Advanced Quantum Chaos and Mesoscopics

Transcription

1 Advanced Quantum Chaos and Mesoscopics SS 2010 J = us z10 z1 0 z2 J z20 z2 J z2 ts tu Priv. Doz. Dr. Heiner Kohler Latest Update: October 19,

2 Contents 2

3 Chapter 1 Motivation and Introduction The lecture s goal is to provide an up to date introduction to some actual topics of chaotic quantum systems, mesoscopics and random matrix theory. Due to the rich nature of either of these topics we will not be able to present them comprehensively. We will not even try. At first glance quantum chaos, mesoscopics and random matrix theory seem to be rather disjoint. An important goal of the lecture is to show what these topics have in common. In fact they have so much in common that it is justified to cover them in one lecture. We have selected the material in an attempt to underpin these common features. Random matrix theory (RMT) was introduced in the sixties of the last century by Wigner and Dyson in nuclear physics. The foundation of RMT in mathematics dates back to the work of Wishard in the twenties of the last century. The time from 1980 to 1995 were the heydays of RMT in physics, triggered by the theoretical and practical interest in chaotic quantum systems and by the groundbreaking developments in mesoscopics. Historically, the three research fields have evolved slightly differently during the last two decades since then. RMT is still a valuable tool in physics but the main activity has shifted more and more towards mathematics as an important field of probability theory. On the other hand quantum chaos and mesoscopic have merged more and more with respect to methods and objectives into one bigger research area. This lecture is intended as a sequel of the Quantum chaos I lecture, so it s an advantage to have attended it, but it is not necessary. The present lecture notes are to a large extent self contained and only in very few occasions reference to the Quantum chaos I is made. Before we start with a detailed exposition let us give an initial preliminary definition of the above topics. 3

4 Quantum chaos: Usually when physicists speak of Quantum chaos they refer to the study of quantum systems whose classic analogs behave chaotically, or more precisely, to quantum systems which behave like chaotic systems in the classical limit of large quantum numbers. Quantum mechanics is a linear theory and therefore, as we will see in more detail briefly, it does not allow for chaotic behavior in the classical sense. However, it is well accepted that Quantum mechanics is the more fundamental theory. This means that it should be possible at least in principle to derive any kind of classical behavior, including chaoticity, from an appropriately chosen quantum system in the properly chosen classical limit. A large bulk of research has been devoted to this question. How can classical chaotic behavior be derived from quantum theory, which is linear but the more fundamental theory? A different question is, if and how we can detect in a quantum system, whether its classical analog is chaotic or not? This question is related with the question of universality. Given a quantum mechanical spectrum, there are many different types of dynamical systems, which can be reached in the classical limit, depending on the choice of operators to become the classical canonical coordinates. So the question arises, which properties of the spectrum a stable under such different choices and therefore have a certain degree of universality. Other questions in quantum chaos are related to the old critisism of Einstein of quantisation, who realised that quantisation in Bohr Sommerfeld quantization is not possible for classical chaotic systems. Mesoscopics: An important trigger for the study of mesoscopics was the seminal work of Anderson on localisation of a one dimensional particle in a disorder potential?. He found that this effect was due to quantum coherence, or in other words it was a purely quantum effect. Since then there has been a large activity in solid state physics to find other coherence effects for electrons or more general for waves in disordered media. Studying theoretically and experimentally these effects has become a research area which is known today as mesoscopics. Roughly speaking, for a system of matter waves, like electrons there are two requisites to be mesoscopic. On the one hand temperatures should be low. For too high temperatures an effect called dephasing occurs which spoils coherence and which ultimately accounts for the fact that we can often consider electrons as classical particles. 4

5 Figure 1.1: random motion of an electron through a disordered conductor. After a few scattering events the initial direction is randomized, the energy of the electron is unchanged. The second requisite is the size of the probe, which is related to the first requisite via the decoherence time. Sooner or later any electron, if not exactly at temperature zero, will loose its coherence. The length of the path it has travelled until this happens is the decoherence length. A probe is called in the mesoscopic regime if this length is larger than the size of the probe. This restricts the latter to a few microns 10 6 m at most. These are extensions, which are not macroscopic but are larger than the typical atomic lengths scale, which explains the name mesoscopic from greek: µɛσoζ = in between. An exact definition of mesoscopics is, as often happens in physics, missing. A good definition might be as follows: Study of quantum mechanic effects on length scales, which are large compared to atomic length scales, but not macroscopic. In this lecture we adopt a slightly more restricted point of view and define mesoscopics as the investigation of waves and interference phenomena in disordered mediums. Random matrix theory (spectral statistics): Random matrix theory deals with the investigation of properties of matrices with random numbers as entries. To illustrate the concept we consider a 2 2 matrix as the simplest possible example of a random matrix [ ] a c/ 2 H = c/, (1.1) 2 b where a, b, c are normal distributed random variables. ( ) 1 a, b, c : p(x) = 2σ2 π exp x2, x = a, b, c. (1.2) 2σ 2 The two eigenvalues E 1,2 of this matrix are rather complicated functions of the three entries E 1,2 = 1 (a + b ± ) (a b) c 2. (1.3) Although the three random numbers a, b, c are uncorrelated the eigenvalues are not. We show in figure (??) the motion of the eigenvalues of the matrix 5

6 H, when the diagonal elements are tuned from 1 to 1 and vice versa. We see that for non vanishing off diagonal element the levels avoid each other at the degeneracy point. This is a general effect. Whenever a matrix has offdiagonal elements they induce a repulsion of the eigenvalues. This holds for random matrices as well. When we look at the joint probability distribution of the two eigenvalues a brief calculation reveals that it can be written as [ p(e 1, E 2 ) E 1 E 2 exp 1 ( ) ] E 2 2σ E2 2, (1.4) which vanishes when both eigenvalues are equal. Only in the case that c = 0 the prefactor E 1 E 2 disappears and a level crossing is allowed. The joint probability distribution of eigenvalues of a given random matrix with independent matrix entries is a typical object of interest in RMT. Many other quantities were invented to measure the degree of correlation between the eigenvalues. In particular interesting is the limit of infinite matrix dimension. As we will see only in this limit smooth curves can be expected for the correlation functions of interest. The application of RMT in physics comes originally from the observation that in many complicated systems, like nuclei avoided crossings, respectively level repulsion, was observed. Many of theses systems are so complicated that they defied other conventional treatment. Therefore the approach of Wigner and Dyson and others was to investigate what properties these systems share with random systems as described by random matrices. If it is proven that certain properties of complex nuclei are reliably reproduced by random systems one can then in a second step replace the complicated many body Hamiltonian by a random Hamiltonian to investigate further the physics of the system. Relation between the areas At first glance the three fields of research activity described before seem to be rather disconnected. The link between them is established through the statistics of energy levels. It turns out that certain statistical quantities of the spectra of chaotic quantum systems and of mesoscopic systems are perfectly described by random matrices. It required much effort of some of the best physicists of the late last century to explain this perfect equivalence from first principles. It is still not yet completely understood. The upshot of these efforts might be summarised in the following two statements 1. The relation between spectral statistics and quantum chaos is established by the Bohigas-Giannoni-Schmit conjecture: Spectral correla- 6

