CHAPTER 34. PROLOGUE 655

34.1 Quantum pinball

In what follows, we will restrict the discussion to the non-relativistic Schr¨odinger equation. The approach will be very much in the spirit of the early days of quan- tum mechanics, before its wave character has been fully uncovered by Schr¨odinger in the mid 1920’s. Indeed, were physicists of the period as familiar with classical Chapter 34 chaos as we are today, this theory could have been developed 80 years ago. It was the discrete nature of the hydrogen spectrum which inspired the Bohr - de Broglie picture of the old quantum theory: one places a wave instead of a particle on a Keplerian orbit around the hydrogen nucleus. The quantization condition is that Prologue only those orbits contribute for which this wave is stationary; from this followed the Balmer spectrum and the Bohr-Sommerfeld quantization which eventually led to the more sophisticated theory of Heisenberg, Schr¨odinger and others. Today we are very aware of the fact that elliptic orbits are an idiosyncracy of the Kepler problem, and that chaos is the rule; so can the Bohr quantization be generalized Anyone who uses words “quantum” and “chaos” in the to chaotic systems? same sentence should be hung by his thumbs on a tree in the park behind the Niels Bohr Institute. The question was answered affirmatively by M. Gutzwiller, as late as 1971: a —Joseph Ford chaotic system can indeed be quantized by placing a wave on each of the infinity of unstable periodic orbits. Due to the instability of the orbits the wave does not (G. Vattay, G. Tanner and P. Cvitanovi´c) stay localized but leaks into neighborhoods of other periodic orbits. Contributions of different periodic orbits interfere and the quantization condition can no longer be attributed to a single periodic orbit: A coherent summation over the infinity of ou have read the first volume of this book. So far, so good – anyone can periodic orbit contributions gives the desired spectrum. play a game of classical pinball, and a skilled neuroscientist can poke rat Y brains. We learned that information about chaotic dynamics can be ob- The pleasant surprise is that the zeros of the dynamical zeta function (1.10) tained by calculating spectra of linear operators such as the evolution operator derived in the context of classical chaotic dynamics, chapter 22 of sect. 20.3 or the associated partial differential equations such as the Liouville = 1/ζ(z) Y (1 − tp) , equation (19.33). The spectra of these operators can be expressed in terms of peri- p odic orbits of the deterministic dynamics by means of periodic orbit expansions. also yield excellent estimates of quantum resonances, with the quantum amplitude But what happens quantum mechanically, i.e., if we scatter waves rather than associated with a given cycle approximated semiclassically by the weight

point-like pinballs? Can we turn the problem round and study linear PDE’s in 1 i = ~ S p−iπmp/2 terms of the underlying deterministic dynamics? And, is there a link between tp 1 e , (34.1) |Λ | 2 structures in the spectrum or the eigenfunctions of a PDE and the dynamical prop- p erties of the underlying classical flow? The answer is yes, but ... things are be- whose magnitude is the square root of the classical weight (22.9) coming somewhat more complicated when studying 2nd or higher order linear 1 PDE’s. We can find classical dynamics associated with a linear PDE, just take ge- t = eβAp−sT p , p |Λ | ometric optics as a familiar example. Propagation of light follows a second order p wave equation but may in certain limits be well described in terms of geometric and the phase is given by the Bohr-Sommerfeld action integral S p, together with rays. A theory in terms of properties of the classical dynamics alone, referred chapter 42 an additional topological phase mp, the number of caustics along the periodic to here as the semiclassical theory, will not be exact, in contrast to the classi- trajectory, points where the naive semiclassical approximation fails. chapter 37 cal periodic orbit formulas obtained so far. Waves exhibit new phenomena, such as interference, diffraction, and higher ~ corrections which will only be partially In this approach, the quantal spectra of classically chaotic dynamical systems incorporated into the periodic orbit theory. are determined from the zeros of dynamical zeta functions, defined by cycle ex- pansions of infinite products of form chapter 23 = = 1/ζ Y (1 − tp) 1 − X t f − X ck (34.2) p f k

