Volume 104A, number 2 PHYSICS LETTERS 13 August 1984

DOES AN IRREGULAR SEQUENCE OF LEVEL SPACINGS PROVIDE

A NEW REPRESENTATION OF QUANTUM CHAOS?

David FARRELLY Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90024, USA

Received 29 May 1984

The recent claim by Hirooka et al. that an irregular sequence of energy level spacings gives an unambiguous representa- tion of quantum chaos is examined. It is shown that observation of an irregular sequence of spacings does not necessarily imply that the dynamics is chaotic.

1. Introduction. The suggestion by Percival [ 1 ] that classical chaos might manifest itself in a random distribution of quantum energy levels has been fol- lowed by a number of investigations into the nature of the spectrum of both integrable and non-integrable hamiltonians [2-10] . Recently, on the basis of a comparison of the sequences of energy level spacings corresponding to a separable and a non-integrable hamiltonian, Hirooka et al. [ 11 ] claim to have found a new representation of quantum chaos;namely that an "irregular sequence of level spacings gives an un- ambiguous representation of chaos in quantum sys- tems". Hirooka et al. base this statement on their ob- servation that the sequence of level spacings they cal- culate is regular for the separable system but irregu- lar in the non-integrable case. Of particular interest is their suggestion that even when the dynamics ap- pears to be almost integrable, the sequence of energy level spacings will be irregular, thus providing an ex- tremely sensitive indicator of the presence of chaos. The purpose of this note is to present evidence that no such clear correspondence exists and to demon- strate that it is in fact possible to observe irregular sequences of level spacings even when the dynamics is not chaotic.

In section 2 we review briefly the calculations of Hirooka et al. [ 11 ] and point out that on the basis of their comparison of the level spacings in a sepa- rable system with those of a non-integrable system it is not possible to draw any conclusions as to the pos-

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sible connection between classical chaos and an irreg- ular spectrum. Rather, since separability is a suffi- cient but not a necessary condition for integrability, it is apparent that the results obtained by Hirooka et al. do not rule out the possibility that an irregular sequence of level spacings is a consequence of non- separability rather than of non-integrability. A nec- essary test of the conjectured sensitivity of level spacings to chaos is therefore a comparison of the lev- el spacings in an integrable but non-separable system with those of a non-integrable problem. In order to make such a comparison as clear cut as possible it is important to choose the integrable and non-integrable hamiltonians to be in some sense "close" to each oth- er. For example we might require that the dynamics of the integrable system closely approximate the non- integrable system when the dynamics of the latter is mainly regular. In fact general techniques exist to construct integrable approximations to non-integrable hamiltonians [12]. Taking the H~non-Heiles prob- lem as an example, it is shown in section 3 that such an approximation to the exact non.integrable dynam- ics leads to a sequence of energy level spacings almost indistinguishable from the exact spacings. Conclu- sions are in section 4.

2. Energy level spectrum. Hirooka et al. [11] cal- culated eigenvalues for the hamiltonian

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Volume 104A, number 2 PHYSICS LETTERS 13 August 1984

_ l ( e 2 2 1 2 H - - ~ + P # ) + 5 ( x + y 2 )

+ ¼ol(ax 4 + by 4 + 2cx2y2) , (1)

with a = 0.088 and for the two sets of parameters; a = b = c = 1 (case A) and a = b = 1, c = 2 (case B). The hamiltonian is separable (and therefore integra- ble) in case A whereas case B is non-integrable. As noted in ref. [ 11 ] this hamiltonian has previously been considered by Pullen and Edmonds [13] for the case a = b = 0, c = 1 which is also non-integrable.

Defining the level spacing,

AE n = En+ 1 - En, (2)

Hirooka et al. examine histograms of the spacing AE n as a function of n and also the sequence of level spac- ings in the (AEn, AEn+I) plane. Observing regular be- havior of the level spacings in the separable case but irregular spacings for the non-integrable problem they conclude that the irregular spacing is due to chaos. However, based on these calculations it is not possible to state conclusively that the observed irregular se-

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Fig. 1. Composite surface of section for (a) case A and (b) case B obtained by running 10 trajectories with different initial conditions in each case, at E = 30.

quence of level spacings in case B is a consequence of non-integrability rather than of non-separability. Put another way, is it possible to construct non-separable but integrable hamiltonians which have an irregular spectrum? Another question might also be raised as to the source of irregularity in the spectrum. In the energy regime where case B is largely regular, the dy- namics, as evidenced by the surface of section plots in fig. 1, is qualitatively very different from case A. Since the dynamics in the two cases is so different it is difficult to pinpoint the particular feature(s) of the classical dynamics which give rise to an irregular spec- trum.