7 E 0.0 c = 0 c = t E c = 0 c = t Figure 1.2: Eigenvalue repulsion at an avoided crossing in a Landau Zener transition, a(t) = 1 2t, b = 1+2t, t [0, 1]. For small off diagonal values c 1 the eigenvalues come close to each other at t = 0, for large c 1 the eigenvalues strongly repell each other. 7

8 tions of quantum systems, whose classical analogs are chaotic, are identical to the correlations of normal distributed random matrices. 2. Efetov s non-linear sigma-model: The energy correlations of weak disordered systems are identical to those of random matrices. Symbolically they can be visualised as follows. quantum chaos mesoscopics 1 2 spectral statistics random matrix theory There have been recent advances towards a proof of the conjecture by Bohigas-Giannoni-Schmit, which will be presented in this lecture. On the mesoscopics side a rigorous proof was provided by Efetov s so called non linear sigma model. Literature 1. Quantum chaos: A comprehesive account on the fundamentals and on recent developments is provided by the textbook by F. Haake, Quantum signatures of Chaos?. The textbook by H. Stöckmann, Quantum chaos, an Introduction? is easier to read and covers experimental aspects as well. The online book by P. Cvitanovic et al., is probably the most comprehensive account on classical and quantum chaos available in the moment. 2. Mesoscopics: A very good textbook on mesoscopic physics was released recently by E Akkermans & G Montambaux, Mesoscopic Physics of Electrons and Photons?, which covers most of the subjects discussed here. A milestone and standard reference is the textbook by K. Efetov, Supersymmetry in Disorder and Chaos?, which however is rather a work of research than an introductory textbook. Another good textbooks, which covers many aspects not discussed here is the book by Imry?. 8

9 3. Random matrix theory: The textbook by M. L. H. Mehta, Random Matrices? is the standard reference on random matrix theory, which however does not serve for a first reading. Good review articles on the application of RMT in quantum chaos, nuclear physics and beyond were provided by Beenakker? and in?. 9

10 Chapter 2 Classical mechanics 2.1 Hamiltonian Mechanics In classical mechanics we study the time evolution of points in phase space under symplectic transformations. Phase space is a symplectic manifold, i. e. a differentiable manifold Γ with a symplectic 2-form ω(x, Y ) : T p Γ T p Γ R, (2.1) where X, Y are vector fields, i. e. elements of the tangent space T p Γ of the manifold at point p. The symplectic two form ω as well as the vector fields are local objects. This means they depend on a given point p of the manifold Γ as illustrated by the picture below. X p Y Γ Figure 2.1: Two vector fields X and Y emanating from a point p Γ. 10

11 2.1.1 Darboux coordinates and equation of motion For the study of symplectic manifolds Darboux s theorem is most important. It states that the symplectic 2 form can be written locally always as follows ω = f dq i dp i. (2.2) i=1 The set of coordinates {q i, p i } is called Darboux coordinates, however in physics the notion canonical coordinates is more common. From equation (??) it is seen that the dimension of symplectic manifolds is always even dim(γ) = 2f and f is the number of degrees of freedom. For the wedge product holds dq i dp i = dp i dq i (2.3) dq i dq i = dp i dp i = 0. (2.4) Given the vector fields X = Y = f ( i=1 f ( i=1 X qi Y qi ) + X pi q i p i ) + Y pi q i p i (2.5) (2.6) in Darboux coordinates, we get for the action of the symplectic 2 form on the vector fields ω(x, Y ) = f (X qi Y pi X pi Y qi ). (2.7) i=1 In many cases it is easier to imagine ω as a matrix [ ] 0 1f ω : I =. (2.8) 1 f 0 With the help of ω we can transform covariant vectors to contravariant vectors and vice versa in a procedure which is well known from special relativity ω(x, ) = ( X qi dp i + X pi dq i ). (2.9) Exercise 1: Verify equations (??) and (??) using the properties of the wedge product. 11

12 as Equation (??) can be written as well in the more familiar matrix language [ ] [ ] [ ] 0 1f Xq Xp = 1 f 0 X p X q (2.10) We can get an important class of vector fields from a potential function H( q, p) : Γ R. (2.11) We can act with ω 1 on the differential one form dh = H q i dq i + H p i dp i (2.12) to get the associated vector field X H X H = H p i H q i q i p i. (2.13) Such a vector field is called Hamiltonian vector field. The derivative of direction of a phase space function f with respect to the vector field X H is the poisson bracket X H [f] = H p i f q i H q i f p i = {f, H}. (2.14) A phase space function will be constant along the vector field X H, if the poisson bracket vanishes. In particular H itself must be a constant. Therefore a point in phase space [ ] q(t) z(t) = (2.15) p(t) which moves along a trajectory of constant H, must move along a Hamiltonian vector field T 0 Γ z(t) = X H. (2.16) These are nothing but the Hamiltonian equations of motion: q i (t) = H p i (2.17) ṗ i (t) = H q i. (2.18) 12

13 2.1.2 Symplectic transformation The time evolution in classical mechanics is propagated by a symplectic transformation, i. e. by a transformation, which keeps the symplectic 2-form invariant f f dq i (t) dp i (t) = dq i dp i (2.19) i=1 i=1 To understand this, we write: ω(t) = ω. (2.20) ω(t) = 1 2 [d z(t)]t I d z(t) (2.21) d z(t) = J (t, z(0))d z(0), (2.22) where q 1 (t) q 1 (0) J (t, z(0)) =. p f (t) q 1 (0) q 1 (t) q f (0) q 1 (t) p 1 (0) is the Jacobi matrix. From equation (??) it follows that q 1 (t) p f (0) (2.23) ω(t) = 1 2 [d z(0)]t J T (t, z(0))ij (t, z(0))d z(0) (2.24) = ω(0), (2.25) iff J T (t, z(0))ij (t, z(0)) = I. (2.26) A matrix, which keeps the symplectic metric invariant according to J T IJ = I is called symplectic. It is obvious that the product of two symplectic matrices J 1 and J 2 is symplectic as well, so the symplectic matrices form a group (the so called symplectic group) under matrix multiplication. Even though the dependence of J on the initial values is highly non linear, it is possible to expand J (t, z(0)) for small times. For small times we expect a linear dependence on the initial values. But before we do so let us first compare the evolution of time in classical and quantum mechanics. H ψ(t) = U(t, 0)ψ(0) (QM) (2.27) T Γ d z(t) = J (t, z(0))d z(0) (classical mechanics) (2.28) 13

