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Quantum chaos

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                                    Martin Gutzwiller (2007), Scholarpedia, 2(12):3146.
   [3]doi:10.4249/scholarpedia.3146      revision #91683 [[4]link to/cite this article]
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   Curator: [7]Martin Gutzwiller
   Contributors:


   0.60 -

   [8]Eugene M. Izhikevich

   [9]Nick Orbeck

   [10]Benjamin Bronner

   [11]Gregor Tanner
     * [12]Dr. Martin Gutzwiller, Yale University, New Haven, CT

   Quantum Chaos describes and tries to understand the nature of the wave-like
   motions for the electrons in atoms and molecules (quantum mechanics), as well as
   electromagnetic waves and acoustics, etc.. To a limited extent, these waves are
   like the chaotic trajectories of particles in classical mechanics, including the
   light rays in optical instruments and the sound waves in complicated containers.

Contents

     * [13]1 Introduction
          + [14]1.1 The need for a scientific connection
          + [15]1.2 Classical Mechanics (CM)
          + [16]1.3 Quantum Mechanics (QM)
          + [17]1.4 Building a bridge between CM and QM
     * [18]2 Two examples from physics
          + [19]2.1 The eigenstates of a quantum system
          + [20]2.2 Energy levels of the donor impurity in a silicon or germanium
            crystal
          + [21]2.3 An ordinary hydrogen atom near ionization in a strong magnetic
            field
     * [22]3 The Path Integral (PI)
          + [23]3.1 The Path Integral of Dirac and Feynman
          + [24]3.2 Simplification of the path integral for complicated problems
     * [25]4 The Trace formula
          + [26]4.1 Connecting the quantum spectrum with a semi-classical spectrum
          + [27]4.2 A chaotic motion where the trace formula is correct
     * [28]5 The results for the 2 examples of Atomic Chaos
          + [29]5.1 The spectrum of a donor impurity
          + [30]5.2 Ordinary hydrogen atom near ionization in a strong magnetic
            field
     * [31]6 Beyond atomic physics
          + [32]6.1 All kinds of ordinary waves inside hard walls
          + [33]6.2 Microwaves in the stadium and light in a oval-shaped cavity
     * [34]7 Spectral Statistics and more Applications
          + [35]7.1 Applications in nuclear physics
          + [36]7.2 Some generalizations of the trace formula
          + [37]7.3 Various technical areas of application
     * [38]8 Some Reading
     * [39]9 References
     * [40]10 See Also

Introduction

The need for a scientific connection

   Quantum Chaos (QC) tries to understand the connection between two phenomena in
   physics, call them Q and C. The word quantum (Q) comes from the physics of small
   systems like atoms and molecules, where the energy very often appears only in a
   well defined amount, called quantum. Very surprisingly, the movement of a small
   particle like the electron in a molecule looks more like a wave on the surface of
   a pond than the scratch of a dot on some plate. Wave phenomena of this kind
   describe the propagation of light, and quite generally most electronagnetism, as
   well as sounds in any kind of medium. These waves obey linear [41]partial
   differential equations, whose solutions have smooth shapes, and are quite
   pleasant to behold.

   C stands for the chaos, i.e. unexpected and nearly unpredictable behavior of very
   simple mechanical devices like the double pendulum, or the motion of a billiard
   ball on an imaginary table with a more complicated than rectangular shape. The
   double pendulum does unexpected turns and loops, and the exact direction of the
   ball after few bounces is difficult to predict. The motion is controlled by
   ordinary differential equations, whose solutions are extremely sensitive to the
   initial conditions. The resulting shape of the trajectories is confusing,
   although it can be computed rather simply to arbitrarily many decimals.

   These two phenomena contradict our expectations because we try to find a simple
   explanation for the behavior of many interesting and useful objects. An electron
   as a wave in a molecule makes a pleasant picture, but its computation is tricky,
   particularly if one has to understand several elctrons acting simultaneously. The
   same is true for elctromagnetic and sound waves. Trajectories for electrons and
   rays for light and sound seem more in touch with our experience, therefore direct
   and satisfying. But for the complete explanation, the trajectories and rays are
   not always helpful. Nevertheless our intuition has to be prepared with the help
   of simple models that fall back on what our senses and intelligence can grasp.
   Quantum and Chaos look unrelated, and almost contradictory in spite of our
   scientific efforts.

