The trace formula

Before 1925/26 quantum mechanics was based on ad hoc quantization procedures such as the Bohr-Sommerfeld quantization method. The disadvantage of this method is that it is coordinate dependent, and usually works only in Cartesian coordinates. In 1917 Einstein suggested a coordinate independent quantization procedure, an important improvement over the Bohr-Sommerfeld quantization scheme. Einstein s quantization method was subsequently extended and improved by Brillouin (1926) and Keller (1958). Therefore, this quantization scheme is referred to as EBK 

In order to exhibit the basic ideas of Gutzwiller s method, we follow an elegant derivation of the trace formula given by Miller (1975). We restrict ourselves to a two degree of freedom bounded autonomous system with Hamiltonian H. The spectrum of H is discrete and determined by 

In the semiclassical approximation the matrix elements (4.1.68) take the form 

This is the trace formula, first derived in a physics context by Gutzwiller in 1971. Similar periodic orbit formulae were also derived in the context of abstract mathematical dynamical systems (Selberg (1956), McKean (1972)). The sum in (4.1.72) extends over all possible periodic orbits 

We define the fluctuating part of the level density according to 

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Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

The interrelations between the propagator, the resolvent, and the level density will be central to our discussion. In particular, the trace formulas referred to in Section I represent semiclassical approximations to the quantities (2.6) or (2.7) and turn out to involve the periodic orbits of the classical dynamics. 

The amplitude of the periodic orbits is therefore determined by the linear stability with respect to perturbations transverse to the orbit. In this sense, the leading term in expression (2.13), obtained by setting C = 0, treats the dynamics transverse to the orbit at the level of the harmonic approximation. The nonlinear stability properties appear thus as anharmonic corrections to the dynamics transverse to the orbit. These anharmonicities contribute to the trace formula by corrections given in terms of series in powers of the Planck constant involving the coefficients C , which can be obtained as Feynman diagrams [14, 31]. 

In this section, we arrive at the quantization condition expressed in terms of periodic orbits. The periodic-orbit contribution to the trace formula can be written as the logarithmic derivative of a so-called zeta function,... 

The periodic orbits (2.31) are referred to as bulk periodic orbits in the sense that all the F actions are nonvanishing. Therefore, all the F degrees of freedom are excited in this periodic motion. On the other hand, there exist edge periodic orbits in the subsystems in which one or several action variables vanish (see Fig. 1). These subsystems have a lower number of excited degrees of freedom, but their periodic orbits also contribute to the trace formula. However, they have smaller amplitudes, related to the amplitude of the bulk periodic orbits as... 

Noting that TrPv = 1 according to Eq. (61) and that Lv is not a matrix but a multiplicative factor, we find that the first term in Eq. (66) is reduced to 2LV, which is independent of the order of Sv in the definition of SSim- The second term was already shown, below Eq. (52), to be equal to 2hdSb/dE. Thus, Eq. (66) recovers the trace formula (65). 

Quantum calculations for a classically chaotic system are extremely hard to perform. If more than just the ground state and a few excited states are required, semiclassical methods may be employed. But it was not before the work of Gutzwiller about two decades ago that a semiclassical quantization scheme became available that is powerful enough to deal with chaos. Gutzwiller s central result is the trace formula which is derived in Section 4.1.3. 

The forward application of (4.1.72) is much more diflScult to perform, but has been accomplished in many cases. Gutzwiller (1971) was the first to apply the trace formula to a real quantum mechanical problem. It consists of electrons with an asymmetric mass tensor moving in a Coulomb potential. This problem is important in semiconductor physics. Gutzwiller was able to compute good approximations to the first few quantum eigenstates of this system. 

For classically chaotic quantum systems the forward application of the trace formula is difiicult because of the following reasons, (i) In a chaotic system the number of periodic orbits proliferates exponentially (ii) the orbits have to be computed numerically and (iii) there are convergence problems with (4.1.72) that have to be circumvented with appropriate resummation prescriptions such as, e.g., cycle expansions (Cvitanovic and Eckhardt (1989), Artuso et al. (1990a,b)). Nevertheless, sometimes valuable information on the structure of atomic states can be obtained by retaining only the shortest orbits in the expansion (4.1.72). This was... 

Wintgen (1987). This section consists of three parts. In part (a) we derive the trace formula for the one-dimensional heUum atom, a system with an odd-even symmetry. In part (b) we use the classical scaUng properties of the one-dimensional helium atom to apply the scaled energy technique. In part (c) we generalize the technique to apply to autonomous systems without scaling symmetries. 

Figure Bl.16.9. Background-free, pseudo-steady-state CIDNP spectra observed in the photoreaction of triethylamine with different sensitizers ((a), antliraquinone (b), xanthone, CIDNP net effect (c), xanthone, CIDNP multiplet effect, amplitudes multiplied by 1.75 relative to the centre trace) in acetonitrile-d3. The stmctiiral formulae of the most important products bearing polarizations (1, regenerated starting material 2, N,N-diethylvinylamine 3, combination product of amine and sensitizer) are given at the top R denotes the sensitizer moiety. The polarized resonances of these products are assigned in the spectra. Reprinted from [21].

A rotating propeller traces out a helix in the fluid, from which a full revolution moves the liquid longitudinally to a fixed distance, depending on its pitch, i.e., the ratio of this distance to the propeller diameter. Pitch may be computed from the following formula ... 

Thus, three types of components can be distinguished in most substances, whether of natural origin or made by humans major, minor, and trace components (see Table 8). The major components, also known as the main or matrix components, are those that determine the chemical nature and properties of a substance. The major components occur in the substance in high concentration, generally exceeding 1 % of the total weight. In minerals and biological substances, for example, the major components are those that appear in the chemical formula that expresses their composition. 

As expected, the EDS data set indicates that the polymeric matrix material (the PE-PP blend) is composed only of carbon (hydrogen is not detectable by this method). The particle, however, appears to be composed mainly of aluminum and oxygen along with small amounts of copper. The ratio of aluminum to oxygen is consistent with the chemical formula for aluminum oxide (A1203). The SEM-EDS results are consistent with aluminum oxide and traces of copper as the primary constituents of the particulate contamination. (Al2O3.3H20 is a commonly used fire-retardant additive in polymeric products.)... 

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

Trace formulas like (8) are a starting point for analysing the statistical properties of quantum spectra. The statistical quantities such as the two-point correlation function can be written in terms of the density of states d(9,N), that is,... 

In fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

For more complicated geometries, the computations become more and more involved as it is the case for the ordinary electromagnetic Casimir effect. However, Casimir calculations of a finite number of immersed nonoverlapping spherical voids or rods, i.e. spheres and cylinders in 3 dimensions or disks in 2 dimensions, are still doable. In fact, these calculations simplify because of Krein s trace formula (Krein, 2004 Beth and Uhlenbeck, 1937)... 

In fact, the latter is the leading contribution to Gutzwiller s trace formula (Gut-zwiller, 1990), namely the contribution of the two-bounce periodic orbit between the two spheres without repetition, with the action Spo(k) = 2(r—2d)k where 2 (r — 2a) is the length of the geometric path. Note that the semiclassical result is suppressed by a factor of 1/4 in comparison to the small-scatterer one. 

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