14 The first equation is the well known time evolution of a quantum state ψ(0), element of a Hilbert space H, under the time evolution operator U. The second equation describes the time evolution of an infinitesimal displacement of the phase space point z(t) under the action of the Jacobi matrix. There are two important differences 1. J symplectic U unitary: Without going into details we mention that the symplectic group is a non compact group. This means that its action on a vector can increase the length of this vector by an arbitrary factor. On the other hand the unitary group is compact and its action onto a state vector leaves invariant the length of this vector as required by conservation of probability. 2. J non linear in z(0) QM linear in ψ(0): Since the Jacobi matrix can depend on the initial phase space point in an arbitrary way, this dependence is in the generic case non linear, in quantum mechanics the time evolution operator U does not depend at all on the initial state. Thus time evolution depends linearly on the initial value. We now perform the announced linearisation of J for small t. We expand q i (t) q i (0) + dq i dt t t=0 q i (0) + H p i t (2.29) t=0 and likewise p i (t) p i (0) H q i t. (2.30) t=0 The Jacobi matrix reads for small times H t H t q 1 p 1 p 1 p 1.. J ( t, z(0)) = 1 2 H t 1 2 H t q 1 q 1 p 1 q 1.. (2.31) = 1 + F 0 t, (2.32) 14

15 e tf 0 z(0) + d z(0) z(0) Figure 2.2: Whereas the point z(0) follows the original trajectory. A slightly displaced initial point follows a different trajectory, which generically deviates from the original one exponentially. where F 0 is given by F 0 = 2 H q i p j 2 H q i q j 2 H p i p j 2 H p i q j. (2.33) t=0 With the help of the marix F 0 we can calculate approximately the time evolution of d z(t). For small times t we write d z(t) = d z(0) + F 0 t d z(0) +... e F 0t d z(0). (2.34) The matrix F 0 is a measure for the stability of trajectory close to a point in phase space. Therefore we call F 0 stability matrix. The stability matrix is crucial for characterising the behavior of a classical dynamical system. We take a closer look on its algebraic structure. From equation (??) it is seen that F 0 can be written symbolically as [ ] A B F 0 = C A T, (2.35) where B = B T and C = C T. One can show that a matrix of the form as given in equation (??) fulfills F T 0 I + IF 0 = 0. (2.36) Exercise 2: a) Verify that a matrix of the form (??) fulfills equation (??). b) Show that if F 0 fulfills equation (??) the matrix U = exp(af 0 ), a C, is symplectic. c) Show that if λ is an eigenvalue of a matrix (??) so is λ. 15

16 The stability matrix F 0 can be diagonalised by a matrix J 0 which is itself a symplectic matrix. Let us call the outcome of this diagonalisation Λ Λ = J 1 0 F 0 J 0. (2.37) Since F 0 is non Hermitean, Λ has in general 2f complex eigenvalues, which come in f pairs consisting of an eigenvalue λ and its negative. In principle we can distinguish four cases 1. λ C, 2. Im(λ) = 0, 3. Re(λ) = 0, 4. Im(λ) = 0 and Re(λ) = 0. We want to obtain a deeper understanding of the physical meaning of the stability matrix and of its eigenvalues. In particular we want to see how chaoticity of a dynamical system can be characterised by the stability matrix. To this end we first look at regular (integrable) systems and try to characterise them by their stability matrix Canonical transformations and integrable systems Canonical transformations are coordinate transformations that leave the symplectic 2-form invariant (q i, p i ) (Q i, P i ) (2.38) dqi dp i dq i dp i (2.39) {q i, p j } = δ ij {Q i, P j } = δ ij (2.40) The Hamiltonian function transforms as H( Q, P ) = H( q( Q, P ), p( Q, P )). (2.41) A properly chosen coordinate transformation can solve a given problem immediately. As an example we solve the equations of motion for the classical harmonic oscillator by canonical transformation. The Hamilton function of the harmonic oscillator is given by H(q, p) = p2 2m + ω2 m 2 q2. (2.42) 16

17 Of course, it can be treated in the original coordinates without problems. We introduce a new pair of coordinates Q and P via 2P q = mω sin Q, p = 2P mω cos Q, (2.43) which likewise might be inverted ( ) Q = arcsin mω2 2P q, P = 1 2mω p mωq2. (2.44) Exercise 3: Show that the new coordinates Q and P have the same Poisson bracket as the original ones {Q, P } = 1. In the new coordinates the Hamiltonian becomes particularly simple H(Q, P ) = ωp. (2.45) We notice that H does not depend on Q in the new coordinates. Such a coordinate is called cyclic. The equations of motion for cyclic pairs of coordinates (Q, P ) become extremely easy to solve Q = H Q(t) = ωt + Q 0, ω = H P P P = 0 P (t) = const.. (2.46) From the above example it is seen that in general canonical transformations connect the old coordinates in a not linear way with the new coordinates. A systematic way to construct a canonical transformation is by defining mixed generating functions F 1 ( q, Q), F 2 ( q, P ), F 3 ( Q, p), F 4 ( p, P ), (2.47) which are functions of new and original coordinates. The symplectic 2-form s invariance requires f dq i dp i = i=1 f dq i dp i. (2.48) i=1 For a generating function F 1 this condition is fulfilled if we set p i = F 1 q i P i = F 1 Q i. (2.49) 17

18 With equation (??) we can write dp i = j F 1 dq j + q i q j j F 1 q i Q j dq j dp i = j F 1 Q i Q j dq j j F 1 Q i q i dq j. (2.50) Using the two equations (??) the original and the transformed symplectic form can be written as f dq i dp i = ij i=1 f dq i dp i = ij i=1 ( F1 dq i dq j + F ) 1 dq i dq j q i q j q i Q j ( F1 dq i dq j + F ) 1 dq i dq j Q i Q j Q q j This is identical with equation (??), since due to the wedge product i,j i,j F 1 q i q j dq i dq j = F 1 Q i Q j dq i dq = 0 F 1 Q i q j dq i dq j = i,j (2.51). (2.52) F 1 q i Q j dq i dq j. (2.53) Likewise, we find for the other generating functions F 2, F 3 and F 4 the equations p i = F 2 q i, Q i = F 2 P i, (2.54) P i = F 3 Q i, q i = F 3 p i, (2.55) q i = F 4 p i, Q i = F 4 P i. (2.56) In principal one can choose an arbitrary function as generating function. In practice however one should use a generating function which generates a system of Darboux coordinates, which render the new Hamiltonian function as simple as possible. We might agree that the simplest possible function is a constant. Further elaboration on this idea leads to the theory of Hamilton- Jacobi. 18