3. Level spacings for the H#non-Heiles problem. For the range of energies considered by Hirooka et al. case B appears to be almost integrable; evidently (see fig. lb), an approximate "third constant of motion" exists. In other words, there exists some integrable system whose (exact) "third constant" is an approx- imate constant for the non-integrable problem. A gen- eral technique for constructing such an integrable hamiltonian (the normal form) is Birkhoff-Gustav- son theory [ 12,14-16]. It would clearly be of inter- est to examine the energy level spacings for the inte- grable but generally non-separable normal form and compare with the sequence obtained from the non- integrable hamiltonian of eq. (1). However, since re- sults already exist in the literature for the H6non- Heiles problem and the quadratic Zeeman effect in hydrogen, it is sufficient to refer to these instances. Using Birkhoff-Gustavson theory Swimm and Delos [14] and Jaff6 and Reinhardt [16] constructed the normal form for the H6non-Heiles problem which after transforming to action-angle variables they quantized semiclassically. Using appropriate uniform semiclassical methods Jaff~ and Reinhardt [16] ob- tained good estimates for all of the levels of the H6non-Heiles hamiltonian, even in the chaotic re- gions of phase space. Taking their published results [16] we calculated energy level histograms and plots of AEn+ 1 as a function of AE n in analogy with ref. [ 11 ]. Examination of figs. 2 and 3 reveals that there is no significant difference between the results for the exact non-integrable problem and the integra- ble normal form hamiltonian. The level sequences ap- pear irregular in both cases. Clearly the irregularity observed is not a consequence of chaos; instead it ap-

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Volume 104A, number 2 PHYSICS LETTERS 13 August 1984

AE

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1 .0-

o . o ' II 0 10 20 30 40 50 60 70

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Fig. 2. Energy level histograms for (a) the non.integrable H~non-Heiles hamiltonian (quantum results) and (b) the integrable normal form (uniform semiclassical results), based on ref. [16].

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Fig. 3. Energy level sequences for (a) the H6non-Heiles hamiltonian (quantum results) and (b) the normal form (uniform semielassical results), based on ref. [ 16].

pears to be due to non-separability. It may perhaps be argued that since the H6non-Heiles hamiltonian strictly supports no bound states these results are in- conclusive. However, the normal form hamiltonian is truly bound and thus provides an example o f an inte- grable system having an irregular sequence of bound levels according to the definition o f Hirooka et al. [11 ]. It is reasonable to expect that analogous re- suits would be obtained for integrable approxima- tions to the hamiltonian o f eq. (1).

4. Conclusion. On the basis o f results already in the literature we have provided a counter example to the claim of Hirooka et al. that an irregular sequence of level spacings is the hallmark of chaos. It is certain- ly tempting to identify irregular or complicated be- havior o f some quantity such as level spacings with chaos. However, it is clear that complicated dynam- ics does not necessarily imply chaos. Other examples, such as the quadratic Zeeman effect in hydrogen [10, 17,18] could also have been put forward as evidence to support the conclusion that the criteria outlined in ref. [11 ] do not provide a new representation of quantum chaos. Interestingly, the Zeeman problem appears to be an example o f a classically chaotic sys- tem [ 17] whose level spacing probability distribution indicates almost integrable dynamics [9,10,18]. In this note we have not attempted to comment on the larger question as to whether any property o f the spectrum (e.g. multiply avoided crossing) [3-5 ,9 ,10, 18] is a consequence of chaotic, rather than of non- separable dynamics. This issue is currently o f consid- erable interest and has been examined in a number of recent reviews [4,10,19].

R eferenc es

[1] l.C. Percival, J. Phys. B6 (1973) L229. [2] M.V. Berry and M. Tabor, Proc. R. Soc. A356 (1977)

375; S.W. McDonald and A.N. Kaufman, Phys. Rev. Lett. 42 (1979) 1i89.

[3] R.A. Marcus, in: Horizons of quantum chemistry, eds. K. Fukui and B. Pullman (Reidel, Dordrecht, 1980) p. 107.

[4 ] D.W. Noid, M.L. Koszykowski and R.A. Marcus, Ann. Rev. Phys. Chem. 32 (1981) 267.

[5 ] D.W. Noid, M.L, Koszykowski and R.A. Marcus, Chem. Phys. Lett. 73 (1980) 269; J. Chem. Phys. 78 (1983) 4018.

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Volume 104A, number 2 PHYSICS LETTERS 13 August 1984

[6] G.M. Zaslavsky, Phys. Rep. 80 (1981) 157. [7] A.R. Edmonds, R.A. PuUen and I.C. Percival, Chem.

Phys. Lett. 91 (1982) 273. [8] M.V. Berry, Ann. Phys. 131 (1981) 163. [9] W.P. Reinhardt, in: Chaotic behavior in quantum me-

chanics, Proc. 1983 Como Conf. on Quantum me- chanics, ed. G. Casati (Plenum, New York), to be pub- lished.

[10] W.P. Reinhardt, in: The mathematical analysis of phys- ical systems, ed. R. Mickens (Van Nostrand-Reinhold, New York), to be published.

[11] H. Hirooka, Y. Yotsuya, Y. Kobayashi and N. Sait6, Phys. Lett. 101A (1984) 115.

[12] S.A. Rice, Adv. Chem. Phys. 47 (1981) 117, and ref- erences therein.

[13] R.A. Pullen and A.R. Edmonds, J. Phys. A14 (1981) L477.

[14] R.T. Swimm and J.B. Delos, J. Chem. Phys. 71 (1979) 1906.

[15 ] R.B. Shirts and W.P. Reinhardt, J. Chem. Phys. 77 (1982) 5204.

[16] C. Jaff6 and W.P. Reinhardt, J. Chem. Phys. 77 (1982) 5191.

[17] W.P. Reinhardt and D. Farrelly, J. Phys. (Paris) 43 Colloq. Suppl. C2 (1982) 29.

[18] D. FarreUy and W.P. Reinhardt, J. Chem. Phys. 78 (1983) 606.

[19] M. Tabor, Adv. Chem. Phys. 46 (1981) 73.

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