Classical Mechanics (CM)

   This difference in appearance has required very different scientific
   explanations. Before the 20-th century, the physical laws of Isaac Newton were
   able to explain the motion of the planets and moons in the solar systems, but
   also solve many problems of everyday life. This approach came to be called
   Classical Mechanics (CM). It is based on the solution of ordinary differential
   equations. They are able to explain what we now call [42]chaos, although finding
   the best methods for each case is not easy. At the end of the 19-th century,
   Henri Poincare invented new treatments for chaotic systems, and his work was
   continued by many outstanding mathematicians and astronomers. E.g. he used
   "[43]Surface of Section" where the same [44]trajectory cuts through a fixed
   surface, over and over again, leaving a dot wherever it crossed. In the simplest
   cases this leads to smooth curves, while chaos produces a wild scatter of
   isolated points.

Quantum Mechanics (QM)

   The idea of the quantum of energy, however, and the electrons moving like waves,
   was found to be incompatible with classical mechanics. During the first 25 years
   of the 20-th century, the best physicists tried to find some compromise with
   classical mechanics, but only with limited success. The big breakthrough came in
   1925, and within four years there was a new kind of mechanics, that is capable of
   solving all atomic, molecular, and optical problems. Schroedinger's equation has
   to be solved to get the wave function of the system, and that is the most
   convenient expression of quantum mechanics (QM). It is a partial differential
   equation very much like the [45]wave equation for the explanation of sound, radio
   and light waves, etc. But in many-body systems quantum mechanics goes way beyond
   our familiar kind of wave phenomena.

Building a bridge between CM and QM

   Quantum Chaos (QC) tries to build a bridge between QM and CM. This bridge
   provides a transition from QM to CM, as well as from CM to QM. The existence of
   such a bridge puts limits on CM and on QM. An ever smaller tennis-ball bounces
   differently from moving surfaces, and it looks more like an electron. Similarly
   an ever larger molecule eventually may become a big crystal that does no longer
   move like a wave.

   Now imagine a tennis ball bouncing off the hard walls in a closed court. The
   trajectory of an ideal ball keeps on going around the court. Two motions with
   initially close directions may eventually have no similarity, depending on the
   exact shape of the court. This process keeps on going in CM as long as we want.
   The only limit to the precision in CM is the size of the computer. In QM,
   however, a built-in lower limit for the description of the motion prevents the
   chaos from getting too wild. Chaos in QM is mild compared to chaos in CM, but its
   appearance is not as clear cut as in CM.

Two examples from physics

The eigenstates of a quantum system

   In order to appreciate the problem of making the connection between QM and CM,
   there will be first a simple presentation of some examples, where the connection
   was established successfully. In describing these examples it is important to be
   aware of some basic differences between CM and QM with respect to atoms and
   molecules. In CM there are almost no conditions where the nuclei and electrons
   with their electrostatic interactions can find some kind of [46]equilibrium,
   because they are bound to collapse.

   In QM there is usually a whole set of eigenstates with precise energies, starting
   with the "ground state" that is absolutely [47]stable. The "excited states" can
   decay only if the system is allowed to interact with the electromagnetic field,
   and emit or absorb photons to change its energy. If these eigenstates are limited
   in space, they can be enumerated with integers, starting with 0 for the ground
   state, and positive integers \(n\) in the order of increasing energy \(E_n\ .\)
   The set of these energies is the spectrum.

Energy levels of the donor impurity in a silicon or germanium crystal

   A donor impurity replaces an atom of Si or Ge in the crystal lattice; it has an
   effective nuclear charge of 1 higher than Si or Ge, and brings along an extra
   electron which tries to stay nearby if there is no outside electric field.
   Therefore, there is a local hydrogen atom in the crystal, with one trouble: the
   inertial mass of the electron in the x-direction is effectively much larger than
   in the y- and z-direction, by a factor 5 in Si and by a factor 20 in Ge.

An ordinary hydrogen atom near ionization in a strong magnetic field

   At first the only electron stays near the nucleus in the ground state. But then
   it is exposed to some ultraviolet light, of sufficient and well controlled
   frequency to almost throw out the electron, i.e. ionize the atom. The electron
   ends up far away, but is still weakly held by the nucleus in one of the great
   number of eigenstates at a great distance. That leads to a measured spectrum that
   looks as if the lines of absorption were arranged totally at random. The
   eigenstates near ionization are random creatures!