19 In this context F 2 is very important. We denote this function with a capital W ( q, P ) and call it Hamilton characteristic function. Requiring that H = const. in the new coordinates and using relation (??) for F 2 yields ( H q 1,..., q f ; W... W ) = E, (2.57) q 1 q f where we called the constant E and identify it with the energy. Equation (??) is the stationary Hamilton-Jacobi equation. It is in general a non linear partial differential equation in f variables. In the general theory of partial differential equations (PDEs) a solution of a PDE of the form (??) depends on f initial conditions which we can chose as the momenta at given position q (0) i, i = 1... f W q 1 q1 =q (0) 1 = p (0) 1,..., W q f qf =q (0) f = p (0) f. (2.58) While this is a valid choice it is not the most convenient one, since the new momenta depend on a rather arbitrary fashion on the initial points q (0) i, i = 1,... f. For periodic motion we might choose P i = 1 2πL period p i dq i = ω L T 0 p i (t) q i (t) dt = I i L (2.59) as new coordinate, where L is the length of the period. This choice for P i is democratic in the initial positions and most conveniently independent of the initial condition. In the last equation we introduced the quantity I i which can serve as constant of motion like P i but with the dimension of an action. Note that not all I 1,..., I f are independent but are connected by the condition (??). We introduce the quantity I(E) = 1 2π = period p( q) d q = 1 2π W ( q, E)d q f I i (2.60) i=1 which is called action or action integral. The time dependent Hamiltonian principal function S( q, P, t) is related to the Hamilton characteristic function in a similar way as a solution of the time 19

20 dependent Schrödinger equation to solutions of the stationary Schrödinger equation. Write S( q, P, t) =W ( q, P ) Et (2.61) Then S( q, P, t) fulfills the Hamilton Jacobi equation ( H q 1,..., q f ; S... S ) + S = 0. (2.62) q 1 q f t The function S( q, P, t) is called Hamilton s principal function. Since by construction ds dt ds dt = = f i=1 f i=1 S q i E dq i p i q i H S(t, t 0 ) = t t 0 dt L(t ), (2.63) where L(t) is the classical Lagrange function. Thus the Hamilton principal function is identical with the classical action integral. On the other hand we obtain for Hamilton characteristic function W ( q, P ) = t t 0 (L + H)dt = q q (0) p( q)d q. (2.64) Thus, in particular for periodic motion at q = q (0) the characteristic function becomes independent of q and equal to the action integral W = 2πI. Exercise 4: Derive the Hamilton Jacobi equation from the Schrödinger equation ψ( q, t) i t = 2 + V ( q) (2.65) 2m by making the ansatz ψ( q, t) = exp(is( q, t)/ ). In what limit the Hamilton Jacobi equation arises? We choose again the harmonic oscillator as example. stationary Hamilton Jacobi equation reads In this case the 1 2m ( W q ) 2 + ω2 m 2 q2 = E. (2.66) 20

21 It can easily be integrated W (q, P (E)) = W q = 2mE ω 2 m 2 q 2 q 0 2mE ω2 m 2 q 2 dq. (2.67) Now the problem is in principle solved. We still have a freeness in the choice of the transformed momentum P (E). Using the choice (??) wet get for the action integral I(E) = 2 π = 4E πω q max mE ω2 m 2 q 2 dq, q max = 2E ω 2 m (2.68) 1 q2 dq = E ω. (2.69) We have therefore in this example a very simple relation between energy and action. I(E) = E ω, I(E) 2Em P (E) = = (2.70) L 8 The generating function of the canonical transformations, which transform H H(I) = ωi H(P ) = 32P 2 m is given by (2.71) W (q, I) = W (q, P ) = q 0 q 0 2mωI ω2 m 2 q 2 dq, (8P )2 ω 2 m 2 q 2 dq (2.72) Exercise 5: Show that for the harmonic oscillator the canonical transformation induced by W (q, I) is the coordinate transformation (q, p) (Q, P ) as given in equations (??) and (??). 21

22 When the action integral is taken as momentum coordinate and its canonical conjugate coordinate Q is cyclic the latter is usually called angle variable and denoted by a lower case greek character. Now we are able to give a definition of an integrable system: If one can transform a Hamiltonian function H( q, p) H( I, θ) (2.73) by a canonical transformation ( q, p) ( I, θ) {I i, I j } = {θ i, θ j } = 0, {I i, θ j } = δ ij, (2.74) into a form such that the transformed Hamilton function does not depend on the angle coordinates H( I, θ)! = H( I), (2.75) the system is called integrable. The Darboux coordinates ( I, θ) in which all θ i, 1 i f are cyclic are called angle action variables. By the action angle variables the phase space is naturally foliated in submanifolds of constant action. These submanifolds are products of one dimensional manifolds, parametrised by the time evolution of the angles θ i (t) = ω i t + θ i (0) (2.76) There are only two topologically distinct one dimensional manifolds, which are described by equation (??). The non compact real line R and the compact circle S 1. If one or more particles can escape to infinity the system is unbound and one or more one dimensional submanifolds have the topology of R. On the other hand if the state is bound the surfaces of constant action I 1,..., I f are f dimensional tori S 1 S 1... S 1. (2.77) We add two remarks. Although in angle action coordinates the trajectory looks simple, the canonical transformation into angle action coordinates might be a highly complicated non linear transformation. Therefore in the original coordinates the trajectory can look very complicated. Second, for generic frequencies ω i = H I i, 1 i f the motion is not periodic and for long times the torus will be filled densely by a single trajectory. The exist infinitely many periodic trajectories. They are also called Resonances. Therefore simply looking at the trajectories of a system and see if it behaves 22

23 Figure 2.3: The phase space trajectories of a bound integrable system twist around an f dimensional torus. In the picture a short periodic orbit (thick black) and a longer periodic orbit (thin blue) are drawn. sufficiently complicated or not is usually not enough to distinguish regular and non regular systems. Even regular systems can behave in a very complicated way. We now can study the stability matrix F0 of an integrable system. We assume, we have been able to transform it into angle action variables. Then 2H ϑi Ij F0 = 2H ϑi ϑj 2H Ii Ij 2H Ii ϑj can be written as 2H 0 Ii Ij F0 =, 0 (2.78) (2.79) 0 because by definition of a cyclic coordinate H = 0. We conclude that all θi eigenvalues of F0 are zero. This property holds for an arbitrary point in phase space. Moreover it holds for an arbitrary coordinate system, although it is only easy to see in angle action coordinates. The fact that all eigevalues of F0 are zero is a characteristic property of an integrable system. 23

