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

Journal of Mathematical Physics

Semiclassical quantization of nonseparable systems

Abraham Klein, Ching‐Teh Li

Published Apr 1, 1979• 18 citations
Publisher

Abstract

The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.

DOI
10.1063/1.524125

Related papers

From Heisenberg matrix mechanics to semiclassical quantization: Theory and first applications

William R. Greenberg, Abraham Klein, I.S. Zlatev +1 more

Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical Einstein-Brillouin-Keller (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. In the special case of a one-dimensional system a different and succinct algebraic derivation of the WKB quantization rule for bound states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several modified applications of EBK quantization. \textcopyright{} 1996 The American Physical Society.

arXiv:chem-ph/9603006•Physical Review A•Sep 1, 1996•12 citations

Invariant tori and Heisenberg matrix mechanics: a new window on the quantum-classical correspondence

William R. Greenberg, Abraham Klein, Ching-The Li

After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.

Physics Reports•Jan 1, 1996•8 citations

Invariant Tori and Heisenberg Matrix Mechanics: A New Window on the Quantum- Classical Correspondence

William R. Greenberg, Abraham Klein, Ching‐Teh Li

Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics, we show that there are fundamental, previously undisclosed aspects of this quantum-classical correspondence. These include a quantum variational principle that implies the classical variational principle for invariant tori and the connection between commutation relations and quantization of action variables. Possible applications are described briefly.

Physical Review Letters•Aug 14, 1995•27 citations

Semiclassical quantization via adiabatic invariance of classical action variables

Tasko P. Grozdanov, Subhash Saini, Howard S. Taylor

The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.

Physical Review A•Jan 1, 1986•39 citations

Quantization of integrable systems

M. Brack, Rajat K. Bhaduri

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.

CRC Press eBooks•Mar 5, 2018•0 citations

Quantum integrability and nonintegrability of Hamiltonian systems with two degrees of freedom

Vyacheslav V. Stepanov

The subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.

Journal of Media Literacy Education•Jan 1, 1999•0 citations
The problem of semiclassical quantization of nonseparable systems with a finite number of degrees of freedom is studied within the framework of Heisenberg matrix mechanics, in extension of previous work on one-dimensional systems. The relationship between the quantum theory and multiply-periodic classical motions is derived anew. A suitably averaged Lagrangian provides a variational basis not only for the Fourier components of the semiclassical equations of motion, but also for the general definition of action variables. A Legendre transformation to the Hamiltonian verifies that these have been properly chosen and therefore provide a basis for the quantization of nonseparable systems. The problem of connection formulas is discussed by a method integral to the present approach. The action variables are shown to be adiabatic invariants of the classical system. An elementary application of the method is given. The methods of this paper are applicable to nondegenerate systems only.
Despite the seminal connection between classical multiply periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical Einstein-Brillouin-Keller (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of action variables. In the special case of a one-dimensional system a different and succinct algebraic derivation of the WKB quantization rule for bound states is given. With the help of several standard models with one or two degrees of freedom, we then illustrate how the methods of Heisenberg matrix mechanics described in this paper may be used to obtain quantum solutions with a modest increase in effort compared to semiclassical calculations. We also describe and apply a method for obtaining leading quantum corrections to EBK results. Finally, we suggest several modified applications of EBK quantization. \textcopyright{} 1996 The American Physical Society.
After a brief review of the extensive work done on the theory of invariant tori and their quantization, we show that nevertheless an important connection between the quantum and classical theories remains to be exploited. This is the relationship between matrix elements of operators in the energy diagonal representation and Fourier components of the corresponding classical dynamical variables that was the basis for Heisenberg's invention of quantum mechanics. We describe a number of previously unknown or little-known aspects of this relationship, with special emphasis on variational principles and the connection between commutation relations and quantization of action variables. As a single illustration of the utility of these ideas we show that it is possible to obtain approximate solutions to the quantum scheme that are more accurate than the semiclassical approximation with little additional effort compared to the latter.
The semiclassical method based on the Einstein-Brillouin-Keller quantization of invariant phase-space tori and the hypothesis of adiabatic invariance of classical action variables is applied in the calculation of highly excited energy levels of two-dimensional coupled oscillator systems. The results obtained are in good agreement with quantum-mechanical calculations even in the regions where the corresponding classical dynamics becomes apparently irregular. General limitations of the method are discussed.
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 subject of the four manuscripts which make up this dissertation is the concept of integrability in quantum mechanical systems with two degrees of freedom, including the relation between quantum integrability and classical integrability in such systems. In the first manuscript, I consider a two-spin model both classically and quantum mechanically. In the six-dimensional parameter space of this model, classical integrability is satisfied on a five-dimensional hypersurface, and level crossings occur on four-dimensional manifolds that are completely embedded in the integrability hypersurface except for some lower-dimensional sub-manifolds. Under mild assumptions, the classical integrability condition can be reconstructed from a purely quantum mechanical study of level degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix. In the second manuscript, I analyze the functional dependence [special characters omitted] of the Hamiltonian operator on the action operators for an integrable quantum system, and I compare it with the corresponding functional relationship H (p1, q1; p2, q 2) = HC(J1, J2) in the classical limit of that system. It is demonstrated that the functional dependence is not the same quantum mechanically and (semi-)classically. The former is shown to converge toward the latter in some asymptotic regime associated with the classical limit, but the convergence may be non-uniform. This analysis is carried out for an integrable one- parameter two-spin model. Additional results presented for the (integrable) circular billiard model exhibit similar quantum corrections to the predictions of semiclassical quantization. In the third and the fourth manuscripts, I investigate the relations between symmetry, integrability, and the assignment of quantum numbers to eigenstates for two-spin and spin-boson models. I calculate quantum invariants in the form of expectation values for selected operators and monitor their dependence on the Hamiltonian parameters along loops within, without, and across the integrability region in parameter space. I find clear-cut signatures of integrability and nonintegrability in the observed traces of quantum invariants evaluated in invariant Hilbert subspaces. The results support the notion that quantum integrability depends on the existence of action operators as constituent elements of the Hamiltonian.