several methods or indicators of quantum chaos have been suggested [ 1,2]. Of these we mention: sensitivity to perturbations, irregularity of the nodal structure of the wavefunction, overlapping avoided crossings, second energy differences with respect to changes in the coupling and the dominant coefficient method proposed by Hose and Taylor [ 2 1. As far as we known the question has not been examined in the context of the adiabatic invariance switching (AIS) method [ 3 ]. In this method the term which causes the Hamiltonian to be non-separable and thus provides the possibility for chaotic behaviour, is switched on adiabatically, i.e. infinitely slowly. According to Ehrenfest’s hypotheses the quantum numbers (the labelling of the wavefunctions) are invariant - but the energy of the state and the wavefunction itself changes. The AIS method has recently been applied by Johnson [ 31 to obtain semiclassical eigenvalues for the H&non-Heiles Hamiltonian. In the present paper we use the AIS method within a purely quanturn-mechanical framework and solve the time-dependent Schr6dinger equation for the system. By 0 009-2614/89/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
comparing the eigenvalues and the time-dependent “memory function” it is possible to gain new insight into the phenomena of quantum chaos and discuss the results in the context of previously suggested criteria. We choose for our investigation a system and an energy regime for which classical chaotic as well as regular motion has been found. Only in this way is it possible to distinguish between classical and quantum chaotic behaviour. As the energy of the system increases criteria found for quantum chaos should become more and more pronounced. In the present paper it is suggested that the memory function delined as the projection of the wavefunction, obtained by the AIS method, onto the initial separable wavefunction yields a “measure” for quantum chaos.
2. The AIS equations Consider the H&ton-Heiles B(r, e)=&+B,
,
Hamiltonian (11
where B.V.
521
Volume
155,number 6
CHEMICAL
PHYSICS LETTERS
17 March 1989
(2) and A, = @r’cos 30 _ In the AIS mechanically
method
(3) one
obtains
g(n)=i
i$&(T)yl,
(4)
where fi( T) = Ei+ S( T)fi, and S(T) is a switching function such that S(0) =O and limT_, S(T) = 1. We have introduced in eqs. ( l)-(4) dimensionless units such that fro= 1 and time is in units of w-l. In order to solve eq. (4) we expand the wavefunction in eigenfunctions of the Hamiltonian fiO, 6) =-%VI%(r, 0) >
(5)
t&,(r,O)=N,,r’exp(-$r*)L!,(r*)exp(+ilO),
(6)
&&(C where
and (n2=0,1,...),
(8)
where n=t(n,-I),
I=&,&-2
)..‘l (or.).
(9)
Thus we use the expansion VI= c a,,(t) a
fi (n-i+s) 1=I
, nal,
(13)
wherecx=(n,I),j?=(n’,P),r=j(5+I’+I)+m’and s= 1+ I- y. We will now follow as a function of time the “memory function” defined as the overlap between the adiabatically propagated wavefunction y,(t) and the initial wavefunction v/y, i.e. P~(T)=I((V,(t)l~10)i2=[aN(T)12
(14)
and the adiabatic energy of the state I, i.e.
&(T)=l-exp(-aT),
the initial
(~czl(O)=~CI,). Inserting ( 10) into eq. (4) we obtain .. la,l= g bWL,+SCT) Hf,~)qt~,
state S,(T)=l-exp(-aT2). (11)
where H~g=f~(W~1r2COS301~/Og).