24 d z z 0 d z λ d z λ Figure 2.4: Illustration of the principal axes at point p = z 0. To every eigenvector to eigenvalue λ corresponds an orthogonal direction to eigenvalue λ. 2.2 Chaotic systems As pointed out at the end of section?? in general the stability matrix F 0 can have non zero eigenvalues. These non zero eigenvalues can be linked to non integrable or chaotic behavior. They are useful for the classification of chaotic systems Characterisation of chaotic systems The algebraic structure of F 0 and their eigenvalues have already been studied in section??. We recall some of the results: All eigenvalues of the stability matrix come in pairs. Accordingly we write the set of principal axes J 0 = [ e λ1, e λ1,..., e λf, e λf ]. (2.80) Although in general F 0 has non zero eigenvalues, at least two eigenvalues are always zero. They are obtained by a variation of z 0 = z(0) into the direction of the Hamiltonian flow. For the non zero eigenvalues one can distinguish four different cases. They are called ioxodrom if λ( z 0 ) C, hyperbolic if λ( z 0 ) is real, elliptic if λ( z 0 ) purely imaginary: marginal if λ( z 0 ) = 0. They are illustrated in figure??. 24

25 d z z 0 d z λ d z λ Figure 2.6: Illustration of the behavior for different displacement directions. The trajectory of the undisplaced point z 0 follows in the beginning the direction of z. A phase space point which is displaced in the direction e λ departs from the original trajectory. A point which is displaced in the direction of e λ initially comes closer to the original trajectory. δp δp δp δq δq δq elliptic marginal hyperbolic Figure 2.5: Illustration of an elliptic, a marginal and of an hyperbolic eigenvalue. Chaotic behaviour is identified with a real positive eigenvalue λ of the stability matrix. In this case an infinitesimal displacement of the initial point z(0) in the direction of e λ leads to an exponentially strong deviation of its trajectory from the original one d z λ (t) e λ( z 0)t d z λ (0). (2.81) However, one should notice that for every positive λ( z 0 ) there is a corresponding negative λ( z 0 ). This means that an infinitesimal displacement of the initial point into to direction of e λ has the opposite effect. The trajectory of this point comes closer and closer to the trajectory of the original one d z λ (t) e λ( z 0)t d z λ (0). (2.82) These different behaviors are illustrated in figure?? 25

26 We notice that λ( z 0 ) and e λ are local quantities. They vary depending on the point z 0 in phase space. The positive λ( z 0 ) at a point z 0 is called local Lyapunov exponent or stretching rate. One can get the (global) Lyapunov exponent, if one calculates the mean of λ( z 0 ) over a trajectory with infinite length 1 L( z 0 ) = lim dt λ( z(t)) T T. (2.83) 0 For many systems L( z 0 ) = L does not depend on the initial value z 0. Such systems are called uniform hyperbolic systems. Hyperbolicity is the most important characterisation of a chaotic system. We enumerate two other generic properties of chaotic systems, which will be needed in the sequel. 1. Ergodicity: Let f be a sufficently well behaved function of phase space f : Γ R. (2.84) Then we can define a time average for this function evaluated along a phase space trajectory as follows: f T = 1 T T 0 f( z(t))dt. (2.85) Ergodicity means that for very large times this average can be replaced by a phase space average f r = f( z)dµ( z), (2.86) where Γ dµ( z) = 1 Ω f dp i dq i δ(h( z) E) (2.87) i=1 is called Liouville measure and Ω is the volume of the energy shell. The Liouville measure is the induced measure by restricting the equidistribution in Euklidean phase space to the energy shell. In formulae ergodicity means lim f T T = f Γ. (2.88) This property is crucial for the semiclassical approach to the proof of the conjecture of Bohigas, Gianonni and Schmit. 26

27 2. Mixing: The mixing property is stronger than ergodicity. Loosely speaking it means that a trajectory after a sufficient long time has lost its memory on the initial state. More precisely it means that in the long time limit a phase space average of two phase space functions f and g, where f is a function of z(0) and g a function of z(t ) can be replaced by the product of two independent phase space averages as lim T Γ f( z(0))g( z(t ))dµ( z(0)) = Γ f( z)dµ( z) Γ g( z )dµ( z ). (2.89) t 0 t 1 t 2 t 0 t 1 t 2... Figure 2.7: Illustration of the mixing property: The upper time evolution is mixing and ergodic, the time evolution in the lower row is ergodic but not mixing Ergodic, mixing systems in which every point has at least one hyperbolic direction are called fully chaotic Poincaré section Poincare surfaces are 2(f 1)dimensional Lagrangian submanifolds of Γ. The precise mathematical definition of a Lagrangian submanifold is involved and requires additional mathematical concepts. The reader is referred for instance to Ref.?. A definition which is sufficient for our purposes sufficient goes as follows: a Lagrangian submanifold is spanned by a position coordinate and by a momentum coordinate. A Poincaré section is constructed as follows. 27

28 p Σ p Σ q Σ q Σ Figure 2.9: For regular systems the Poincaré section is pierced on a loop, corresponding to a deformed torus. If the trajectory is resonant the number of points are finite, otherwise the loop will be filled densely. Figure 2.10: For a chaotic system the Poincaré section is expected to be filled uniformly, due to the ergodicity and mixing assumption. However there might show up stable islands. σ 3 σ 3 σ 1 q Σ σ 2 p Σ σ 1 σ 2 p Σ q Σ Figure 2.8: Sketch of the construction of a Poincaré plot. A trajectory pierces the Poincaré section, spanned by the momentum p Σ and by the position q Σ. Only the downwards piercings contribute to the Poncareé plot below. Poincaré sections are very useful to find chaotic behavior. In a regular system the trajectory will pierce Σ R in a regular way, according to the invariant tori. The Poincaré section will look as indicated in figure??. On the other hand for a chaotic system the Poincareé section should be filled with piercing points densely. 28

29 θ k σ 1 s=0=l 1 cos θ σ 1 k σ 2 σ s L Figure 2.11: Sketch of the construction of a Poincaré section for an irregular billiard. The first two points are constructed. By construction the surface is periodic in the s axis with period L Examples for classical chaos We now give a few examples of classical chaotic systems. A more extensive account is given in the Script to Quantum Chaos I. The prime and best studied example for a chaotic system are two dimensional billards with an irregular border. Their phase space dimension is dim Γ = 4. Here and in the following we refer to billiards always as two dimensional billiards if not stated otherwise. This is in agreement with our daily life intuition of a billiard table. We will use the direction of the momentum cos θ k at the boundary as the momentum axis of the Poincaré section. For the position axis we choose the distance s of the point where the ball hits the boundary, measured from an arbitrarily chosen point s = 0 (see figure??). One can indeed show, that for billiards the two coordinates cos θ k and s are canonical conjugate phase space variables. Limaçon billiard (Robnik billiard): The Limacon billiard is a good example for a dynamical system with a mixed phases space. The circumference of the billiard is parametrised in polar coordinates by ρ(ϕ) = 1 ε cos ϕ, φ [ π, π] (2.90) For ε = 0 it is identical with the integrable circular billiard and regular. As ε increases the tori will loose their shape according to the KAM theorem, but the billiard remains regular. For a critical value ε crit the billiard develops a cusp. It becomes more and more chaotic but still has integrable islands. 29