The [48]Path Integral (PI)

The Path Integral of Dirac and Feynman

   In order to make the transition from CM to QM, a very general procedure is
   required. A natural concept of "physical length" \(L\) for a trajectory in CM was
   found about a century after Newton's time. Then two of the most imaginative
   theoretical physicists, P.A.M. Dirac and Richard Feynman, before and after WWII,
   suggested a new approach to QM, and a bridge to CM. A short explanation of their
   idea has to do the job at this point.

   You can ask the question in CM: How does the electron get from the place \(x\) to
   the place \(y\) in the fixed time \(t\) while it is subject to some known forces.
   Answer: Consider any smooth path \(z(s)\) with \(0 < s < t\) from \(x\) to \(y\),
   compute for this path the "physical length" \(L\ .\) In order to calculate the
   physical length of a particular path, the total time available is divided into
   small intervals. For each interval the difference of the kinetic energy minus the
   potential energy is multiplied with the duration of the time interval, and all
   these contributions are added up for the whole path. The path with the smallest
   length, say \(L_0\ ,\) is the simplest classical trajectory that connects the two
   fixed endpoints in the given time. Let me confess that this idea of the physical
   length \(L\ ,\) based on the difference between kinetic and potential energy,
   does not catch my intuition.

   For QM: Any path from \(x\) to \(y\) in the given time \(t\) carries a wave,
   where the phase is the physical length \(L\ ,\) divided by Planck's constant \(h\
   .\) Then let all these waves interfere with each other, and add up. This "path
   integral" (PI) is difficult to calculate. If the lengths \(L\) are large compared
   to \(h\ ,\) however, most contributions cancel one another. Any classical
   trajectory is then favored, because paths with small deviations from \(L_0\) are
   numerous in its neighborhood.

Simplification of the path integral for complicated problems

   In order to get the spectrum without the wave functions, the time \(t\) is
   replaced by the energy \(E\) with the help of a Fourier transform. The dependence
   on the space coordinates \(x\) and \(y\) is eliminated be setting \(x = y\ ,\)
   and then integrating over all available space \(x=y\ .\) The result in QM is the
   trace, simply the sum over the [49]resonance denominators \(1/(E-E_n)\) over the
   spectrum.

   In semi-classical evaluation of the PI, all the trajectories from \(x\) to \(y\)
   in time \(t\ ,\) i.e. all stationary points in the variation of the physical
   length \(L\ ,\) are used. If \(x = y\ ,\) the classical trajectories close
   themselves, but initial and final momentum do not agree. After the summation over
   available space \(x=y\ ,\) the trace accepts only those closed orbits where
   initial and final momenta agree. The result is a [50]periodic orbit (PO).

The Trace formula

Connecting the quantum spectrum with a semi-classical spectrum

   The result of the whole program in the preceding section is expressed in a
   relatively simple formula, now generally called the trace formula (TF). On the
   left is the trace \(g(E)\) as obtained from QM. It is the sum of the resonance
   denominators for the spectrum of the quantum system, \[g(E)=\Sigma_n 1/(E-E_n)\
   .\] On the right is the semi-classical approximation \(g_C(E)\) of \(g(E)\ ,\)
   i.e. the sum over all periodic orbits (PO) in the corresponding classical system,
   \[g_C(E) = \Sigma_\nu A_\nu exp(iL_\nu/h + i\lambda_\nu\pi/2)\ .\] The amplitude
   \(A_\nu\) for each PO reflects its stability; the phase depends on the length
   \(L_\nu\) of the PO, and a multiple \(\lambda_\nu\) ([51]Morse index) of
   \(\pi/2\) for each classical bounce off a dynamical wall. These are all classical
   quantities.

   The TF can be given an intuitive interpretation: The open parameter \(E\)
   represents a small perturbation with a constant frequency \(\mu = E/h\) that
   works on the system from the outside, where \(h\) is always Planck's constant.
   The reaction of the system is a forced motion of the same frequency, with the
   amplitude \(g(E)\ .\) The closer \(E\) is to one of the eigenvalues \(E_n\ ,\)
   the larger is the response of the system; we get a resonance! The external
   perturbation of frequency \(\mu\) can be described also by its period \(\tau\ ,\)
   the reciprocal of \(\mu\ .\) The classical particle gets chased around in its
   space, and it is critical where it lands after one period \(\tau\ .\) The effect
   on the classical particle will be larger if it comes back to its starting point
   after one, or perhaps two or three such periods. Therefore, the classical
   description of a quantum resonance depends on the PO's. The physical length of a
   PO, \(L_\nu\) in the TF, yields the period in time by taking the derivative
   w.r.to the energy \(E\) of the PO.