Note that the quantity Pr( T) in the limit T-m approaches the Hose-Taylor weight. This weight was used to determine the “quantum periodicity”. It was argued that weights larger than 0.5 would give a convergent iteration for an effective Hamiltonian. The switching function in eq. ( 11) is in principle arbitrary but investigations made bp Johnson [ 31 have shown that the convergence rates depend strongly upon the choice of S( 2’). In the present work we have used four different functions
(10)
&CC 0) I
where LY=(n, I) and I denotes
(12)
This matrix element may easily be evaluated giving
522
n=o,
= c EE I a,l I ’+ S( V C G&&p. a a8
(7)
E,,=nz+l
g(n)= 1,
quantum
The second and third choices are Johnson’s fimction S, and & whereas S, and S, contain a parameter a which for a-+0 makes the switching more adiabatic. Another difference is #at S2 and S, operate on a finite interval [0, T,,,,,], i.e. S,( T,,,,) = 1.0 whereas the other two only approach this limit for T-+co. Hence the integration is terminated when S(T) exceeds some number close to unity. In the present analyses we use a basis set consisting of levels from n=O to n= 16, i.e. altogether 153
Volume 155, number 6
Table 1 Eigenvalues of the H&ton-Heiles
CHEMICAL
Hamiltonian
PHYSICS LETTERS
17 March 1989
as a function of basis set size
nl
91 a’
12oa’
153 8’
465 D,
4
4.8702 4.8987 4.9863 5.8172
4.8702 4.8987 4.9863 5.8171
4.8702 4.8987 4.9863 5.8170
4.8702 4.8987 4.9863 5.8170
4A, 4E 5E 6E
5.8673, 5.8815 5.9914
5.9913
5.8670,5.8815 5.9913
5.8670, 5.8815 5.9913
5A,, 2A, 7E
6.7404 6.1672 6.8546
6.7382 6.7652 6.8536
6.7380 6.7649 6.8535
6.7379 6.7649 6.8534
6A, SE 9E
6.9992, 6.998 7.6742
7.6624
6.9990,6.9995 7.6600
6.9989, 6.9994 7.6595
3A1,7A, 10E
+3 +5 +7
7.7162, 7.1410 7.8396 8.0139
7.8336 8.0098
7.6984, 7.7370 7.8329 8.0095
7.6977, 7.7369 7.8327 8.0094
IlE 12E
0 +2 54 +6 k8
8.6170 8.6399 8.7103 8.8324, 8.8379 9.0321
8.5581 8.5808 8.6792 8.8122, 8.8165 9.0222
8.5541 8.5764 8.6779 8.8113, 8.8152 9.0217
9A, 13E 14E 5A2, lOA, 15E
0 *2 +4 5+1 f3
* f5 6
0 +2 *4
26 7+1
8
‘) Present work.
8.5723 8.5958 8.6852 9.0234
gA,,4A1
b, Ref. [2].
states. Table 1 shows that this number of functions is suffkient to obtain good agreement with the HoseTayIor values [ 21. For smaller perturbations (S(T) < 1) the basis set will of course be even better and in any case sufficiently large for the present discussion. Table 2 shows that both S, and S, are superior to the S, switching function with ~7~0.2. Decreasing a
and increasing T,,,,, will improve the agreement as shown in table 3 for the S, function. Note also that the A,, AZ splitting for the I= IL3, f 6 etc. levels are not recovered by the adiabatic propagation. This point has been thoroughly discussed by Johnson. But except for this “problem” we may with the adiabatic propagation get arbitrarily close to the exact eigenvalues. Thus there exists a method of changing adi-
Table 2 Eigenvalues obtained using the AIS method (eq. (4) ) and various switching functions compared with the exact quantum-mechanical values
Table 3 Comparison of adiabatic and exact eigenvalues forf( T) =0.9999. The switching function isf( 7’)= 1 -exp( -aT’)
nl
S,(T)
S,(T)
S, (T]
Exact a)
RI
a-0.005555
a=O.OOl
Exact ‘)
751 7t3 7+5 7*7 8 0 8f2 Sk4 856 8+8
7.6686 1.7244 7.8372 8.0131 8.5719 8.5996 8.6825 8.8218 9.0279
7.6690 7.7247 7.8374 8.0133 8.5726 8.6002 8.6829 8.8222 9.028 1
1.6985 7.7500 7.8543 8.0172 8.6167 8.6418 8.7176 8.8449 9.0346
7.6600 1.6984, 7.7370 7.8329 8.0095 8.5581 8.5808 8.6792 8.8122, 8.8165 9.0222
7+1 723 725 7*7 8 0 81t2 8f4 8?6 s+fl
7.6733 7.7285 7.8405 8.0154 8.5797 8.6069 8.6886 8.8267 9.03 I4
7.6655 7.7222 7.836 I 8.0120 8.5671 8.5937 8.6812 8.8197 9.026 I
7.6600 7.6984, 7.7370 7.8329 8.0095 S-5581 8.5808 8.6792 8.8122, 8.8165 8.0222
T,,,,, = 50
T,,,=50
T-.=36
T,,,=40
T,,, =
” Obtained
with 153 states in expansion
(I 0).
