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 Letters

Globally Uniform Semiclassical Expressions for Time-Independent Wave Functions

Dafna Zor, Kenneth G. Kay

Published Mar 18, 1996• 42 citations
Publisher

Abstract

A semiclassical approximation is presented which describes the time-independent wave function as an integral over the Lagrangian manifold associated with the state. The function produced is free of caustic singularities and satisfies the Schr\"odinger equation with an error that vanishes everywhere uniformly in the classical limit. Illustrative calculations are presented for one-dimensional systems.

DOI
10.1103/physrevlett.76.1990

Related papers

Integral expressions for the semiclassical time-dependent propagator

Kenneth G. Kay

Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.

The Journal of Chemical Physics•Mar 15, 1994•408 citations

Global, Uniform, Semiclassical Approximation to Wave Equations

John R. Klauder

The usual configuration-space semiclassical approximation to wave equations (e.g., Schr\"odinger's equation) needs refinement at caustics. When expressed in a phase-space coherent-state representation, such equations admit a semiclassical solution that properly captures the effects of caustics. Transformation of this expression to the configuration-space representation yields a global, uniform, semiclassical approximation.

Physical Review Letters•Mar 3, 1986•50 citations

Approximate solutions of the Schrödinger equation in the semiclassical limit via generalized Fourier integrals

Ira B. Bernstein

It is shown in the limit \ensuremath{\Elzxh}\ensuremath{\rightarrow}0 that an approximate solution of the time-dependent Schr\"odinger equation can be obtained by the use of generalized Fourier integrals. These are constructed by the continuous superposition of eikonals, each generated by solving a problem in classical mechanics. The solution thus generated is valid even in the presence of caustics, provided that they are not too dense. The method applies to localized wave functions and provides a short-time solution. The nature of the solution and its relation to an exact solution is demonstrated in the case of the one-dimensional problem of the surface-state electron. The Wigner distribution f arises in a natural way, leading to the interpretation of the semiclassical limit as classical kinetic theory.

Physical Review A•Jul 1, 1985•10 citations

Global, uniform semiclassical approximations for quantum systems on the half-line

John R. Klauder

Motivated by recent experiments on highly excited Rydberg atoms in electrically polarized states, we consider the semiclassical analysis of quantum systems defined on the interval $0<x<\ensuremath{\infty}$. Caustics plague the conventional approach, and Maslov's technique cannot be used. Instead we introduce an affine phase-space formulation that leads to a global, uniform semiclassical approximation to the wave function which fully captures the effects of caustics.

Physical Review A•Nov 1, 1986•6 citations

Globally uniform semiclassical wave functions for multidimensional systems

M. Madhusoodanan, Kenneth G. Kay

The globally uniform semiclassical approximation for energy eigenstates developed by D. Zor and K. G. Kay [Phys. Rev. Lett. 76, 1990 (1996)] is derived explicitly for the case of multidimensional systems and is applied to two test cases. The adiabatic switching approximation is used to obtain various quantities that enter the semiclassical expression. Two formulations of the method are examined: one requiring several trajectories for each desired state and another requiring only a single trajectory per state. The multitrajectory version yields accurate results for all states investigated (overlap between semiclassical and quantum eigenstates >0.98), including some influenced by classical chaos. The single-trajectory treatment, however, is more efficient, gives accurate results for regular states, and is even applicable for certain chaotic states, although the multiple-trajectory method is preferred in such cases. Despite the substantial resemblance of the present theory to the frozen Gaussian approximation (FGA), it is a true semiclassical approximation and is found to produce wave functions that are significantly more accurate than those obtained from the FGA for all states examined.

The Journal of Chemical Physics•Aug 15, 1998•39 citations

Semiclassical propagation: Phase indices and the initial-value formalism

Giuseppe Campolieti, Paul W. Brumer

Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.

Physical Review A•Aug 1, 1994•132 citations
A semiclassical approximation is presented which describes the time-independent wave function as an integral over the Lagrangian manifold associated with the state. The function produced is free of caustic singularities and satisfies the Schr"odinger equation with an error that vanishes everywhere uniformly in the classical limit. Illustrative calculations are presented for one-dimensional systems.
Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.
It is shown in the limit \ensuremath{\Elzxh}\ensuremath{\rightarrow}0 that an approximate solution of the time-dependent Schr\"odinger equation can be obtained by the use of generalized Fourier integrals. These are constructed by the continuous superposition of eikonals, each generated by solving a problem in classical mechanics. The solution thus generated is valid even in the presence of caustics, provided that they are not too dense. The method applies to localized wave functions and provides a short-time solution. The nature of the solution and its relation to an exact solution is demonstrated in the case of the one-dimensional problem of the surface-state electron. The Wigner distribution f arises in a natural way, leading to the interpretation of the semiclassical limit as classical kinetic theory.
The globally uniform semiclassical approximation for energy eigenstates developed by D. Zor and K. G. Kay [Phys. Rev. Lett. 76, 1990 (1996)] is derived explicitly for the case of multidimensional systems and is applied to two test cases. The adiabatic switching approximation is used to obtain various quantities that enter the semiclassical expression. Two formulations of the method are examined: one requiring several trajectories for each desired state and another requiring only a single trajectory per state. The multitrajectory version yields accurate results for all states investigated (overlap between semiclassical and quantum eigenstates >0.98), including some influenced by classical chaos. The single-trajectory treatment, however, is more efficient, gives accurate results for regular states, and is even applicable for certain chaotic states, although the multiple-trajectory method is preferred in such cases. Despite the substantial resemblance of the present theory to the frozen Gaussian approximation (FGA), it is a true semiclassical approximation and is found to produce wave functions that are significantly more accurate than those obtained from the FGA for all states examined.
Initial-value phase-space integral representations for a time-dependent propagator are obtained in the coordinate and momentum representations. To do so we first derive nonuniform semiclassical propagators for the various representations, obtaining the global-time semiclassical phase indices (Maslov indices) due to caustics. Results include readily implementable general phase index formulas for any type of caustic, including cases where the Morse index theorem is inapplicable. The method of obtaining the indices is general and based simply on concatenating uniform short-time propagators which also gives rise to alternative path-integral forms. Initial-value integral representations are then derived by introducing a method of extending short-time initial-value propagator formulas to global times via a simple stationary-phase asympotoic-equivalence approach. The integrals reduce to the nonuniform semiclassical propagators within the stationary-phase approximation, are uniform about caustics, and have integrand phases which properly account for the global-time phases in terms of appropriate Maslov indices. The initial-value integrals are also consistently derived via a canonical mapping procedure on the coordinate-space path integral. Initial-value integrals for time-dependent wave-function propgation are also given. Evaluation of the initial-value integral expressions do not require trajectory root searches for propagation.