654 introQM - 10jul2006 ChaosBook.org version15.9, Jun 24 2017 10 ) 2 chapter 41 chapter 41 p , 8 1 0 p , ≥ 657 2 i r 6 r (34.3) , 1 r 1 ), r 2 r 4 , 1 r ether remark- 2 ow what cycles bjects of interest, is an example of ome basic notions ortest cycles up to n the ( ectrum to shocking he Coulomb poten- mensional Poincar´e 0 lium spectrum. This 6 0 energies are good to 8 4 2 e the game of pinball 10 coordinates ( nted, we forge ahead ated with the periodic ost all of the classical for the survivors turns ChaosBook.org version15.9, Jun 24 2017 2 r (34.1) which is expected ues; even though the cycle , the helium nucleus charge - ∞ axis 1 r . 2 r 1 + axis. 1 2 r r + 2 2 r ) plane is topologically similar to the pinball motion − semiclassical approximation 2 helium, with zero total angular momentum, and the r 1 2 , r 1 r − 2 2 ++ A typical collinear helium trajectory in p 1 2 collinear plane; the trajectory enters along the 2 + r 34.1: 2 1 – p 1 34. PROLOGUE r 1 2 - = Figure the and escapes to infinity along the , we will finally be in position to accomplish something altog H p Before we can get to this point, we first have to recapitulate s The motion in the ( We set the electron mass to 1, the nucleus mass to to 2, the electron charges to -1. The Hamiltonian is a given symbol string length then yields an estimate of the he in a 3-disk system, excepttial. that the The motion classical is collineartrajectories not escape. helium free, is but Miraculously, in also the t symbolic a dynamics repeller; alm need to be computed for the cycle expansion (1.11). A set of sh 1% all the way down to the ground state. of quantum mechanics; after having defined the main quantum o introQM - 10jul2006 out to be binary, just as in the 3-disk game of pinball, so we kn to be good only in the classical large energy limit, the eigen quadrant, figure 34.1, or,section. better still, by a well chosen 2-di Due to the energy conservation, only three of the phase space 34.2 Quantization of helium Once we have derived the semiclassical (34.1) weight associ orbit simple calculation yields surprisingly accurate eigenval expansion was based on the accuracy. From the dreadedPoincar´e’s classical and dynamics intractable point 3-body of problem.and view, Undau consider helium the unpredictable, and compute a “chaotic” part of the helium sp able. We are now able to put together all ingredients that mak two electrons on the opposite sides of the nucleus. CHAPTER are independent. The dynamics can be visualized as a motion i chapter 38 chapter 23 chapter 29 chapter 39 ) the 656 cycle or in a quan- of cycle ex- race formulas to demonstrate s over periodic anized around a m mechanically ich we shall not is reason the cy- ction leaks out of ginary part of the hts related to their ? l, except who truly ermittent dynamics ieced together from ated by short cycles han the probabilities orbits, periodic orbit ace formula) fit into nenergies. For open itates development of the periodic orbit sums ChaosBook.org version15.9, Jun 24 2017 nto (34.2), and estimates tings, are also a powerful ths and stabilities of a fi- ince the invariant density er from the classical ones ff erences may be traced back ff resonances of classically chaotic systems. erent types of dynamical behavior such as, on one ff contributions, with longer cycles contributing rapidly corrections. Computations with dynamical zeta functions quantum cycles. These short cycles capture the skeletal topology of from such polynomial approximations. associated to every prime (non-repeating) periodic orbit ( fundamental /ζ p t 34. PROLOGUE curvature fundamental The key observation is that the chaotic dynamics is often org From the vantage point of the dynamical systems theory, the t First, we shall warm up playing our game of pinball, this time Where is all this taking us? The goal of this part of the book is The type of dynamics has a strong influence on the convergence erent approaches for di ff . p few (both the exact Selberga and general the framework semiclassical oforbits. Gutzwiller replacing For tr classical phase hyperbolic space systems this averages is by possible s sum motion in the sense that anythe long fundamental orbit can cycles. approximately be Incle p expansion chapter (34.2) 23 is it agrouped was highly into shown convergent expansion that domin for th decreasing are rather straightforward; typically onenite determines number leng of shortest periodic orbits,the substitutes zeros them of i 1 with weight can be represented byinstability. sum over The all semiclassical periodic periodic orbits, orbit with weig sums di only in phase factors and stability weights; such di tum version. Were theone game would of determine pinball its asystems stationary closed one eigenfunctions system, seeks and quantu instead eige eigenenergy complex describes resonances, the where rate the atthe ima which central the scattering quantum region.wants wave to fun This know will accurately the turn resonances out of to a quantum work pinball wel hand, the strongly hyperbolic and, on the other hand, the int of chapters 23 and 29.discuss here), with mixed For phase generic spacesummations are and nonhyperbolic hard marginally systems to stable (wh control, andshould it necessarily is be still the not computational clear method that of choice. that the cycle expansions, developed so fartool in for classical set evaluation of to the fact that inare quantum added. mechanics the amplitudes rather t pansions and the propertiesdi of quantal spectra; this necess CHAPTER introQM - 10jul2006 659 (Springer, (Springer- ChaosBook.org version15.9, Jun 24 2017 (Addison-Wesley, New , 3. edition (Springer-Verlag, New Semiclassical Physics (Springer-Verlag, New York 1992). Chaos in Classical and Quantum Mechanics Mathematical Methods in Classical Mechanics The Transition to Chaos in Conservative Classical Systems: Quantum Signatures of Chaos Verlag, Berlin 1978). York 1997). York 2010). New York 1990). Quantum Manifestations refsIntroQM - 13jun2008 [34.2] M.C. Gutzwiller, [34.5] V.I. Arnold, [34.3] L.E. Reichl, [34.4] F. Haake, REFERENCES References [34.1] M. Brack and R.K. Bhaduri, 658 ycle expansion ate the quantum to chaos in quantal ects in chapter 42. subsequent chapters ff amics required for the aos. More suitable as a ian dynamics starting r Madelung’s ‘quantum l system. We will then ida’s monograph [A2.9] haotic dynamics both in anics. For that, Arnol’d ChaosBook.org version15.9, Jun 24 2017 uantum propagator and unction as a sum and as on quantum and celestial 926. e goal is to milk the deter- 4.4], among others. ntrates on the periodic orbit n chapter 39, the spectrum on the random matrix theory sical methods. Gutzwiller’s ion. The book by Brack and with emphasis on periodic or- antization. This book is worth raction e ff ered along the way. The derivation ff A key prerequisite to developing any theory of croigrsfirstSchr¨odinger’s wave mechanics paper [37.3] (hydrogen The dates. Guide to literature.

