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Chao-Dyn/9402001 7 Feb 94 chao-dyn/9402001 7 Feb 94 DESY ISSN Quantum Chaos January Einsteins Problem of The study of quantum chaos in complex systems constitutes a very fascinating and active branch of presentday physics chemistry and mathematics It is not wellknown however that this eld of research was initiated by a question rst p osed by Einstein during a talk delivered in Berlin on May concerning Quantum Chaos the relation b etween classical and quantum mechanics of strongly chaotic systems This seems historically almost imp ossible since quantum mechanics was not yet invented and the phenomenon of chaos was hardly acknowledged by physicists in While we are celebrating the seventyfth anniversary of our alma mater the Frank Steiner Hamburgische Universitat which was inaugurated on May it is interesting to have a lo ok up on the situation in physics in those days Most I I Institut f urTheoretische Physik UniversitatHamburg physicists will probably characterize that time as the age of the old quantum Lurup er Chaussee D Hamburg Germany theory which started with Planck in and was dominated then by Bohrs ingenious but paradoxical mo del of the atom and the BohrSommerfeld quanti zation rules for simple quantum systems Some will asso ciate those years with Einsteins greatest contribution the creation of general relativity culminating in the generally covariant form of the eld equations of gravitation which were found by Einstein in the year and indep endently by the mathematician Hilb ert at the same time In his talk in May Einstein studied the BohrSommerfeld quantization R conditions p dq n i l for systems with l degrees of freedom i i i Abstract where the q are the co ordinates and the p their conjugate momenta de i i notes Plancks constant divided by and the n are integer quantum numbers i A short historical overview is given on the development of our Einstein calls the quantum conditions SommerfeldEpsteinsche Quantenbedin knowledge of complex dynamical systems with sp ecial emphasis on gung He emphasized that the pro ducts p dq are in general not invariant and i i ergo dicity and chaos and on the semiclassical quantization of inte thus the quantum conditions have no invariant meaning but rather dep end on grable and chaotic systems The general trace formula is discussed the choice of the co ordinate system in which the classical motion is separable if as a sound mathematical basis for the semiclassical quantization at all By analyzing a simple example the twodimensional motion of a particle of chaos Two conjectures are presented on the basis of which it is under an attractive central force Einstein found a general co ordinateinvariant argued that there are unique uctuation prop erties in quantum me formulation of the quantum conditions k l chanics which are universal and in a well dened sense maximally random if the corresp onding classical system is strongly chaotic I l X These prop erties constitute the quantum mechanical analogue of I p dq n k i i k the phenomenon of chaos in classical mechanics Thus quantum i L k chaos has been found P noticing that the line integrals of the oneform p dq taken over a complete i i i set of top ologically inequivalent irreducible closed lo ops L are invariant In k contrast to the original version of the quantization conditions it is not necessary to p erform explicitly the separation of variables indeed one need not require the motion to b e separable but only to b e multiply p erio dic However Einstein Invited contribution to the Festschrift UniversitatHamburg Schlaglichter p ointed out that conditions can only b e written down in the case of very der Forschung zum Jahrestag Ed R Ansorge published on the o ccasion sp ecial systems for which there exist l integrals of the l equations of motion of the th anniversary of the University of Hamburg Dietrich Reimer Verlag of the form R p q const where the R are algebraic functions of the p k i i k i Hamburg such that the relevant manifolds in l dimensional phase space have the shap e Supp orted by Deutsche Forschungsgemeinschaft under contract No DFGSte of l dimensional tori In mo dern terminology these systems are called integrable 1 Frank Steiner Quantum Chaos systems Here and in the following we are only considering Hamiltonian sys talk which was published without delay by the German Physical So ciety had tems that is motion governed by Newtons equation without dissipation The l no inuence at all on the development of physics during the next fourty years constants R are assumed to b e smo oth enough and to b e in involution ie In Bohrs famous pap er in which the principle of correspondence was k their Poisson brackets with each other vanish See Arnold and Lichtenberg exp osed one nds no reference