100
‘I Obtained with 153 states in the expansion of the wavefunction. 523
abatically from the separable to the non-separable system. From a practical point of view one would of course never calculate the eigenvalues in this manner - since the propagation of the solution to eqs. ( 11) by the Billing-Baer propagator method [ 41 requires diagonalization of the matrix E”+S( r> H’ a number of times as S(T) is switched on. The integration steplength which can be used in this propagation method is typically AT= 1, i.e. much larger than conventional integrator methods. Thus the propagation of the solution to eqs. ( I1 ) typically requires from 40 to 100 diagonalizations whereas the exact eigenvalues are obtained simply by diagonalizing E”+ H ‘. Thus for obtaining eigenvalues the adiabatic switching method is only relevant if trajectory methods can be invoked [3]. The only reason for discussing eqs. ( 11) at all is that they have some interesting bearing upon the discussion of quantum chaos. Thus we may follow the adiabatic energy E,(T) as a function of time and compare it with the exact eigenvalue obtained by diagonalizing EO+S( T) H’. This is done in table 4 for levels n, 1 where n = 8. Note that the levels 8 IL8 are those which follow the exact values the best when the perturba-
Table 4 Comparison of the exact and the adiabatic tion of the perturbation S(T)& nl
&(exact)
eigenvalues
E(adiabatic)
?-=10,S(T)=0.09516s’ 8.99689 8 0 8.99710 8+-2 E&4 8.99775 8.99884 826 8+8 9.00036
X90( 5.80( 5.00( 3.70( 1.90(
r=20,S(T)=0.3297 8 0 8.95799 8+2 8.96090 8+_4 8.96966 8k6 8.98423 8+_8 9.00464
2.85(-3) 2.75( -3) 2.43(-3) l-93(-3) 1.241-3)
r=30,S(T)=0.5934 8 0 8.85635 8.86617 8f2 a+4 8.89596 8.94416,8.94487 8+6 B&8 9.0138 I
6.01( -3) 5.74( -3) 4.67(-3) 3.89( -3) 2.63(-3)
ai) S(T)=&(T)
524
witho=O.OOl.
17 March 1989
CHEMlCAL PHYSICS LEITERS
Volume 155, number 6
tion is switched on and that the difference between this level and the others is more pronounced for I”= 10 and 20, i.e. for smaller perturbations. Furthermore we note that there is a gradual change in the value of AE=E( adiabatic) -E( exact) on going from I= f 6 to l= 0. The difference in AE reflects the ability of the various levels to “resist” an external perturbation. An even more pronounced difference is found when considering the time dependence of the “memory function” P,(T) (see fig. 1). We see that the states l=O, + 2, + 4 and + 6 in the n= 8 level loose their memory about the initial and separable wavefunction much &ster than the I= + 8 state. But again we see that there is a gradual change in the behaviour of Z’,( T) as a function of 1. According to the Hose-Taylor criterium P,( 00 ) < 0.5 would designate the levels I= 0 to ? 6 as “chaotic”. But for all levels one can recover the separable initial situation by switching off the function S( T). Thus the value 0.5 is no “magic number” in the adiabatic switching discussion. In principle any number between zero and one could have been chosen [ 5 1. However, the value 0.5 may give the largest correlation between classical and quantum chaos. Fig. 1 clearly shows that the
as a func-
-E(exact)
-4) -4) -4) -4) -4)
0
10
20
30
T
Fig. 1. The initial state memory function P,(T) (eq. ( 14) ) as a function of Tfor the (n, I) = (8, I) levels. The non-separable part of the Hamiltonian is adiabatically switched on using the switchingfunction &(‘i-) with a=0.005555.
Volume 155, number 6
CHEMICAL
iz 0.9
E 0.8
t
0.7 t 0.6C
0.5 t 0.4 0.3 0.2 0.1 0.0 0
10
20
Fig.2.Sameasfig.lbutforthelevelsZ=n=1-12andI=O,n=l2 and 10.
