Foundations of Physics

doi: 10.1023/A:1017556707359pmid: N/A

doi: 10.1023/A:1017508824198pmid: N/A

Aurich, R.; Steiner, F.

doi: 10.1023/A:1017560808268pmid: N/A

Sum rules are derived for the quantum wave functions of the Hadamard billiard in arbitrary dimensions. This billiard is a strongly chaotic (Anosov) system which consists of a point particle moving freely on a D-dimensional compact manifold (orbifold) of constant negative curvature. The sum rules express a general (two-point)correlation function of the quantum mechanical wave functions in terms of a sum over the orbits of the corresponding classical system. By taking the trace of the orbit sum rule or pre-trace formula, one obtains the Selberg trace formula. The sum rules are applied in two dimensions to a compact Riemann surface of genus two, and in three dimensions to the only non-arithmetic tetrahedron existing in hyperbolic 3-space. It is shown that the quantum wave functions can be computed from classical orbits. Conversely, we demonstrate that the structure of classical orbits can be extracted from the quantum mechanical energy levels and wave functions (inverse quantum chaology).

Kleppner, Daniel; Delos, John

doi: 10.1023/A:1017512925106pmid: N/A

A complete quantum solution provides all possible knowledge of a system, whereas semiclassical theory provides at best approximate solutions in a limited region. Nevertheless, semiclassical methods based on the work of Martin Gutzwiller can provide stunning physical insights in regimes where quantum solutions are opaque. Furthermore, they can provide a unique bridge between the quantum and classical worlds. We illustrate these ideas with an account of a theoretical and experimental attack on the paradigm problem of the hydrogen atom in strong magnetic and electric fields.

Braun, Peter; Gnutzmann, Sven; Haake, Fritz; Kuś, Marek; Życzkowski, Karol

doi: 10.1023/A:1017564909177pmid: N/A

The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we show that equilibrium statistical mechanics for the fictitious gas of particles associated with the parametric motion of levels yields spectral fluctuations of the random-matrix type. Previously known clues to that goal are an appropriate equilibrium ensemble and a certain ergodicity of level dynamics. We here complete the reasoning by establishing a power law for the ħ dependence of the mean parametric separation of avoided level crossings. Due to that law universal spectral fluctuations emerge as average behavior of a family of quantum dynamics drawn from a control parameter interval which becomes vanishingly small in the classical limit; the family thus corresponds to a single classical system. We also argue that classically integrable dynamics cannot produce universal spectral fluctuations since their level dynamics resembles a nearly ideal Pechukas–Yukawa gas.

Choquard, Philippe

doi: 10.1023/A:1017517026015pmid: N/A

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations St $${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}$$ S x 2 =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the terms of these expansions and this suggests that S(x, t) should be written as a product of the elementary solution x2/(2t) and a function f=f(φ) where φ=φ(x, |t|) owing to the symmetry property which is that S(x, −t)=−S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of φ(x, |t|)=φ0 |t/t0|κ (x/x 0)λ≥0 with κ, λ∈ ℝ results in a soluble ordinary differential equation, of first order in u=ln φ and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called “trivial.” The phase plane (fu, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the “trivial” fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid.

Palla, Gergely; Vattay, Gábor; Voros, André; Søndergaard, Niels; Dettmann, Carl Philip

doi: 10.1023/A:1017569010085pmid: N/A

We review studies of an evolution operator ℒ for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of ℒ yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator ℒ in the weak noise limit can be computed in several ways. A method using a local matrix representation of the operator allows to push the corrections to the escape rate up to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actually, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators ℒn. Using an integral representation of the evolution operator ℒ, we then investigate the high order corrections to the latter traces. Their asymptotic behavior is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of trace formula.

Berry, Michael

doi: 10.1023/A:1017521126923pmid: N/A

Complex superpositions of degenerate hydrogen wavefunctions for the n th energy level can possess zero lines (phase singularities) in the form of knots and links. A recipe is given for constructing any torus knot. The simplest cases are constructed explicitly: the elementary link, requiring n≥6, and the trefoil knot, requiring n≥7. The knots are threaded by multistranded twisted chains of zeros. Some speculations about knots in general complex quantum energy eigenfunctions are presented.

Connors, R.; Keating, J.

doi: 10.1023/A:1017573110994pmid: N/A

It is shown that both universal and non-universal correlations must exist between classical periodic orbits in order that Gutzwiller's semiclassical trace formula is consistent with a real, discrete quantum energy spectrum. Formulae for the two-point correlations are derived. The universal correlations are consistent with those conjectured by Argaman et al. (1993). Likewise, both universal and non-universal correlations must exist between quantum energy levels in order that the trace formula be consistent with the fact that periodic orbit actions are real and discrete. In this case, the two-point correlations implied are consistent with random matrix theory and previous semiclassical calculations. These ideas are illustrated with reference to the primes and the Riemann zeros.

Briggs, John; Rost, Jan

doi: 10.1023/A:1017525227832pmid: N/A

Few have done more than Martin Gutzwiller to clarify the connection between classical time-dependent motion and the time-independent states of quantum systems. Hence it seems appropriate to include the following discussion of the origins of the time-dependent Schrödinger equation in this volume dedicated to him.