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IEEE Journal of Quantum Electronics

Correspondence principles: the relationship between classical trajectories and quantum spectral

J. B. Delos, M. L. Du

Published Jul 1, 1988• 16 citations
Publisher

Abstract

The general correspondence principle specifies a relationship between wave functions and families of trajectories. For bounded systems with regular trajectories, this general correspondence principle leads to the principle of quantization of action. The general principle also leads to a correspondence principle for finite-resolution spectra: oscillation in the average density of states, or in the average oscillator strength as a function of energy, are connected to periodic or to closed orbits. This principle holds independently or whether classical trajectories are orderly or chaotic.>

DOI
10.1109/3.983

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The general correspondence principle specifies a relationship between wave functions and families of trajectories. For bounded systems with regular trajectories, this general correspondence principle leads to the principle of quantization of action. The general principle also leads to a correspondence principle for finite-resolution spectra: oscillation in the average density of states, or in the average oscillator strength as a function of energy, are connected to periodic or to closed orbits. This principle holds independently or whether classical trajectories are orderly or chaotic.>
The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrödinger equation with a potential, where the orbits depend on the energy.In special cases the simplicity has been restored by rescaling the size of the orbit, and perhaps a coupling constant at the same time.We point out that the goal can be achieved for any system of this class simply by rescaling the overall coupling constant.In each of these situations we inspect critically the relation between the oscillation frequency and the period of the orbit; in many cases it is observed that a characteristic length of the orbit is a better indicator.When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity.Finally, we comment on the alleged 'paradox' that semiclassical periodic-orbit theory is more effective in calculating low energy levels than high ones.
The eigenvalue density of a quantum-mechanical system exhibits oscillations, determined by the closed orbits of the corresponding classical system; this relationship is simple and strong for waves in billiards or on manifolds, but becomes slightly muddy for a Schrodinger equation with a potential, where the orbits depend on the energy. We discuss several variants of a way to restore the simplicity by rescaling the coupling constant or the size of the orbit or both. In each case the relation between the oscillation frequency and the period of the orbit is inspected critically; in many cases it is observed that a characteristic length of the orbit is a better indicator. When these matters are properly understood, the periodic-orbit theory for generic quantum systems recovers the clarity and simplicity that it always had for the wave equation in a cavity. Finally, we comment on the alleged "paradox" that semiclassical periodic-orbit theory is more effective in calculating low energy level...
Bohr’s correspondence principle is that associated with each observable in classical mechanics is an observable in quantum mechanics that follows certain quantization axioms which can be used to explain the energy spectra of atoms and quantum oscillators. In Schrodinger’s wave mechanics, the state of the system at any time is described by a density operator in a Hilbert space. The chapter describes how the Pauli equation can be used to calculate the average electric dipole moment and average magnetic dipole moment of an atom in an external electromagnetic field. This calculation enables to calculate the electric polarization field and the magnetization field of a material.
In order to have a complete description of the spatio-temporal evolution of a quantum state, the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry. Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry. In the former case, we have regular motion with the general character which tends to that of the corresponding classical case in the limit ℏ → 0. While in the latter case, we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.
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.
In this paper we show what seems to be the (very simple) key for quantization of classical systems. Given a manifold M , each pair given by a riemannian metric (nondegenerate, of arbitrary signature) and a linear connection, canonically determine a quantization rule or 'Correspondence Principle', which assigns to each classical magnitude (function in TM , subject to certain conditions) a differential operator in . The issue about the order in which the p ' and q ' are to be taken in quantization loses all meaning, when the general rule has been fixed. Once specified the Correspondence Principle, each 'classical state' of the system, understood as a vector field on M , determines a wave equation for each magnitude. The Schrödinger equation is a particular example of these wave equations.