A chaotic motion where the trace formula is correct

   This correspondence between the set of energies \(E_n\) in QM and the set of
   periodic orbits in CM is a deep mathematical result, even if the proposed
   derivation of the TF is sloppy by mathematical standards. The result was first
   derived as an equality by the mathematician Atle Selberg in 1952 for the motion
   on a 2-dimensional surface of constant negative curvature.

   Surfaces of constant negative curvature are products of the non-Euclidean
   geometry, starting in the first half of the 19-th century. It was then discovered
   at the end of the 19-th century that their geodesics, equivalent to the
   trajectories of a small ball rolling freely on the surface, were very chaotic.
   But it was also understood that these surfaces came in very many, very symmetric
   varieties, i.e. like polygones, they were tiling all the available space. The
   Euclidean plane has relatively few regular triangles, squares, hexagones, without
   any chaotic behavior of the straight lines. The sphere, of constant positive
   curvature, is trivial.

   Selberg tried to find a relation between Riemann's [52]zeta-function, which holds
   all the secrets of the prime numbers, and geometry. The zeroes of the
   zeta-function would play the role of the eigenvalues, and the logarithm of the
   primes are the corresponding PO's, unstable as on Selberg's surfaces. But no real
   quantum problem for the zeta-function is known.

   With 2 as well as with 3 dimensions, with constant negative curvature, there is
   an incredible variety of geometric models. They have different topologies, and
   then within each topology there are continuous parameters available to generate
   surfaces that are metrically different. With constant positive curvature,
   however, there is only one surface up to a scale factor, the sphere of 2 or 3
   dimensions. Evidently chaotic motions are much more numerous, than the regular
   motion, even in pure geometry.
   Figure 1: The 2 simplest periodic orbits for the electron in the neighborhood of
   the donor impurity in Si at the center of the circle; its radius corresponds to
   the energy of the electron set to 1.
   Figure 2: The 2 next simplest PO's, each characterized by a binary code that
   indicates the order of intersection with the horizontal axis.
   Figure 3: A periodic orbit of code length 10, without much symmetry, and
   therefore hard to find.

The results for the 2 examples of Atomic Chaos

The spectrum of a donor impurity

   The extra electron does not stay very close to the place of the donor impurity,
   because the neighboring atoms of Si and of Ge get pushed out of their ordinary
   positions by the presence of the impurity. The effective attraction of the
   electron gets weakened by factors 11 for Si and 15 for Ge. The ordinary Coulomb
   force gets divided by 11 or 15, and the radius of the impurity increases by that
   factor. Figure [53]1 and Figure [54]2 show the 4 shortest PO´s. and their codes,
   i. e. intersections with the x-axis. All of them have some symmetry, and finding
   them is easy. Figure [55]3 shows a PO of code length of 10, and no symmetry.
   Finding it requires patience because this PO is very unstable.
   Figure 4: The semi-classical spectrum of the donor impurity in Si, plotted as
   intersections with the horizontal E-axis; upper diagram on the basis of only the
   8 shortest PO's, and lower diagram with the 71 shortest PO's.
   Figure 5: Numerical computations for the spectrum of the donor-impurity: Names
   for the levels in first column, QM in 1969 in second, then trace formula in 1980,
   and QM in high precision in 1988.

   The trace \(g(E)\) can be written as a converging product of factors \((E-E_n)\
   ,\) and \(g_C(E)\) becomes something similar with respect to the PO's. Figure
   [56]4 shows \(g_C(E)\ ,\) the upper diagram for only the 8 shortest PO's, and the
   lower for the 71 PO's. The approximate energies \(E_n\) are the intersections of
   the curve with the \(E\)-axis. The correct values are indicated by short lines
   crossing the \(E\)-axis. The low energies come out very well. Figure [57]5 shows
   the energy levels E_n , each with the usual description in the hydrogen atom,
   level and angular momentum in the first column, the second column computed with
   QM in 1969, third column computed with trace formula in 1980, fourth column
   computed with QM and high precision.
   Figure 6: Resurgence Spectroscopy of an "Ordinary hydrogen atom near ionization
   in a strong magnetic field", cf. main text. The coordinates are in the upper
   diagram: relative absorption versus increasing energy, and in the lower:
   correlation versus time (period).