III= n levels preserve most memory about the initial separable situation and fig. 2 shows that this memory gradually decreases with increasing n. For 1~0 and increasing n the memory function drops rapidly (fig. 2). Fig. 3 shows the number of expansion coefficients with Ia, I*a 0.0 1 as a function of I at T= T,,,. Thus
n.6 o_ 20
I
n -7
17 March 1989
PHYSICS LETTERS
the n, I= 8,O state is expanded using 14 zeroth-order states & with coeffkients larger than 0.01 but only 5 of these 14 states have Iao,,J‘> 0.10, representing 71% of the wavefunction. Thus those states which one would designate chaotic remain localized on relatively few zeroth-order states [ 61. But as n increases and/or 1 decreases the wavefunction expansion requires an increasing number of eigenfunctions of I?@ The results shown in figs. 1 and 2 show that one can characterize the quantum levels according to the rate at which they lose memory about their initial situation. Thus for the (n, I) = ( 12,O) state the P,( T) function drops below 0.5 at T= 10, i.e. after 10 vibrational periods, but the (n, I) = ( 12, _+12) levels have P[( T) > 0.5 for T-a. As mentioned, the value 0.5 suggested by Hose and Taylor does not have special significance in the adiabatic switching method but it may from a practical point of view be a reasonable choice. For the n=8 state we see that P,(T)=03 for 1=46 if T,,2221. Hence in a case where the perturbation is not an anharmonic potential but, say, an external field (e.g. an appropriate laser) to excite the molecule such that Pr( T) stays above a certain value optimal for the oscillator strengths. In any case we see that the definition of quantum chaos has to be seen in the context of the perturbation which is applied to the system. Xn our opinion a quantum state can be said to be chaotic if its memory function P,(T) drops below a certain
n:8
n:9
II
I
I
30
LO
50
n=lO
I
I I 60 level III
Fig. 3. The number (N) of zeroth-order basis functions 9: with weights /a,,/ * larger than 0.01 in the adiabatically propagated wavefunctions 9,(T) for T= T,.,. The switching function S,(T) and 153 levels were used in the propagation. The I quantum number is indicated for n = 9.
525
Volume 155, number 6
CHEMICAL PHYSICS LETTERS
value (e.g. OS) in a time interval shorter than the experimental measuring time. This is of course a rather loose and may be even an obvious “definition” but the question should be seen in a dynamical rather than a static framework and expressed in terms of rate of memory loss of the initial system. Note that in this discussion we have not constrained ourselves to systems which are initially separable. The adiabatic switching method allows us to study individual quantum states rather than a distribution of states. But the gradual change in P,(T) or in the survival function S,( I”) = 1 -P,(T) makes it impossible to distinguish between chaotic and non-chaotic behaviour by looking at S,(T) as the perturbation is switched on. In agreement with Hutchinson and Wyatt [ I ] we do not find any sudden change in any of the observed quantities as a function of quantum level - rather a smooth transition from regular to “non-regular” behaviour is observed.
3. Conclusion (a) Those states which are “chaotic” in a quantum mechanical sense lose their memory about the initial, separable situation rapidly when an external perturbation is switched on. (b) There is gradual transition from the regular to the chaotic situation. Thus in the non-separable case no state is completely chaotic or regular. (c) This must lead to the conclusion that in quantum mechanics the concept “chaos” does not have a unique definition, and in itself is not so important
526
17 March 1989
as in classical mechanics. What is important is how localized in function space the wavefunction of a given set of levels stays when the energy increases. But this discussion can be held without talking about quantum chaos.
Acknowledgement This research was granted computer time on the CRAY2 by the Centre de Calcul Vectoriel pour la Recherche (CNRS).
References [ 1) I.C. Percival, J. Phys. B 6 (1973) L222; R.M. Stratt, NC. Handy and W.H. Miller, J. Chem. Phys. 7 1 (1979) 3311; D.W. Noid, M.L. Koszykowski and R.A. Marcus, Chem. Phys. Letters 73 (1980) 269; R.A. Marcus, Faraday Discussions Chem. Sot. 75 (1983) 103; N. Pomphrey, J. Phys. B 7 ( 1974) 1909; SC. Farantos and J. Tennyson, I. Chem. Phys. 82 (1985) 800; K.S.J. Nordholm and S.A. Rice, J. Chem. Phys. 61 (1974) 203. [Z] G. Hose and H.S. Taylor, I. Chem. Phys. 76 (1982) 5356. [ 31 B.R. Johnson, J. Chem. Phys. 83 (1985) 1204. [4] G.D. Billing and M. Baer, Chem. Phys. Letters 48 (1979) 327. [ 51 S. Mukamel, J. Chem. Phys. 78 (1983) 5843. [6 ] N. Moiseyev, R.C. Brown, R.E. Wyatt and E. Tzidoni, Chem. Phys.Letters 127 (1986) 37. [7] J.S. Hutchinson and R.E. Wyatt, Phys. Rev. A 23 (1981) 1567.
