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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Closed orbits and the regular bound spectrum

Michael Berry, M. Tabor

Published Apr 13, 1976• 514 citations
Publisher

Abstract

Abstract The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K. conditions into a representation of the spectrum in terms of a ‘topological sum’ involving only the closed classical orbits; the theory applies equally to separable and non-separable systems; stability parameters are not involved. Significant contributions come from complex closed orbits which however have real constants of the motion. Clustering of levels on different scales is demonstrated by smoothing the spectrum using the formal device, due to Balian & Bloch, of adding a variable imaginary part to the energy. The topological sum is shown to agree very well with exactly-computed spectra for circular and spherical potential wells with repulsive core.

DOI
10.1098/rspa.1976.0062

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Abstract The energy levels of systems whose classical motion is multiply periodic are accurately given by the quantum conditions of Einstein, Brillouin & Keller (E. B. K.). We transform the E. B. K. conditions into a representation of the spectrum in terms of a ‘topological sum’ involving only the closed classical orbits; the theory applies equally to separable and non-separable systems; stability parameters are not involved. Significant contributions come from complex closed orbits which however have real constants of the motion. Clustering of levels on different scales is demonstrated by smoothing the spectrum using the formal device, due to Balian & Bloch, of adding a variable imaginary part to the energy. The topological sum is shown to agree very well with exactly-computed spectra for circular and spherical potential wells with repulsive core.
The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
The phase integral approximation for the Green's function is investigated so as to yield an approximate expression for the density of states per unit interval of energy. This quantity is shown for negative energies (bound states) to depend only on the periodic orbits, i.e., the smoothly closed trajectories, unlike the approximate wavefunctions which depend on all possible trajectories. A particle in a periodic box of one, two, and three dimensions is discussed first to demonstrate how the approximate density of states contains a continuous background besides the δ-function spikes of the discrete spectrum. Then we examine the situation in a spherically symmetric potential where special problems arise because the quasiclassical propagator has to be evaluated at a focal point of the classical trajectory. With the help of the Helmholtz-Kirchhoff formula of diffraction theory, the amplitude is shown to remain finite at the focus. The orbits which remain entirely in a region of Coulombic potential yield a spectrum of Balmer terms with appropriately reduced degeneracy. However, the orbits which penetrate the screening charge give discrete levels obeying the Bohr-Sommerfeld conditions with the correct degeneracy. The continuous background in the approximate density of states can be discussed on the basis of the formulas derived in this paper. This is necessary as an introduction to the problem of a particle in a potential where the motion is not multiply periodic.
The Einstein-Brillouin-Keller (EBK) quantization equation is used to determine the energy levels of a two-body system with an arbitrary central potential that allows for bound states. The treatment is based on the conservation laws and avoids both the Newtonian and Schrödinger differential equations. Because analytic solutions for the energy levels do not exist in general, the EBK condition is applied using the Newton-Raphson method and the radial probability density is computed. Potentials appropriate for a diatomic molecule are considered and the effect of the angular momentum on the radial distribution, the nature of the classical orbits, and the possibility of closed orbits is studied.
The semi-classical method has become a necessary instrument to study the classical movement of the particle.Periodic orbit theory is rapidly becoming one of most useful semi-classical tools which can be used to make direct connections between the quantized energy eigenvalues of a bound state and the classical motions for the corresponding point particle.We use a quantum spectral function which contain rich information of classical orbits in well.We study the correspondence between quantum spectra and classical orbits in the circular.Two-dimensional billiard systems have provided easily visualization examples relevant for both types of analyses.As a simple example of the application to a billiard or infinite well system of Periodic orbit theory,we compute the Fourier transform(ρL) of the quantum mechanical energy level density of two-dimensional circular billiard systems.The resulting peaks in plots of ρL2 versus L are compared to the lengths of the classical trajectories in these geometries.The locations of peaks in ρL agree with the lengths of classical orbits perfectly,which testifies the correspondence of quantum mechanics and classical mechanics.This examples show evidently that semi-classical methods provides a bridge between quantum and classical mechanics.
We examine the semiclassical limit of the quantum energy spectrum in many dimensions: by means of a WKB-like ansatz leading to Einstein-Brillouin-Keller (EBK) quantization, by means of a path integral, hence associating a bound state with a particular classical periodic trajectory, and by the Birkhoff-Gustavson (BG) transformation to action-angle variables. We extend the EBK method to many-fermion systems using coherent states; and apply both EBK using surfaces of section, and the BG transformation to an SU(3) schematic nuclear shell model. We describe a new algorithm for finding periodic trajectories of a Lagrangian system with polynomial potential. It is applied to the Henon-Heiles system with good results, and these trajectories are used to quantize the system. The EBK and BG methods have some success, while periodic trajectory quantization fails. We discuss possible reasons for this failure and future approaches to these problems.
We study detailed classical-quantum correspondence for a cluster system of three spins with single-axis anisotropic exchange coupling. With autoregressive spectral estimation, we find oscillating terms in the quantum density of states caused by classical periodic orbits: in the slowly varying part of the density of states we see signs of nontrivial topology changes happening to the energy surface as the energy is varied. Also, we can explain the hierarchy of quantum energy levels near the ferromagnetic and antiferromagnetic states with Einstein-Brillouin-Keller quantization to explain large structures and tunneling to explain small structures.