Ordinary hydrogen atom near ionization in a strong magnetic field

   The upper diagram in Figure [58]6 shows the measured absorption in a high
   precision experiment. The width of the individual lines depends on the stability
   of the laser light. There are no sensible names for the lines in this spectrum,
   like we had in the donor impurity. The question arises whether this spectrum is
   truly random. The answer depends on all kinds of tests one could try; and then
   one would have to interpret the result. It then came as a great surprise: the
   trace formula suggests that the Fourier transform of this spectrum, from energy
   \(E\) to time \(t\ ,\) yields strong lines whenever there is a PO with that time
   for its period. The lower diagram in Figure [59]6 shows the PO's for this random
   spectrum. This method of explaining a random looking spectrum was only discovered
   by the work on the trace formula; it is now called Resurgence Spectrocopy.
   Although this analysis does not always work, it is marvelous result of QC.
   Figure 7: A liquid is contained in an open container with the shape of a stadium,
   which is lightly shaken at various fixed frequencies; photographic pictures of
   the vibrating surface show the same well defined patterns as can be found in the
   stadia of figures 8 to 11.

Beyond atomic physics

All kinds of ordinary waves inside hard walls

   The history of optics is well known for the battles between rays and waves as the
   fundamental way of propagating light. The mathematics of these waves and their
   relation to the corresponding rays is almost identical to the relation between CM
   and QM. Chaos in the optical rays is just as complicated as in the motion of
   electrons. A popular model in 2 dimensions is a flat area surrounded by a hard
   wall. The rays inside such a cavity are straight lines with ideal reflection at
   the wall. Chaos comes from the shape of the wall, a simply closed curve. The
   equations for an electron in such a model are the same as for light, sound, fluid
   motion as in Figure [60]7. It shows the surface waves of a liquid due to the
   shaking of its container.

   Russian mathematicians distinguished themselves after WWII by studying in great
   detail certain classes of geometric models to determine the nature of the
   trajectories. The measure of chaos is called its [61]entropy, and the main
   results show that it is not zero. Among them is the "[62]stadium", 2 parallel
   lines of equal length that are connected with half circles at each end. It is
   instructive to look at some work with this system.

   Figure 8: Resonances in a microwave cavity between 17 and 18 GHz at temperatures
   of 300 K (upper) and 2 K (lower diagram) to reduce the damping in the metal of
   the cavity, increasing the Q-factor and the resolution of the spectrum.
   Figure 9: Fourier analysis (resurgence spectroscopy) of the resonances for the
   lower part of figure 8. Each one of the various peaks can be associated with the
   period of the various periodic orbits in the stadium.

Microwaves in the stadium and light in a oval-shaped cavity

   Microwaves with a wave length of several centimeters are interesting to watch in
   a stadium-like cavity of about 1 m, but no more than 2 cm thick. At room
   temperature the resistance of the metal of the cavity only allows subdued
   resonances, while at 2 degrees Kelvin they are very clear as in Figure [63]8.
   With [64]Resurgence Spectroscopy, i.e. Fourier transform from \(E\) to \(t\ ,\)
   the PO's of the stadium are shown in Figure [65]9, just as in Figure [66]6.
   Figure 10: Resonances in the conductivity for electrons inside a two-dimensional
   stadium (upper) and circle (lower diagram) of mesoscopic size, as a function of
   an applied very large magnetic field and at very low temperature.
   Figure 11: Picture explaining the laser function inside a glas cavity; the oval
   shape (not an ellipse) has been carefully chosen for the light to escape at
   either end tangentially, as shown in the lower picture.

   ===The stadium in the real world===Figure [67]10 shows the electric resistance
   versus an applied magnetic field in a conducting layer between two semiconductors
   in two configurations. The mean free path of the electrons is larger than the
   stadium or the circle; the temperature is extremely low, and the resonances are
   very sharp. The statistical distribution for fhe chaotic stadium has only one
   broad peak, whereas the nonchaotic circle has many resonances beyond 2 symmetric
   minima. Figure [68]11 shows laser light caught inside a stadium of glass with an
   oval cross section. The light is forced out at the ends tangentially by the
   curvature, and only there.