30 Exercise 6: Calculate the critical value ε crit from equation (??) ε = ε = ε = ε = Figure 2.12: Limaçcon billiard for four different values of ε. For values ε > ε crit a cusp emerges. 30

31 Figure 2.13: Poincaré surface of section of the Limaçon billard for parameter ε = 0.3. We see that most of the phase space is chaotic, but some regular islands can clearly be seen. Hénon-Heiles Hamiltonian: The Hamiltonian was introduced by the astronomer Michel Hénon and a graduate student Carl Heiles as a mathematical model to numerically investigate the so called third integral problem of celestial mechanics?. It is a celebrated and historically important example for a chaotic system. The Hénon-Heiles Hamiltonian is given by H = 1 2 (p2 x + p 2 y) (x2 + y 2 ) + x 2 y y3 3 = 1 2 (p2 x + p 2 y) + V (x, y) (2.91) In figure?? the potential is plotted. It is seen that for small energies the potential felt by the particle is almost circular, the motion of the particle is regular. As the energy of the particle increases the potential, which is felt by the particle has a more and more triangular shape. The motion of the particle becomes chaotic. For a critical dimensionless energy ε = 1/6 the particle can escape. Exercise 7: Show that for energies higher than ε = 1/6 the particle can escape from the potential well. 31

32 y x Figure 2.14: Plot of the equipotential lines of the potential V (x, y) = 1 2 (x2 + y 2 ) + x 2 y y3 for the values V (x, y) = 0.01, 0.04, 0.1, 0.15, Figure 2.15: Poincaré section for the Hénon-Heiles Hamiltonian. We see that as the energy ε increases and approaches the critical value the chaotic sea increases. Figures taken from HeilesEquation.html. In the Poincaré surfaces of section of the Limaçon billiard?? and of the Hénon-Heiles Hamiltonian?? it is nicely seen that the transition from regu- 32

33 lar to chaotic dynamics with increasing perturbation parameter ε is neither abrupt nor smooth but of remarkable complexity. It is clear that for a completely integrable system the Poincaré surface of section is densly foliated by invariant curves stemming from the tori for different values of the action variable. In the first panel of Fig.?? we see that many of theses invariant curves persist for small perturbation parameter ε. As the perturbation increases theses invariant curves gradually disappear, yet certain resonant (periodic) orbits persist and develop secondary invariant curves around them. The invariant curves are separated by the resonance through a small chaotic sea. The ensemble of an isolated resonance with a invariant curve separated by a chaotic sea is called stable island. Stable islands are abundantly seen in Fig.?? and also in the second and third panels of Fig.??. Around these parameterpoincaré Figure 2.16: Transition to chaos. 2.3 Classical perturbation Theory Perturbation theory is the first and standard approach to any non trivial quantum mechanical system. It gives a first intuition on what physical phenomena van be expected and serves as a valuable benchmark for other non perturbative methods. Nevertheless in classical mechanics perturbation is much more complicated. We first focus on one degree of freedom Perturbation theory for f = 1 Let us consider the Hamilton function of a pendulum of length l H(p φ, φ) = p2 φ 2ml 2 + ml2 ω 2 (1 cos(φ)), (2.92) where ω = g/l, and g the gravitational constant. A straightforward but naive way to go beyond the harmonic approximation is to expand the cosine up to 33

34 fourth order and consider this fourth order term as a perturbation H(p, q) = p2 2m + mω2 q 2 εmω2 q 4. (2.93) 4! Here we made the substitutions ml 2 m p φ p φ q (2.94) in order to stick with the usual notation in terms of position and momentum coordinates. Moreover ε is formally a small parameter which is set ε = 1 at the very end. Following the steps of quantum mechanical perturbation theory as close as possible, we expand the solution to the equation of motion in powers of ε q(t) = q 0 (t) + εq 1 (t) + ε 2 q 2 (t) (2.95) Plugging this into the equation of motion and comparing terms with the same power in ε yields in zeroth order the solution of the unperturbed harmonical oscillator q 0 (t) = A cos(ωt). (2.96) In first order in ε the equation of motion for q 1 (t) reads q(t) + ω 2 q 1 (t) = A3 ω 2 cos 3 (ωt) 6 = A3 ω 2 ( ) cos(3ωt) + 3 cos(ωt). (2.97) 24 The equation of motion for q 1 (t) is that of a driven oscillator. While the driving frequency 3ω is harmless the second driving term is in resonance with the oscillator and leads to the secular solution q 1 (t) = A3 192 A3 ( ) cos(3ωt) + ωt cos(ωt) + 2 cos(ωt). (2.98) 64 The secular term t in the solution of q 1 (t) has the unpleasant consequence that even for very small perturbation at a time scale t ε ε the perturbative term dwarfs the unperturbed solution. The secular term is clearly unphysical and leads to the conclusion that naive perturbation theory miserably fails in classical mechanics. 34

35 The correct perturbative approach to classical mechanics is due to Poincaré and von Zeipel. It makes use angle- action coordinates and of the fact that the unperturbed Hamilton function is a function of action coordinates only. We first focus on the one dimensional case. Write the full Hamilton function as H(I, θ) = H 0 (I) + εh 1 (I, θ). (2.99) To zeroth order I(t) = I 0, θ(t) = ω 0 t + β where ω 0 = H 0 / I 0 I0 A canonical transformation is introduced by W (Ĩ, θ) = Ĩθ + εw 1 (2.100) such that in lowest order in ε this transformation is just the identical transformation I Ĩ and θ θ. To first order in ε one obtains the relations 1(Ĩ, θ) I = Ĩ + ε W θ θ = θ W1(Ĩ, θ) ε θ (2.101) between new and old coordinates. These can be plugged into the Hamilton function H(Ĩ, θ) = H 0 (Ĩ) + ε H 1 (Ĩ, θ) H 1 (Ĩ, θ) H0(Ĩ) W 1 (Ĩ, θ) = Ĩ θ + H 1 (Ĩ, θ) W 1 (Ĩ, θ) = ω 0 θ + H 1 (Ĩ, θ) (2.102) We still can choose W 1 at will. We choose it in such a way that H 1 (Ĩ, θ) = H 1 (Ĩ), i. e. only depends on the action coordinate. If we write expand H 1(I, θ) in Fourier components H 1 (I, θ) = H 1,n (I)e i nθ H 1,n (I) = 1 2π then obviously W 1 (Ĩ, θ) = n= 2π 0 dθe i nθ H 1 (I, θ) (2.103) n= n 0 H 1,n (Ĩ) n θ ei i nω (2.104) 35

