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Chinese Physics C

Quantization of the Periodic Orbits and Long-Range Correlations in Quantum Spectra

Xi Li, SongJJ, XingYZ, ZuoW

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Using the classical quantization way in a two dimensional integrable system and quantized energies periodic orbits correspondence relationship, the long range correlation among the quantum lavels has been discussed in detail. Making use of Berry Tabor trace formula, the quantization conditions of the action of the periodic orbits in two dimensional integrable systems have been obtained. Furthermore, considering the periodicity conditions for the periodic orbits on reasonable torus, the correspondence relations between quantum levels and classical periodic orbits in the two dimensional uncoupled oscillators have been established. It is also shown that there exist the long range correlations among these quantum levels which correspond to the group of the classical periodic orbits with same topology M (M 1,M 2) . To concert state the new points, an example has been given.

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Ludong University Journal•Jan 1, 2008•0 citations
Using the classical quantization way in a two dimensional integrable system and quantized energies periodic orbits correspondence relationship, the long range correlation among the quantum lavels has been discussed in detail. Making use of Berry Tabor trace formula, the quantization conditions of the action of the periodic orbits in two dimensional integrable systems have been obtained. Furthermore, considering the periodicity conditions for the periodic orbits on reasonable torus, the correspondence relations between quantum levels and classical periodic orbits in the two dimensional uncoupled oscillators have been established. It is also shown that there exist the long range correlations among these quantum levels which correspond to the group of the classical periodic orbits with same topology M (M 1,M 2) . To concert state the new points, an example has been given.
This chapter discusses in some detail the connection between the classical and quantum mechanical equations of motion, largely using examples that are solvable. It recounts how the old quantum theory was formulated by quantizing the action. Choosing irreducible cycles around the tori leads naturally to defining the action and angle variables of motion, and the quantization of action around each cycle. The chapter shows that families of these periodic orbits contribute to the oscillating part of the density of states of an integrable system in the form of a trace formula. It discusses the transition from integrability to chaos in such systems when perturbations are added that couple resonantly to these natural frequencies. The chapter elaborates with a view to develop the semiclassical approximation, and the associated quantization condition. It brings out the similarity between the Schrodinger equation of wave mechanics, and the HJ equation.
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.
Abstract One of the fundamental results of semiclassical theory is the existence of trace formulae showing how spectra of quantum mechanical systems emerge from massive interference among amplitudes related with time-periodic structures of the corresponding classical limit. If it displays the properties of Hamiltonian integrability, this connection is given by the celebrated Berry–Tabor trace formula, and the periodic structures it is built on are tori supporting closed trajectories in phase space. Here we show how to extend this connection into the domain of quantum many-body systems displaying integrability in the sense of the Bethe ansatz, where a classical limit cannot be rigorously defined due to the presence of singular potentials. Formally following the original derivation of Berry and Tabor (1976 Proc. R. Soc. A 349 101), but applied to the Bethe equations without underlying classical structure, we obtain a many-particle trace formula for the density of states of N interacting bosons on a ring, the Lieb–Liniger model. Our semiclassical expressions are in excellent agreement with quantum mechanical results for <?CDATA N=2,3N = 2,3N=2,3?> N = 2 , 3 and 4 particles. For N = 2 we relate our results to the quantization of billiards with mixed boundary conditions. Our work paves the way towards the treatment of the important class of integrable many-body systems by means of semiclassical trace formulae pioneered by Michael Berry in the single-particle context.
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.
Using the periodic orbit theory,the quantum spectra in the two dimensional rectangular billiard are calculated and analyzed.The results show that each peak in the Fourier transformed spectrum of this system corresponds to the contribution of one classical periodic orbit;and with the increase of the length of the orbit,the number of the classical periodic orbit increases,and the peaks in the corresponding quantum spectra have also increased.This study suggests the good correspondence between the quantum spectra and the classical orbits in the rectangular billiards.Therefore,the correctness of the periodic orbit theory has been verified,which further deepen people's awareness of the relations between the classical and quantum mechanics.