Spectral Statistics and more Applications

Applications in nuclear physics

   In contrast to the use of QM in atomic and molecular physics, the atomic nucleus
   is not well understood, because the forces between the nucleons, i.e. proton and
   neutron, are much more complicated than the simple Coulomb forces between nuclei
   and electrons. Nuclear physicists have to work with empirical models.
   Nevertheless the spectrum of nuclear energy levels is very rich, and therefore,
   complicated.

   Neutron resonance spectroscopy provides a unique situation where, in a narrow
   energy window, successive eigen-energies (in the compound nucleus region) around,
   say, the one-hundred-thousandth level in a heavy nucleus, can be detected very
   accurately one by one (cf. fig. 12). It is then natural to adopt a statistical
   approach. Such statistics were discussed ever after WWII under the assumption
   that the [69]fluctuation properties of the energy levels come from finite, but
   large matrices of various kinds. The random choice of the matrix elements was
   investigated and compared with the experiments. This "[70]random matrix theory"
   became the foundation for understanding large parts of the nuclear spectra. In
   the beginning of the 1980's the origin of these empirical random matrices was
   finally explained by the important conjecture that the origin of the
   distributions is the result of Quantum Chaos.

   If such a connection was in fact to be expected, one could check it in other
   systems with a rich spectrum. A partial proof of this general conjecture in some
   special cases has since been found on the basis of the trace formula. Some
   special features of the PO's in CM are limiting the statistics of the system in
   QM. In the 1970's, some mathematicians observed that the statistics of the
   "mysterious" zeroes for Riemann´s zeta-function have strong similarities with the
   eigenvalues of random-matrices. Some physicists like to talk about "Riemannium"
   as a new element with characteristic features in the "spectrum" of its zeroes.
   Figure 12: Total cross section for the reactions n + 232 Th as a function of the
   neutron energy (from the compilation 'Neutron Cross Section', 1964). Notice the
   neutron energies, which are given to single eV, as well as the sharpness of the
   lines.

Some generalizations of the trace formula

     * Sofar we have studied only how the spectrum of some wave phenomenon arises
       approximately with the help of the PO's. The path-integral also tells us how
       a particle starts in the point \(x\) and ends up in the point \(y\ .\) One
       can even give to \(x\) and \(y\) certain distributions to reflect the
       conditions of the experiment.

     * A particular PO can depend on the energy or on the time available. For
       instance, it can appear or disappear as one increase the time of the energy.
       In that case the TF needs some additional details to be worked out in the
       neighborhood of the transition in time or energy.

     * In the case of light rays, but just as well in the presence of steep rises in
       the potential energy, the ray or the trajectory may simultaneously split into
       reflection and into refraction on a wall. Such a possibility increases the
       number of PO's greatly.

     * Many simple problems in molecular physics require the electron to tunnel,
       i.e. overcome a mountain of potential energy that is higher than the
       available total energy. In the simplest chemical bond, two protons being held
       together by either one or two electrons, the electron cannot move
       "classically" from the neighborhood of one proton to the neighborhood of the
       other proton. Therefore we have to allow classical trajectories with
       stretches of negative kinetic energy, where the time is a purely imaginary
       quantity, i.e. its square is negative.

     * The angular momentum with a spin of h/2 is a very important attribute for the
       electron. The description by the Pauli matrices characterizes its local
       direction by 3 real components, i.e. a vector S of fixed length at each point
       in space. There are essentially 2 waves spreading at the same time over the
       same volume; together they determine exactly the 3 components of S. The
       motion of the electron through any electric or magnetic field will then lead
       to a motion of S along its motion in space.

     * Scattering of electrons and photons from atoms and molecules can be treated.
       For this purpose the process is most usefully considered as in a Feynman
       diagram, where a light ray hits the electron trajectory. Energy and momentum
       have to be conserved at the point of collision.

     * For quite a while it was not clear whether it is possible to get reliable
       results from TF. But with some better understanding, the precision of the
       bound states depends on a chosen upper limit \(E_n\) of the energy. Quite
       unexpectedly, if the upper limit is chosen relatively low, the TF will yield
       a few of the lowest states quite well, contrary to the general assumption
       that semiclassical results are good only for large energies.