36 does the job. coordinates Thus the Hamilton function reads in the new action angle ε H(Ĩ) = H0(Ĩ) + 2π 2π 0 d θh 1 (Ĩ, θ) (2.105) Let us treat with this method the former example. The Hamilton function reads in the angel action coordinates (??) H(I, θ) = ω 0 I εi2 6m sin4 (ωt). (2.106) Expanding sin 4 θ = 3 1 cos(2θ) + 1 cos(4θ) yields H(Ĩ) = ω 0Ĩ εĩ2 16m ε=1 ω = 1 Ĩ ω 0 8ω 0 m = 1 E 8ω0m. (2.107) 2 The frequency becomes energy/amplitude dependent. This feature is not captured in naive perturbation theory. The frequency of the pendulum can be calculated exactly ω = π ( π/2 ) ω 0 2 K 1 E/2mω0 2 dφ, K(x) = 1 x2 sin 2 (φ), (2.108) for x < 1. Here K is the complete elliptic function of the first kind. The series expansion of K(x) recovers the result (??). One might conclude that the method of Poincaré and von Zeipel is the correct perturbative approach to classical mechanics. While this holds true in one dimension, unfortunately in higher dimension, respectively for more than one degree of freedom, canonical perturbation theory is plagued by singularities. Exercise 8: Derive Eq. (??) from the Hamilton function of the pendulum Eq. (??) Canonical perturbation theory for f > 1 The method described in the previous section for one degree of freedom can in principle applied to systems with two or more degrees of freedom. The generating function (??) can be generalized to f 2 by W (Ĩ1,..., Ĩf, θ 1,..., θ f ) = f n=1 (2.109) 36

37 ω ω 01 E E crit 1 Figure 2.17: Energy dependence of the frequency of the pendulum. For small energies first order canonical perturbation theory (red curve) yields a good approximation. As the energy approaches its critical value E crit = 2mω0 2 the time for one period becomes infinite T. 4 Π Π Figure 2.18: Phase space diagram of the pendulum. Regions of periodic motion (clear grey) are separated by regions of unbound motion (dark grey) by the so called separatrix. At the separatrix The energy at the separatrix is E crit. 37

38 Chapter 3 Spectral statistics of quantum systems Quantum mechanics is more fundamental than classical mechanics. In principle it should therefore be possible to derive a theory of classical chaos from quantum mechanics. In practice we choose to go the other way. We quantize a classical chaotic system and look at the properties of the quantum system. In the statistical approach to quantum spectra the concept of universality becomes important. We might consider many different classical chaotic systems, leading to different quantum systems. We are only interested in quantities of the quantum systems which are shared by a wide class, which we will call universality class. In the classical limit all these quantum systems should have at least one common property, they should be chaotic. 3.1 Quantization In a quantization procedure the classical phase space is coarse grained by dividing it into volumes of magnitude h f. For a compact phase space a phase space volume V Γ can be defined and we will get V Γ /h f unit cells. This number h h Γ Figure 3.1: Coarse graining of the phase space into unit cells of volume h f 38

39 of unit cells is identified with the Hilbert space dimension of the quantum mechanical system. In many cases Γ is not compact, then arbitrary high energies are possible. In this case the Hilbert space of the quantum mechanical system is infinite-dimensional. If some phase space function I( q, p) is conserved under the Hamiltonian flow one might take surfaces of constant I for the coarse graining to obtain a finite dimensional Hilbert space representation of the system. Generically there are few fundamental guiding principles on how to choose the form of the unit cells of volume h f and how to distribute them in the phase space. Should they all have the same shape? The same number of neighboring cells? These questions usually can only be answered by our physical intuition and knowledge about the classical system. An exception is the case of integrable systems. In this case the set of action variables {I 1,... I f } yields a natural foliation of the phase space by the quantization condition I n = n, where n N. A unit block is spanned by the f unit vectors e n in the directions di n. This is the Bohr Sommerfeld quantization. However, as already observed by Einstein it works only for integrable systems. An unsatisfactory feature of the Bohr Sommerfeld quantization is its dependence on the Hamiltonian. The prescription how phase space is foliated is given by the Hamiltonian, i. e. by the problem under consideration. Depending on what dynamical system we consider, we foliate one and the same phase space in many different ways. It would be more desirable to have a quantisation procedure which only depends on the geometry of the phase space but not on any additional input. The canonical quantization avoids this problems. The phase space is independently of the Hamiltonian under consideration divided in blocks defined by the infinitesimal volume element f i=1 dq i dp i of volume h, where q i, p i, 1 i f are the particle s canonical position and momentum. This method, however has other drawbacks. In more complicated Hamilton functions there might appear non Hermitean products of non commuting operators like p i q i, which leads to an ordering problem in the quantum Hamiltonian. The canonical quantization works without modifications only in the case that the phase space is topologically equivalent to R 2f. Any quantization scheme is flawed by the fundamental problem that it takes the road into the wrong direction. A first principle approach should derive classical mechanics from quantum mechanics and make any quantization scheme spurious. 39

40 3.2 Quantum mechanics Lets us briefly recall the principles of quantum mechanics 1. A quantum mechanical system is determined by a Hilbert space H of dimension N which is spanned by a complete set of state vectors ψ H. The completeness condition and orthogonality relation hold ϕ i ϕ j = δ ij (3.1) N ϕ i ϕ i = 1. (3.2) i=1 2. The time evolution of a state is governed by a time evolution operator Û(t) = exp( i Ĥt/ ). 3. Physical observables are self-adjoint operators  = Â. Only expectation values ψ  ψ are amenable to measurement. The spectral decomposition follows as a consequence of the first point above Ĥ = E i ϕ i ϕ i Û(t) = e i E jt/ ϕ j ϕ j. (3.3) Heisenberg uncertainty principle follows from the non commutativity of two operators  and ˆB obeying the Heisenberg algebra [Â, ˆB] = i. Define the variance of an operator A = ψ (A A ) 2 ψ. (3.4) Then Heisenbergs uncertainty relation holds A B 2 4. (3.5) States for which the equality A B = 2 /4 holds are called coherent states. From the second point Heisenberg equations of motion follow for an arbitrary Heisenberg operator d dtâ(t) = 1 [Â, Ĥ]. (3.6) i In particular Heisenberg operators fullfil the classical equations of motion. 40