17 March 1989
CHEMICAL PHYSICS LETTERS
ON THE APPLICATION OF THE ADIABATIC INVARIANCE METHOD FOR THE IDENTIFICATION OF “QUANTUM CHAOS” Gert Due BILLING Chemistry Laboratory III, H. C. Brstedlnsfitute. University of Copenhagen, DIG2100 Copenhagen 0, Denmark
and Georges JOLICARD Laboraroire de Physique MolPnrlaire, WA 722 du CNRS, Fact& des Sciences et des Techniques, La Bouloie. 25030 BesancoanCedex. France
Received 11 October 1988; in final form 14 January 1989
The quantum-mechanical adiabatic invariance method is applied to the H&non-Heiles Hamiltonian and used to discuss the criteria for a quantum analog to classical chaotic motion.
1. Introduction There has been much interest in exploring the quantum-mechanical analog to classical chaos, and several methods or indicators of quantum chaos have been suggested [ 1,2]. Of these we mention: sensitivity to perturbations, irregularity of the nodal structure of the wavefunction, overlapping avoided crossings, second energy differences with respect to changes in the coupling and the dominant coefficient method proposed by Hose and Taylor [ 2 1. As far as we known the question has not been examined in the context of the adiabatic invariance switching (AIS) method [ 3 ]. In this method the term which causes the Hamiltonian to be non-separable and thus provides the possibility for chaotic behaviour, is switched on adiabatically, i.e. infinitely slowly. According to Ehrenfest’s hypotheses the quantum numbers (the labelling of the wavefunctions) are invariant - but the energy of the state and the wavefunction itself changes. The AIS method has recently been applied by Johnson [ 31 to obtain semiclassical eigenvalues for the H&non-Heiles Hamiltonian. In the present paper we use the AIS method within a purely quanturn-mechanical framework and solve the time-dependent Schr6dinger equation for the system. By 0 009-2614/89/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
comparing the eigenvalues and the time-dependent “memory function” it is possible to gain new insight into the phenomena of quantum chaos and discuss the results in the context of previously suggested criteria. We choose for our investigation a system and an energy regime for which classical chaotic as well as regular motion has been found. Only in this way is it possible to distinguish between classical and quantum chaotic behaviour. As the energy of the system increases criteria found for quantum chaos should become more and more pronounced. In the present paper it is suggested that the memory function delined as the projection of the wavefunction, obtained by the AIS method, onto the initial separable wavefunction yields a “measure” for quantum chaos.
2. The AIS equations Consider the H&ton-Heiles B(r, e)=&+B,
,
Hamiltonian (11
where B.V.
521
Volume
155,number 6
CHEMICAL
PHYSICS LETTERS
17 March 1989
(2) and A, = @r’cos 30 _ In the AIS mechanically
method
(3) one
obtains
g(n)=i
i$&(T)yl,
(4)
where fi( T) = Ei+ S( T)fi, and S(T) is a switching function such that S(0) =O and limT_, S(T) = 1. We have introduced in eqs. ( l)-(4) dimensionless units such that fro= 1 and time is in units of w-l. In order to solve eq. (4) we expand the wavefunction in eigenfunctions of the Hamiltonian fiO, 6) =-%VI%(r, 0) >
(5)
t&,(r,O)=N,,r’exp(-$r*)L!,(r*)exp(+ilO),
(6)
&&(C where
and (n2=0,1,...),
(8)
where n=t(n,-I),
I=&,&-2
)..‘l (or.).
(9)
Thus we use the expansion VI= c a,,(t) a
fi (n-i+s) 1=I
, nal,
(13)
wherecx=(n,I),j?=(n’,P),r=j(5+I’+I)+m’and s= 1+ I- y. We will now follow as a function of time the “memory function” defined as the overlap between the adiabatically propagated wavefunction y,(t) and the initial wavefunction v/y, i.e. P~(T)=I((V,(t)l~10)i2=[aN(T)12
(14)
and the adiabatic energy of the state I, i.e.
&(T)=l-exp(-aT),
the initial
(~czl(O)=~CI,). Inserting ( 10) into eq. (4) we obtain .. la,l= g bWL,+SCT) Hf,~)qt~,
state S,(T)=l-exp(-aT2). (11)
where H~g=f~(W~1r2COS301~/Og).