     * The TF arises from the second order correction to the propagator, or path
       integral PI, because we took into account the second order variation to the
       appropriate classical trajectories in the PI. The lowest order is the famous
       formula of Hermann Weyl, which yields the density of the eigenstates for any
       linear differential operator. For quantum mechanics the TF implies a correct
       term of order 2 in Planck's quantum h. By including third- and higher order
       variations in the PI, one can get a formal expansion to higher order for the
       spectrum and other properties.

     * The time dependence in QM has not as yet been studied in great detail for
       many systems. It turns out to be a very difficult mathematical problem, with
       many unexpected features even in very simple systems such as the reflexion of
       the wave from a steep wall of finite height. Strangely, the PI is defined for
       a fixed time interval t; the energy E arises only with the help of a Fourier
       integral. The time dependence in QM should be easy to obtain directly from
       the PI, or its semiclassical approximation. But it is quite tricky, even
       numerically in an oval-shaped stadium. There is still much work to do that
       might have many practical applications, and compare directly with
       experiments.

Various technical areas of application

     * Bound states and scattering in chemistry.
     * Intra- and inter-molecular dynamics due to molecular vibrations.
     * 2-dimensional electron traps on a metal surface.
     * Shell structure of crystals depending on the lattice vibrations.
     * Magnetic susceptibility in anti-dot arrays.
     * Spin-orbit coupling for electrons in GaAs/GaAlAs interface.
     * Cohesion and stability of metal nanowires.
     * Concert halls, drums, church bells, tsunamis, etc.

Some Reading

   Postmodern Quantum Mechanics, by Eric J. Heller and Steven Tomsovic, Physics
   Today (American Institute of Physics) July 1993, 38-46.

   Einstein's Unknown Insight and the Problem of Quantizing Chaos, by A. Douglas
   Stone, Physics Today (American Institute of Physics) August 2005 37-43.

   [71]Celestial Mechanics on a Microscopic Scale, by T.Uzer, D.Farrelly,
   J.A.Milligan, P.E.Raines, and J.P.Skelton, Science 253 (1991) 42-48.

   The Culture of Quantum Chaos, by M. Norton Wise and David C. Brock, Stud. Hist.
   Phil. Mod. Phys., Vol.29, No.3 (1998) 369-389.

References

   Resource Letter ICQM-1: The Interplay between Classical and Quantum Mechanics, by
   Martin C. Gutzwiller, American Journal of Physics 66 (1998) 304-324. Same Title
   with Same Editor, Collection of reprints, published 2001 by AAPT (American
   Association of Physics Teachers), College Park, MD 20740-3845.

   Quantum Chaology (The Bakerian Lecture 1987), by M. V. Berry, in Dynammical
   Chaos, Proceedings of the Royal Society, edited by Michael V. Berry, I.C.
   Percival, and N.O. Weiss, A 413, 1-198.

   Semiclassical Physics, by Matthias Brack and Rajat K. Bhaduri, Addison-Wesley
   Inc., Reading MA, 1997, 444 p.

   Quantum Signatures of Chaos, by Fritz Haake, Springer-Verlag, Berlin-Heidelberg,
   2nd ed 2001, 479 p.

   Quantum Chaos - An Introduction, by Hans-Juergen Stoeckmann, Cambridge University
   Press, 1999.

   Quantum Chaos Y2K, Proceedings of Nobel Symposium 116, edited by Karl-Fredrik
   Berggren and Sven Aberg, in Physica Scripta, Kungl. Vetenskapsakademien and World
   Scientific, Singapore, 2001.

   Internal references
     * James Meiss (2007) [72]Dynamical systems. [73]Scholarpedia, 2(2):1629.
     * Eugene M. Izhikevich (2007) [74]Equilibrium. Scholarpedia, 2(10):2014.
     * Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) [75]Periodic orbit.
       Scholarpedia, 1(7):1358.
     * Philip Holmes and Eric T. Shea-Brown (2006) [76]Stability. Scholarpedia,
       1(10):1838.

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   [77]Dynamic Billiards, [78]Chaos, [79]Dynamical Systems, [80]Periodic Orbit,
   [81]Quantum Mechanics
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