41 In theory a quantum mechanical system is fully specified by the spectrum of the Hamilton operator and its eigenfunctions. In practice a quantum mechanical system is complemented by a set of operators, for which we know their classical limit, since they can be measured. This point is most important. Only with the knowledge of some operators, which we can in principle measure and by a measurement of this operator, respectively its expectation values, a quantum system becomes meaningful for us. In many cases we are interested in a set of operators {ˆq i, ˆp i }, which fulfill the Heisenberg algebra. The set of expectation values { ψ ˆq i (t) ψ, ψ ˆp i (t) ψ } z(t) (3.7) might then be interpreted as Darboux coordinates of classical phases space in the classical limit 0. If we leave aside this additional input, a quantum mechanical system is nothing but a set of eigenvalues and eigenvectors and the time evolution of any state vector is determined uniquely by this. Two operators with the same spectrum Spec(Â) = Spec( ˆB) (3.8) but different eigenfunctions are called unitary equivalent. There are many examples of systems having the same spectrum but describing rather different physical situations. However it is natural to assume that unitary equivalent Hamilton operators generate in the classical limit dynamical systems with similar global properties. Therefore the spectrum of a Hamilton operator plays the most fundamental role. It is the skeleton of the quantum system, flesh and blood is added by the eigenfunctions and an appropriate set of operators. We then can ask the fundamental question: Can we detect already from the spectrum of a Hamilton operator, whether it generates a dynamical system with chaotic behavior or not in the classical limit? 3.3 Spectral Statistics The basis of spectral statistics is an ergodicity argument. An unbound quantum mechanical spectrum is a large set N of real numbers. Let R M be a spectral quantity which is a function of a sequence of M eigenvalues taken out of this infinite series. Here M might be very large, but since the spectrum is infinite N M it contains infinite many sequences of this sort. Denote them R Mn, n = 1,... N/M. Each sequence can be considered a representative of an ensemble of spectra of length M. Thus the function R N 41

42 which depends on the whole spectrum is equivalent to an average over an ensemble over smaller spectra. lim R N(E 1,..., E N ) = N N/M MR Mi lim N N, (3.9) i=1 Spectral quantities R for which equation (??) holds are called self averaging Unfolding To compare different quantum mechanical spectra we must compare them on the same energy scale. Every quantum mechanical Hamilton operator has a typical energy scale, which is determined by the parameters of the operator. This energy scale corresponds to a length scale. The length scale gives an estimate of the extension of the bound states of the system. We give two examples: 1. In Gaussian units the Hydrogen atom is defined by the Hamilton operator Ĥ Hydrogen = 1 ˆp 2. (3.10) 2m e e2ˆr With the three parameters m e, e and a typical length scale can be constructed, which is the celebrated Bohr s atom radius a 0 2 m e e 2 = a 0 5, m. (3.11) With this a typical energy scale can be constructed as well E(a 0 ) = 1 2m e e 2 a 2 0 e2 a 0 = 1 e 4 m e 13, 4eV, (3.12) 2 h 2 which is the ionisation energy of a hydrogen atom in the ground state. 2. As a second example we consider a rectangular billard. Its Hamilton operator has as parameters the particle s mass m and the length L and width B of the billiard. If the billiard is approximately quadratic, i. e. L and B are of the same order of magnitude, we can construct a typical energy scale of the billiard by 1 2 2m A, (3.13) 42

43 where A = BL is the area of the billiard. If we want for some reason compare the spectra of the hydrogen atom and the rectangular billiard, we should do this with the rescaled energies E n /E typical. If we want to compare spectra of different quantum mechanical systems, their different energy scales are not the only problem we have to face. Even within one and the same spectrum the number of energy levels per energy interval many vary wildly. In fact in most systems with an unbound spectrum the density of states increases with increasing energy. We define the spectral density ϱ(e) = δ(e E n ) (3.14) n=0 and the integrated spectral density (aka staircase function) N(E) = E ϱ(e )de = N θ(e E n ). (3.15) n=0 From figure?? it is seen that N(E) contains a smooth part and a fluctuating part N(E) = N(E) + N fl (E), (3.16) where N(E) is a smooth curve and N fl (E) fluctuates around zero. The same decomposition into a smooth and a fluctuating part is valid for the spectral density ρ(e) = ρ(e) + ρ fl (E), ϱ(e) = d N(E) de. (3.17) If we want to calculate an expectation value of a spectral quantity, say f(e) f(e) = N f(e n ) = n=0 E max E min f(e)ϱ(e)de E max E min f(e) ϱ(e)de (3.18) we face the problem that ϱ(e) might vary from intervall to intervall and from system to system. One says ϱ(e) is not universal. We want to suppress this non universal part and have a look at spectral quantities where the mean spectral density is normalized to unity. We define the new variable ξ = N(E) dξ = ϱ(e)de. (3.19) 43

44 Then it follows for the averaged quantity f f(e) ξ max ξ min f( N 1 (ξ))dξ, (3.20) and the influence of mean level density ϱ has been eliminated. This procedure is called unfolding. A problem arises in the calculation of the function N which is needed according to equation (??) for the unfolding procedure. Usually it is hard to be determined from data of the complete spectrum. 44

45 10 8 N(E) E Nfl(E) E Nfl(ξ) ξ Figure 3.2: Staircase function for the hydrogen atom (Coulomb potential) and the smooth integrated density of states. Exercise 9: In figure?? the staircase function of the hydrogen atom (Coulomb potential) is depicted. It is well known that the spectrum is given in this case by E n = Ry 1 n 2. Calculate the smooth (integrated) density of states ϱ(e). In practice one proceeds as follows: One takes from the staircase function of a spectrum an interval, which is large enough to contain sufficiently many levels to obtain good statistics. On the other hand the interval is chosen small enough that in the interval N(E) is well approximated by a straight 45

46 N(E) E Figure 3.3: Practical way of unfolding: Not the whole spectrum is unfolded but a region where N(E) is approximated well by a straight line. line N(E) E D + const.. (3.21) Then it follows that in this interval the smooth density of states is constant ϱ = D 1. Inverting N becomes trivial. We define as unfolded energy ξ = E D and obtain f(e) ξ max ξ min f(dξ)dξ (3.22) D is called mean level spacing. After the unfolding one looks at spectral properties on the scale of the mean level spacing. In figure?? the spectra of some rather distinct systems taken from physics or other are compared on the scale of the mean level spacing. 46

47 Figure 3.4: Examples for series of energy levels and of number series on the unfolded scale. Observe that the number of sticks equals hundred in all examples. From left to right the spectra are: a periodic array of evenly spaced lines; a random sequence; a periodic array perturbed by a slight random jiggling of each level; energy states of the erbium-166 nucleus, all having the same spin and parity quantum numbers; the central 100 eigenvalues of a 300-by-300 random symmetric matrix; positions of zeros of the Riemann zeta function lying just above the 1022nd zero; 100 consecutive prime numbers beginning with 103,613; locations of the 100 northernmost overpasses and underpasses along Interstate 85; positions of crossties on a railroad siding; locations of growth rings from 1884 through 1983 in a fir tree on Mount Saint Helens, Washington; dates of California earthquakes with a magnitude of 5.0 or greater, 1969 to 2001; lengths of 100 consecutive bike rides. Taken from? As mentioned already, usually the smooth density of states can not be determined exactly over the whole spectrum. An exception are billiards. If A is the area of the billiard, their typical energy scale is 2. This is the inverse ma of the number of unit blocks with volume h 2 fitting in the energy shell to energy E of the phase space. It is the leading term in a power expansion of the spectral density in powers of E N(E) = 1 ( ) ma m 2π E 2 2 l E (3.23) 2 The next leading term is governed by the circumference length of the billiard and the higher order terms contain more and more details about the shape 47