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

Semiclassical approximations in the coherent-state representation

Jorge Kurchan B, P. Lebœuf, Marcos Saraceno

Published Dec 1, 1989• 94 citations
Publisher

Abstract

We analyze the semiclassical limit of the stationary Schr\"odinger equation in the coherent-state representation simultaneously for the groups ${W}_{1}$, SU(2), and SU(1,1). A simple expression for the first two orders for the wave function and the associated semiclassical quantization rule is obtained if a definite choice for the classical Hamiltonian and expansion parameter is made. The behavior of the modulus of the wave function, which is a distribution function in a curved phase space, is studied for the three groups. The results are applied to the quantum triaxial rotor.

DOI
10.1103/physreva.40.6800

Related papers

Large- N phase-integral approximation for SU(1,1) coherent states

Christopher C. Gerry, James B. Togeas, S. J. Silverman

From our recently developed representation of the path integral in terms of coherent states of the dynamical group SU(1,1), we obtain the semiclassical limit as a large-$N$ limit. Specifically, the parameter $k=\frac{1}{2}(l+\frac{N}{2})$ which labels the relevant SU(1,1) representations plays the role of ${\ensuremath{\hbar}}^{\ensuremath{-}1}$. A phase-integral quantization rule is given and used to calculate the energy eigenvalues for various isotropic anharmonic oscillators. We obtain results comparable to the first-order JWKB calculations even at $N=1$.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Oct 15, 1983•18 citations

Semiclassical approximations in phase space with coherent states

Michel Baranger, Marcus A. M. de Aguiar, F Keck +2 more

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

arXiv:quant-ph/0105153•Journal of Physics A Mathematical and General•Sep 3, 2001•176 citations

A semiclassical treatment of path integrals for the spin system

Hiroshi Kuratsuji, Yutaka Mizobuchi

Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.

Journal of Mathematical Physics•Apr 1, 1981•40 citations

Semiclassical description of quantum rotator in terms ofSU(2) coherent states

Dmitri M Gitman, D. A. Petrusevich, A. L. Shelepin

We introduce coordinates of the rigid body (rotator) using mutual positions between body-fixed and space-fixed reference frames. Wave functions that depend on such coordinates can be treated as scalar functions of the group SU(2). Irreducible representations of the group SU(2) × SU(2) in the space of such functions describe their possible transformations under independent rotations of the both reference frames. We construct sets of the corresponding group SU(2) × SU(2) Perelomov coherent states (CS) with a fixed angular momentum j of the rotator as special orbits of the latter group. Minimization of different uncertainty relations is discussed. The classical limit corresponds to the limit j → ∞. Considering Hamiltonians of rotators with different characteristics, we study the time evolution of the constructed CS. In some cases, the CS time evolution is completely or partially reduced to their parameter time evolution. If these parameters are chosen as Euler angles, then they obey the Euler equations in the classical limit. Quantum corrections to the motion of the quantum rotator can be found from exact equations on the CS parameters.

Physica Scripta•Sep 17, 2013•1 citation

Generalized Gaussian wave packet dynamics, Schrödinger equation, and stationary phase approximation

Daniel Huber, Eric J. Heller, Robert G. Littlejohn

We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.

The Journal of Chemical Physics•Aug 15, 1988•118 citations

Generalized coherent states for a three-dimensional relativistic model of the oscillator

Natig M. Atakishiyev, R. M. Mir‐Kasimov

The SU(1,1) coherent states for a relativistic model of the oscillator in the configurational r representation are considered. Classical equations of motion in the generalized phase space are obtained with the help of the path integral for the transition amplitude between SU(1,1) coherent states. It is shown that the use of the semiclassical Bohr–Sommerfeld quantization rule yields the exact expression for the energy levels.

Journal of Mathematical Physics•Jul 1, 1989•2 citations
We analyze the semiclassical limit of the stationary Schr"odinger equation in the coherent-state representation simultaneously for the groups W1{W}_{1}W1​, SU(2), and SU(1,1). A simple expression for the first two orders for the wave function and the associated semiclassical quantization rule is obtained if a definite choice for the classical Hamiltonian and expansion parameter is made. The behavior of the modulus of the wave function, which is a distribution function in a curved phase space, is studied for the three groups. The results are applied to the quantum triaxial rotor.
From our recently developed representation of the path integral in terms of coherent states of the dynamical group SU(1,1), we obtain the semiclassical limit as a large-NNN limit. Specifically, the parameter k=12(l+N2)k=\frac{1}{2}(l+\frac{N}{2})k=21​(l+2N​) which labels the relevant SU(1,1) representations plays the role of ${\ensuremath{\hbar}}^{\ensuremath{-}1}$. A phase-integral quantization rule is given and used to calculate the energy eigenvalues for various isotropic anharmonic oscillators. We obtain results comparable to the first-order JWKB calculations even at N=1N=1N=1.
We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.
Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.
We introduce coordinates of the rigid body (rotator) using mutual positions between body-fixed and space-fixed reference frames. Wave functions that depend on such coordinates can be treated as scalar functions of the group SU(2). Irreducible representations of the group SU(2) × SU(2) in the space of such functions describe their possible transformations under independent rotations of the both reference frames. We construct sets of the corresponding group SU(2) × SU(2) Perelomov coherent states (CS) with a fixed angular momentum j of the rotator as special orbits of the latter group. Minimization of different uncertainty relations is discussed. The classical limit corresponds to the limit j → ∞. Considering Hamiltonians of rotators with different characteristics, we study the time evolution of the constructed CS. In some cases, the CS time evolution is completely or partially reduced to their parameter time evolution. If these parameters are chosen as Euler angles, then they obey the Euler equations in the classical limit. Quantum corrections to the motion of the quantum rotator can be found from exact equations on the CS parameters.
We have recently published a new semiclassical method, generalized Gaussian wave packet dynamics, which extends Gaussian wave packet dynamics into complex phase space. Although we were able to give an accurate formulation of the method, we had at the time of writing that paper only an intuitive, heuristic understanding of the deeper causes which make the method work. A more mathematical understanding was needed. To close this gap we show in this paper the equivalence of the new method with a first order expansion of ℏ of the Schrödinger equation. We further prove that the new method is equivalent to the stationary phase approximation, using the usual WKB formula for the propagator. The latter equivalence enables us to show that all the symmetry properties of time-dependent quantum mechanics also hold in the new semiclassical theory. Finally, we provide some elaboration of the method, and clarify several issues that were not discussed before. With this new insight we are able to formulate a simple rule for the calculation of semiclassical wave functions that contain contributions from more than one branch. This corrects for the divergence of semiclassical wave functions near caustics, a problem that we encountered in the preceding paper.