Note that the quantity Pr( T) in the limit T-m approaches the Hose-Taylor weight. This weight was used to determine the “quantum periodicity”. It was argued that weights larger than 0.5 would give a convergent iteration for an effective Hamiltonian. The switching function in eq. ( 11) is in principle arbitrary but investigations made bp Johnson [ 31 have shown that the convergence rates depend strongly upon the choice of S( 2’). In the present work we have used four different functions
(10)
&CC 0) I
where LY=(n, I) and I denotes
(12)
This matrix element may easily be evaluated giving
522
n=o,
= c EE I a,l I ’+ S( V C G&&p. a a8
(7)
E,,=nz+l
g(n)= 1,
quantum
The second and third choices are Johnson’s fimction S, and & whereas S, and S, contain a parameter a which for a-+0 makes the switching more adiabatic. Another difference is #at S2 and S, operate on a finite interval [0, T,,,,,], i.e. S,( T,,,,) = 1.0 whereas the other two only approach this limit for T-+co. Hence the integration is terminated when S(T) exceeds some number close to unity. In the present analyses we use a basis set consisting of levels from n=O to n= 16, i.e. altogether 153
Volume 155, number 6
Table 1 Eigenvalues of the H&ton-Heiles
CHEMICAL
Hamiltonian
PHYSICS LETTERS
17 March 1989
as a function of basis set size
nl
91 a’
12oa’
153 8’
465 D,
4
4.8702 4.8987 4.9863 5.8172
4.8702 4.8987 4.9863 5.8171
4.8702 4.8987 4.9863 5.8170
4.8702 4.8987 4.9863 5.8170
4A, 4E 5E 6E
5.8673, 5.8815 5.9914
5.9913
5.8670,5.8815 5.9913
5.8670, 5.8815 5.9913
5A,, 2A, 7E
6.7404 6.1672 6.8546
6.7382 6.7652 6.8536
6.7380 6.7649 6.8535
6.7379 6.7649 6.8534
6A, SE 9E
6.9992, 6.998 7.6742
7.6624
6.9990,6.9995 7.6600
6.9989, 6.9994 7.6595
3A1,7A, 10E
+3 +5 +7
7.7162, 7.1410 7.8396 8.0139
7.8336 8.0098
7.6984, 7.7370 7.8329 8.0095
7.6977, 7.7369 7.8327 8.0094
IlE 12E
0 +2 54 +6 k8
8.6170 8.6399 8.7103 8.8324, 8.8379 9.0321
8.5581 8.5808 8.6792 8.8122, 8.8165 9.0222
8.5541 8.5764 8.6779 8.8113, 8.8152 9.0217
9A, 13E 14E 5A2, lOA, 15E
0 *2 +4 5+1 f3
* f5 6
0 +2 *4
26 7+1
8
‘) Present work.
8.5723 8.5958 8.6852 9.0234
gA,,4A1
b, Ref. [2].
states. Table 1 shows that this number of functions is suffkient to obtain good agreement with the HoseTayIor values [ 21. For smaller perturbations (S(T) < 1) the basis set will of course be even better and in any case sufficiently large for the present discussion. Table 2 shows that both S, and S, are superior to the S, switching function with ~7~0.2. Decreasing a
and increasing T,,,,, will improve the agreement as shown in table 3 for the S, function. Note also that the A,, AZ splitting for the I= IL3, f 6 etc. levels are not recovered by the adiabatic propagation. This point has been thoroughly discussed by Johnson. But except for this “problem” we may with the adiabatic propagation get arbitrarily close to the exact eigenvalues. Thus there exists a method of changing adi-
Table 2 Eigenvalues obtained using the AIS method (eq. (4) ) and various switching functions compared with the exact quantum-mechanical values
Table 3 Comparison of adiabatic and exact eigenvalues forf( T) =0.9999. The switching function isf( 7’)= 1 -exp( -aT’)
nl
S,(T)
S,(T)
S, (T]
Exact a)
RI
a-0.005555
a=O.OOl
Exact ‘)
751 7t3 7+5 7*7 8 0 8f2 Sk4 856 8+8
7.6686 1.7244 7.8372 8.0131 8.5719 8.5996 8.6825 8.8218 9.0279
7.6690 7.7247 7.8374 8.0133 8.5726 8.6002 8.6829 8.8222 9.028 1
1.6985 7.7500 7.8543 8.0172 8.6167 8.6418 8.7176 8.8449 9.0346
7.6600 1.6984, 7.7370 7.8329 8.0095 8.5581 8.5808 8.6792 8.8122, 8.8165 9.0222
7+1 723 725 7*7 8 0 81t2 8f4 8?6 s+fl
7.6733 7.7285 7.8405 8.0154 8.5797 8.6069 8.6886 8.8267 9.03 I4
7.6655 7.7222 7.836 I 8.0120 8.5671 8.5937 8.6812 8.8197 9.026 I
7.6600 7.6984, 7.7370 7.8329 8.0095 S-5581 8.5808 8.6792 8.8122, 8.8165 8.0222
T,,,,, = 50
T,,,=50
T-.=36
T,,,=40
T,,, =
” Obtained
with 153 states in expansion
(I 0).