34. PROLOGUE 34.2 34.1 This book does not discuss the random matrix theory approach ers a compact introduction to the aspects of Hamiltonian dyn ff quantization of integrable andbits, nearly normal integrable forms, systems, catastrophyBhaduri theory [34.1] and is torus an quantizatmonograph excellent [A1.13] introduction is to anclassical the advanced Hamiltonian semiclas introduction settings and focusingbrowsing in on through the c semiclassical for qu itsmechanics many even insights if and one erudite isgraduate not comments course working text on is problems Reichl’s of exposition quantum [34.3]. ch spectra; no randomness assumptions areministic made chaotic here, dynamics for rather its th theory. full For worth. an The introduction book to conce “quantumthe chaos” reader that is focuses referred to the excellent monograph by Haake [3 Remark spectrum) was submitted 27theory January in 1926. hydrodynamical form’ Submission paper [37.2] date was fo 25 October 1 introQM - 10jul2006 the quantum propagator andpropagation to the the classical Green’s flow of function, theproceed underlying to dynamica we construct will semiclassical approximations rel tothe the Green’s q function. Afrom rederivation the of Hamilton-Jacobi classical equation Hamilton will be o Commentary Remark “quantum chaos” is solid understanding of Hamiltonian mech monograph [34.5] is theo essential reference. Ozorio de Alme CHAPTER of the Gutzwiller trace formula anda the semiclassical product zeta over f periodic orbitswe will be buttress given our in chapter casecalculation 38. of by In scattering applying resonances in and aof 3-disk extending helium billiard i in the chapter theory: 41, and the a incorporation c of di