to Einsteins talk Einsteins torus quantization and Lieb erman for further details As a result the tra jectories of integrable for integrable systems was rediscovered by the mathematician Joseph Keller only systems wind round these l dimensional tori which in turn causes the motion of in the fties It was Fritz ReichePlancks assistant at the University of integrable systems to b e very regular in the sense that even the longtime b e Berlin from to who drew Kellers attention to Einsteins talk haviour is well under control Indeed in integrable systems tra jectories with One can only sp eculate why Einsteins deep insight into the structure of classi neighbouring initial conditions separate only as some p ower of time cal phase space and his recognition of the latters imp ortance for quantization Einstein was the rst physicist who realized the imp ortant roleplayed by the in has b een ignored for such a long time It seems quite obvious that the main variant tori in phase space which he called Trakte He said Man hat sich reason lies in the development of quantum mechanics a few years later starting den Phasenraum jeweilen in eine Anzahl Trakte gespalten zu denken die langs with Heisenbergs matrix mechanics in summer Schrodingerswave equa l dimensionaler Flachen zusammenhangen derart da in dem so entstehen tion in spring and Heisenbergs derivation of the uncertainty principle in den Gebilde interpretiert die p eindeutige und auch b eim Ub ergange von einem spring Already in b efore the discovery of the Schrodingerequation i Trakt zum anderen stetige Funktionen sind diese geometrische Hilfskonstruk Pauli in Hamburg was able to calculate rigorously the energy levels of the tion wollen wir als rationellen Phasenraum b ezeichnen Der Quantensatz soll hydrogen atom from Heisenbergs quantum mechanics which was considered as sich auf alle Linien b eziehen die im rationellen Ko ordinatenraume geschlossene a great success and decisive test of the new theory Paulis rst p osition in Ham sind burg was wissenschaftlicher Hilfsarb eiter in theoretischer Physik at the Institut However the integrable systems forming the standard textb o ok systems with f urTheoretische Physik the rst holder of the chair for theoretical physics was their clo ckwork predictability are not typical that is almost all dynamical Wilhelm Lenz from Octob er to August see ref p on Jan systems are nonintegrable in the sense that there exist no constants of motion uary Pauli received the venia legendi f urtheoretische Physik from the b esides the energy and therefore no invariant tori in phase space Ergodicity mathematician Erich Hecke Whereas Heisenberg had completely eliminated the implies that almost all tra jectories llin the absence of invariant torithe whole classical orbits from his theory Schrodingerwas very much inuenced by classical l dimensional energysurface densely Today our knowledge of classical mechanics and the analogy b etween the wellknown mechanical principle due to dynamics is very rich and most natural scientists b egin to appreciate the and named after Hamilton and the wellknown optical principle of Fermat imp ortance of chaos in complex systems It is now commonly recognized that In his pap er where he discovered what was later p opularized as coherent generic systems execute a very irregular chaotic motion which is unpredictable states Schrodingerwanted to illustrate by the example of Plancks linear oscil that is the tra jectories dep end sensitively on the initial conditions such that lator that it is always p ossible to nd solutions of his undulatory mechanics in neighbouring tra jectories in phase space separate at an exp onential rate the form of welllocalized wave packets whose center of gravity oscillates without Einstein made the crucial remark that the absence of tori excludes the formu change of shap e with the p erio d of the corresp onding classical motion and thus lation of the quantum conditions and furthermore that this applies precisely describ es the classical tra jectory of a p oint particle like for example the Kepler to the situation encountered in classical statistical mechanics where one describ es orbit of the electron in the Hatom For a detailed discussion of Schrodingers the motion of colliding atoms or molecules in a gas denn nur in diesem Falle ist pap er and the roleit played in Heisenbergs discovery of the uncertainty princi die mikrokanonische Gesamtheit der auf e i n System sich b eziehenden Zeitgesamt ple see On Octob er Schrodingergave a talk in Cop enhagen where heit aquivalent In his Nachtrag zur Korrektur he referred as an example Bohr had invited him together with Heisenberg The almost fanatic discussions to Poincarein connection with the threeb o dy problem and he
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