Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Roles of phase cancellation removing spuriously quantized states in semiclassical mechanics

Atsuko Inoue-Ushiyama, Kazuo Takatsuka

Published Jul 1, 1999• 13 citations
Publisher

Abstract

Two types of destructive interference among quantum phases are discussed that play an essential role in canceling out the unquantized states and thereby leaving only the correct projection of true eigenstates behind. Without these mechanisms, the Fourier spectra of the autocorrelation function, from which the energy spectra can be deduced numerically, can suffer from spurious features. We show two such examples: One is a spurious spectrum that appears in the negative energy domain of the Fourier frequencies for a positive potential. The other is a family of harmonic and subharmonic frequencies that are born from the ``quantizing orbits,'' which are responsible for forming eigenstates like the periodic orbits or those satisfying the Einstein-Brillouin-Keller condition. The mechanisms for the spurious spectra to appear are analyzed, emphasizing their relationship to the quality of semiclassical wave functions.

DOI
10.1103/physreva.60.112

Related papers

Nature of semiclassical spectrum in terms of classical trajectories(4) Quantum chaos and semiclassical theory in molecular science and nuclear theory, Chaos and Nonlinear Dynamics in Quantum-Mechanical and Macroscopic Systems)

Hiroshi Ushiyama, Kazuo Takatsuka

The mechanism of semiclassical quantization is examined in detail. We found an important role of destructive interference in building quantum spectrum. Based on this analysis, we propose an efficient method to calculate the semiclassical spectrum. This method is particularly promising in multidimensional chaotic systems and also very useful to identify which trajectories dominate a quantum spectrum. We numerically examine this method by applying to a two-dimensional chaotic system and show an illustrative example of identifying such a dominant trajectory that corresponds to an eigenstate.

物性研究•Jun 20, 2005•0 citations

USE OF HARMONIC INVERSION TECHNIQUES IN SEMICLASSICAL QUANTIZATION AND ANALYSIS OF QUANTUM SPECTRA

Jörg Main

Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories -- for both the Berry-Tabor formula and Gutzwiller's trace formula -- and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for v...

•Jan 1, 1999•33 citations

Interference in phase space and phase-only reconstruction

André Knoll, Leon Cohen, Wolfgang P. Schleich

We demonstrate that the Fourier transform of a function leads to interference effects in phase space which allow the reconstruction of the essential properties of the original function, even when we restrict the inverse Fourier transform to the phase factor rather than the complete Fourier transform. We analyze this phase-only reconstruction, known from signal processing, in quantum mechanics for the elementary example of a superposition of two real-valued Gaussians. We employ the Wigner function as well as techniques of semiclassical quantum mechanics to show that their key features, such as their separation and their width, are indeed encoded in the phase of the Fourier transform. For this example, the amplitude-only reconstruction performing the inverse Fourier transform of the absolute value of the Fourier transform fails.

Physical Review A•Nov 11, 2025•0 citations

Semiclassical mechanics of periodic motion: I. General scheme

Peter Kasperkovitz, Momtchil Peev

A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.

Journal of Physics A Mathematical and General•Mar 6, 1998•2 citations

The Wigner representation of quantum mechanics

Valerian I. Tatarskii

Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.

Soviet Physics Uspekhi•Apr 30, 1983•483 citations

From Quantum Spectra to Classical Orbits:the Circular Billiards Systems

Zhang Ye-bing

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.

Chongqing Shifan Daxue xuebao. Ziran kexue ban•Jan 1, 2011•0 citations
Two types of destructive interference among quantum phases are discussed that play an essential role in canceling out the unquantized states and thereby leaving only the correct projection of true eigenstates behind. Without these mechanisms, the Fourier spectra of the autocorrelation function, from which the energy spectra can be deduced numerically, can suffer from spurious features. We show two such examples: One is a spurious spectrum that appears in the negative energy domain of the Fourier frequencies for a positive potential. The other is a family of harmonic and subharmonic frequencies that are born from the ``quantizing orbits,'' which are responsible for forming eigenstates like the periodic orbits or those satisfying the Einstein-Brillouin-Keller condition. The mechanisms for the spurious spectra to appear are analyzed, emphasizing their relationship to the quality of semiclassical wave functions.
The mechanism of semiclassical quantization is examined in detail. We found an important role of destructive interference in building quantum spectrum. Based on this analysis, we propose an efficient method to calculate the semiclassical spectrum. This method is particularly promising in multidimensional chaotic systems and also very useful to identify which trajectories dominate a quantum spectrum. We numerically examine this method by applying to a two-dimensional chaotic system and show an illustrative example of identifying such a dominant trajectory that corresponds to an eigenstate.
Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories -- for both the Berry-Tabor formula and Gutzwiller's trace formula -- and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for v...
We demonstrate that the Fourier transform of a function leads to interference effects in phase space which allow the reconstruction of the essential properties of the original function, even when we restrict the inverse Fourier transform to the phase factor rather than the complete Fourier transform. We analyze this phase-only reconstruction, known from signal processing, in quantum mechanics for the elementary example of a superposition of two real-valued Gaussians. We employ the Wigner function as well as techniques of semiclassical quantum mechanics to show that their key features, such as their separation and their width, are indeed encoded in the phase of the Fourier transform. For this example, the amplitude-only reconstruction performing the inverse Fourier transform of the absolute value of the Fourier transform fails.
A unified treatment of quantum mechanical oscillatory motion in one dimension is presented in a phase space formalism which is especially adapted to the semiclassical energy domain and closely related to a `naive' quantization of action and angle variables. Characteristics of this scheme are non-classical symmetry properties of the phase space functions representing density operators and observables and the inclusion of half-Bohr orbits besides the familiar Bohr orbits. Over long time intervals the quantum evolution can be well approximated by a Hamiltonian flow along these distinguished classical orbits. The interplay of this reduced classical evolution and the symmetry properties of the phase space functions results in a consistent quantitative description of quantum interference effects which are most clearly seen in the revivals of wavepackets.
Correct use of the Wigner representation of quantum mechanics, which is realized with joint distributions of quasiprobabilities in phase space, requires the use of certain specific rules and attention to a number of properties which distinguish the quasiprobability from a true probability. This paper is mainly concerned with these problems. In the Wigner representation the quantum Liouville equation appears instead of the Schroedinger equation. The solution may have no physical meaning unless it is subjected to a necessary and sufficient condition which selects an allowed class of distributions which describe quantum-mechanical pure states. This condition contains Planck's constant and imposes, besides the uncertainty relations, severe restrictions on the possible form of the Wigner function. When this condition is satisfied, one can reconstruct the wave function from the Wigner function. In the case of an oscillator, the quantization condition for the energies of the stationary states does not follow from the Liouville equation, but from this supplementary condition. Unlike the true probability density, any Wigner function (except Gaussian ones) of a pure state takes on negative values. Another important peculiarity is that the quasiprobability is not concentrated on certain hypersurfaces, but is smeared out over the entire phase space. These and other features considered in this paper should be kept in mind when using the Wigner representation in quantum-mechanical problems.
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.