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Journal of Physics A Mathematical and General

Phase-space projection identities for diffraction catastrophes

Michael Berry, Francis J Wright

Published Jan 1, 1980• 40 citations
Publisher

Abstract

For each standard 'diffraction catastrophe' wavefunction psi , describing the interference pattern near a stable caustic, the authors derive two nonlinear identities. These relate the intensity mod psi mod 2 to an integral over the wavefunction psi corresponding to the same catastrophe, or to a less singular one. The identities are interpreted in terms of projections of Wigner functions from phase space on to coordinate space.

DOI
10.1088/0305-4470/13/1/016

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For each standard 'diffraction catastrophe' wavefunction psi , describing the interference pattern near a stable caustic, the authors derive two nonlinear identities. These relate the intensity mod psi mod 2 to an integral over the wavefunction psi corresponding to the same catastrophe, or to a less singular one. The identities are interpreted in terms of projections of Wigner functions from phase space on to coordinate space.
We have investigated the dynamics of a quantum particle in the optical lattice potential. Initially, the quantum particle was represented by a Gaussian wave packet, located in the center of the well. The corresponding Schrödinger equation was solved explicitly by the method of the Chebyshev global propagation. Obtained solutions were also used for the construction of the Wigner functions. We found a great number of local abrupt changes of the solution shape. To explain this behavior, we used the fact that structurally stable systems, which form the largest class of the low dimensional dynamical systems, can be modeled and classified according to the catastrophe theory. All important features of the exact solution were explained on the basis of the mathematical properties of the catastrophic model. Such an approach enabled us to extract relevant information out of numerical solutions without employing any kind of approximations. We have investigated the influence of the Wigner catastrophes on the details of the quantum-classical correspondence breakdown. The wave packet was found to expand rapidly, filling the whole classically available area of the phase space. It was found that its self-interference pattern saturates quickly. A region of the phase space emerges in which the Wigner function oscillations transform into the singularity driven fluctuations. Once this region covers the whole area of the phase space, a wave packet dynamics enters into the new regime where its Wigner function fluctuates around the ergodic average. It will be shown that all mentioned processes are caused by the proliferation of the catastrophes and their mutual interactions.
A detailed framework for a new formalism of phase-space path integral is presented, the outline of the theory of which has been reported elsewhere [K. Takatsuka, Phys. Rev. Lett. 61, 503 (1988)]. This path integral is described in terms of a phase-space distribution function, which was proposed earlier by the present author under the name of the dynamical characteristic function (DCF). The DCF is characterized by two independent wave functions and with two phase spaces. In the present paper, we extend the DCF by utilizing the Feynman kernel in place of the two independent wave functions. This is called the identity DCF. The dynamics of the DCF for general wave packets is reduced to that of the identity DCF. Some characteristics of the identity DCF are presented. For example, the quantum-mechanical time-evolution operator is represented in terms of the identity DCF and of a complete set of quantum q (coordinate) and p (momentum) operators. A semiclassical theory for the identity DCF is also developed, its primary aim being the study of heavy-particle dynamics such as molecular collisions.It is shown that the propagation of the identity DCF is represented by classical trajectories, each of which is associated with an amplitude factor and with the action integral as the quantum phase factor. This amplitude factor is free from singularity, unlike the primitive semiclassical expression of the Feynman kernel, which is well known to diverge at caustics. The physical significance of the identity DCF is analyzed in the context of chaotic dynamics. In particular, it is shown that its amplitude factor is an explicit function of the eigenvalues of a Jacobian matrix [\ensuremath{\partial}(qf{\mathit{q}}_{\mathit{f}}qf​,pf{\mathit{p}}_{\mathit{f}}pf​)/\ensuremath{\partial}(qi{\mathit{q}}_{\mathit{i}}qi​,pi{\mathit{p}}_{\mathit{i}}pi​)], where (qi{\mathit{q}}_{\mathit{i}}qi​,p){\mathit{p}}_{)}p)​ and (qf{q}_{f}qf​,pf{p}_{f}pf​) are the initial and final points, respectively, of a classical trajectory. These eigenvalues are well known in the study of the classical chaos of Hamilton systems and have been used to construct a measure of the linear (and local) stability of an orbit, the Liapounov characteristic exponents, and so on. Further, if the identity DCF is applied to a periodic orbit, the amplitude factor becomes equivalent to the square root of the determinant of a matrix, which is the so-called monodromy matrix minus the unit matrix. This matrix is one of the standard quantities used in bifurcation theory to investigate the stability of fixed points. In fact, the amplitude factor is closely related to the so-called residue, which was proposed by Greene [J. Math. Phys. 20, 1183 (1979)] in order to predict the occurrence of global chaos in classical systems. It is emphasized that all these classical quantities arise naturally in our representation of quantum mechanics.
One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a δ\deltaδ-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point.
Grossmann and Royer have recently shown that the Wigner functions are closely related to the set of all translations and inversions of phase space. This allows the phase space representation of quantum mechanics to be constructed directly on the group of phase space translations and inversions. Starting from this observation, we have derived analytical expressions for the matrix elements of the translation and inversion operators, in the harmonic oscillator representation, without introducing coordinate or momentum wavefunctions.