100
‘I Obtained with 153 states in the expansion of the wavefunction. 523
abatically from the separable to the non-separable system. From a practical point of view one would of course never calculate the eigenvalues in this manner - since the propagation of the solution to eqs. ( 11) by the Billing-Baer propagator method [ 41 requires diagonalization of the matrix E”+S( r> H’ a number of times as S(T) is switched on. The integration steplength which can be used in this propagation method is typically AT= 1, i.e. much larger than conventional integrator methods. Thus the propagation of the solution to eqs. ( I1 ) typically requires from 40 to 100 diagonalizations whereas the exact eigenvalues are obtained simply by diagonalizing E”+ H ‘. Thus for obtaining eigenvalues the adiabatic switching method is only relevant if trajectory methods can be invoked [3]. The only reason for discussing eqs. ( 11) at all is that they have some interesting bearing upon the discussion of quantum chaos. Thus we may follow the adiabatic energy E,(T) as a function of time and compare it with the exact eigenvalue obtained by diagonalizing EO+S( T) H’. This is done in table 4 for levels n, 1 where n = 8. Note that the levels 8 IL8 are those which follow the exact values the best when the perturba-
Table 4 Comparison of the exact and the adiabatic tion of the perturbation S(T)& nl
&(exact)
eigenvalues
E(adiabatic)
?-=10,S(T)=0.09516s’ 8.99689 8 0 8.99710 8+-2 E&4 8.99775 8.99884 826 8+8 9.00036
X90( 5.80( 5.00( 3.70( 1.90(
r=20,S(T)=0.3297 8 0 8.95799 8+2 8.96090 8+_4 8.96966 8k6 8.98423 8+_8 9.00464
2.85(-3) 2.75( -3) 2.43(-3) l-93(-3) 1.241-3)
r=30,S(T)=0.5934 8 0 8.85635 8.86617 8f2 a+4 8.89596 8.94416,8.94487 8+6 B&8 9.0138 I
6.01( -3) 5.74( -3) 4.67(-3) 3.89( -3) 2.63(-3)
ai) S(T)=&(T)
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tion is switched on and that the difference between this level and the others is more pronounced for I”= 10 and 20, i.e. for smaller perturbations. Furthermore we note that there is a gradual change in the value of AE=E( adiabatic) -E( exact) on going from I= f 6 to l= 0. The difference in AE reflects the ability of the various levels to “resist” an external perturbation. An even more pronounced difference is found when considering the time dependence of the “memory function” P,(T) (see fig. 1). We see that the states l=O, + 2, + 4 and + 6 in the n= 8 level loose their memory about the initial and separable wavefunction much &ster than the I= + 8 state. But again we see that there is a gradual change in the behaviour of Z’,( T) as a function of 1. According to the Hose-Taylor criterium P,( 00 ) < 0.5 would designate the levels I= 0 to ? 6 as “chaotic”. But for all levels one can recover the separable initial situation by switching off the function S( T). Thus the value 0.5 is no “magic number” in the adiabatic switching discussion. In principle any number between zero and one could have been chosen [ 5 1. However, the value 0.5 may give the largest correlation between classical and quantum chaos. Fig. 1 clearly shows that the
as a func-
-E(exact)
-4) -4) -4) -4) -4)
0
10
20
30
T
Fig. 1. The initial state memory function P,(T) (eq. ( 14) ) as a function of Tfor the (n, I) = (8, I) levels. The non-separable part of the Hamiltonian is adiabatically switched on using the switchingfunction &(‘i-) with a=0.005555.
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CHEMICAL
iz 0.9
E 0.8
t
0.7 t 0.6C
0.5 t 0.4 0.3 0.2 0.1 0.0 0
10
20
Fig.2.Sameasfig.lbutforthelevelsZ=n=1-12andI=O,n=l2 and 10.
III= n levels preserve most memory about the initial separable situation and fig. 2 shows that this memory gradually decreases with increasing n. For 1~0 and increasing n the memory function drops rapidly (fig. 2). Fig. 3 shows the number of expansion coefficients with Ia, I*a 0.0 1 as a function of I at T= T,,,. Thus
n.6 o_ 20
I
n -7
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the n, I= 8,O state is expanded using 14 zeroth-order states & with coeffkients larger than 0.01 but only 5 of these 14 states have Iao,,J‘> 0.10, representing 71% of the wavefunction. Thus those states which one would designate chaotic remain localized on relatively few zeroth-order states [ 61. But as n increases and/or 1 decreases the wavefunction expansion requires an increasing number of eigenfunctions of I?@ The results shown in figs. 1 and 2 show that one can characterize the quantum levels according to the rate at which they lose memory about their initial situation. Thus for the (n, I) = ( 12,O) state the P,( T) function drops below 0.5 at T= 10, i.e. after 10 vibrational periods, but the (n, I) = ( 12, _+12) levels have P[( T) > 0.5 for T-a. As mentioned, the value 0.5 suggested by Hose and Taylor does not have special significance in the adiabatic switching method but it may from a practical point of view be a reasonable choice. For the n=8 state we see that P,(T)=03 for 1=46 if T,,2221. Hence in a case where the perturbation is not an anharmonic potential but, say, an external field (e.g. an appropriate laser) to excite the molecule such that Pr( T) stays above a certain value optimal for the oscillator strengths. In any case we see that the definition of quantum chaos has to be seen in the context of the perturbation which is applied to the system. Xn our opinion a quantum state can be said to be chaotic if its memory function P,(T) drops below a certain
n:8
n:9
II
I
I
30
LO
50
n=lO
I
I I 60 level III
Fig. 3. The number (N) of zeroth-order basis functions 9: with weights /a,,/ * larger than 0.01 in the adiabatically propagated wavefunctions 9,(T) for T= T,.,. The switching function S,(T) and 153 levels were used in the propagation. The I quantum number is indicated for n = 9.
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value (e.g. OS) in a time interval shorter than the experimental measuring time. This is of course a rather loose and may be even an obvious “definition” but the question should be seen in a dynamical rather than a static framework and expressed in terms of rate of memory loss of the initial system. Note that in this discussion we have not constrained ourselves to systems which are initially separable. The adiabatic switching method allows us to study individual quantum states rather than a distribution of states. But the gradual change in P,(T) or in the survival function S,( I”) = 1 -P,(T) makes it impossible to distinguish between chaotic and non-chaotic behaviour by looking at S,(T) as the perturbation is switched on. In agreement with Hutchinson and Wyatt [ I ] we do not find any sudden change in any of the observed quantities as a function of quantum level - rather a smooth transition from regular to “non-regular” behaviour is observed.
3. Conclusion (a) Those states which are “chaotic” in a quantum mechanical sense lose their memory about the initial, separable situation rapidly when an external perturbation is switched on. (b) There is gradual transition from the regular to the chaotic situation. Thus in the non-separable case no state is completely chaotic or regular. (c) This must lead to the conclusion that in quantum mechanics the concept “chaos” does not have a unique definition, and in itself is not so important
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as in classical mechanics. What is important is how localized in function space the wavefunction of a given set of levels stays when the energy increases. But this discussion can be held without talking about quantum chaos.
Acknowledgement This research was granted computer time on the CRAY2 by the Centre de Calcul Vectoriel pour la Recherche (CNRS).
References [ 1) I.C. Percival, J. Phys. B 6 (1973) L222; R.M. Stratt, NC. Handy and W.H. Miller, J. Chem. Phys. 7 1 (1979) 3311; D.W. Noid, M.L. Koszykowski and R.A. Marcus, Chem. Phys. Letters 73 (1980) 269; R.A. Marcus, Faraday Discussions Chem. Sot. 75 (1983) 103; N. Pomphrey, J. Phys. B 7 ( 1974) 1909; SC. Farantos and J. Tennyson, I. Chem. Phys. 82 (1985) 800; K.S.J. Nordholm and S.A. Rice, J. Chem. Phys. 61 (1974) 203. [Z] G. Hose and H.S. Taylor, I. Chem. Phys. 76 (1982) 5356. [ 31 B.R. Johnson, J. Chem. Phys. 83 (1985) 1204. [4] G.D. Billing and M. Baer, Chem. Phys. Letters 48 (1979) 327. [ 51 S. Mukamel, J. Chem. Phys. 78 (1983) 5843. [6 ] N. Moiseyev, R.C. Brown, R.E. Wyatt and E. Tzidoni, Chem. Phys.Letters 127 (1986) 37. [7] J.S. Hutchinson and R.E. Wyatt, Phys. Rev. A 23 (1981) 1567.