© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

The Journal of Chemical Physics

Globally uniform semiclassical wave functions for multidimensional systems

M. Madhusoodanan, Kenneth G. Kay

Published Aug 15, 1998• 39 citations

Abstract

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.

DOI: 10.1063/1.476875

Related papers

Multidimensional wave functions from classical trajectories

Michael J. Davis, Eric J. Heller

A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.

The Journal of Chemical Physics•Oct 15, 1981•98 citations

Semiclassical tunneling splittings for arbitrary vibrational states in multidimensional double wells

Kenneth G. Kay

, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.

The Journal of Chemical Physics•Oct 9, 2018•2 citations

Semiclassical quantization via adiabatic switching. II. Choice of tori and initial conditions for multidimensional systems

Subhash Saini, Jakub Zakrzewski, Howard S. Taylor

The formulation of a general strategy for implementing semiclassical quantization using the adiabatic-switching method, presented in the preceding paper for two-dimensional systems, is generalized to multidimensional systems. The method is shown to produce accurate semiclassical eigenvalues for the realistic polynomial potential describing ${\mathrm{SO}}_{2}$ as well as for model potentials for other multidimensional systems. The results are compared with semiclassical results obtained with use of other methods.

Physical Review A•Oct 1, 1988•16 citations

Semiclassical treatment of quantum propagation with nonlinear classical dynamics: A third-order thawed Gaussian approximation

Lucas Kocia, Anna Klales

The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.

Physical review. E•Sep 14, 2016•3 citations

Semiclassical quantization of classically chaotic systems

John R. Klauder

It is argued that the global, uniform semiclassical approximation to quantization afforded by a coherent-state approach is well suited for an analysis of classically chaotic systems. Special attention is given to the interpolating action which incorporates in a single expression the classical action for each and every contributing classical trajectory. Approximate expressions for the configuration-space propagator are presented for small, intermediate, and large propagation times.

Physical Review Letters•Aug 17, 1987•17 citations

A new uniform semiclassical wave function for single surface and multisurface scattering

Michael F. Herman

A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.

The Journal of Chemical Physics•Sep 15, 1983•19 citations

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.

A new technique is developed to generate semiclassical wave functions. The method uses only information already available from a standard semiclassical quantization of a system. Linear superpositions of Gaussian coherent states that lie along quantizing classical trajectories are used, with phases given by the action integrals plus a Maslov-type correction. Wave functions generated in this way suffer from none of the problems with caustics that primitive semiclassical wave functions encounter. The semiclassical wave functions are convenient for subsequent use in applications, e.g., molecular spectra. By generating wave functions for several simple systems, we show that under most circumstances these wave functions are very accurate approximations to the true quantum states.

, 042116 (2017)] is applied to calculate tunneling splittings for arbitrary vibrational states of model two-dimensional double-well systems. Cases in which the classical dynamics for the wells is chaotic, mixed, and regular are considered. A perturbative treatment, based on the condition of small tunneling amplitudes, is found to be sufficiently accurate for the cases studied and is applied for most of the calculations. Treatments that approximate certain imaginary-time trajectories in the classically forbidden region by linearization about a variety of judiciously selected reference trajectories yield good results for all systems treated. These calculations can be greatly simplified by approximating all imaginary-time trajectories as linearizations about a single reference trajectory. A simple way to determine optimal reference trajectories for this purpose is presented. It is found that their use yields splittings of satisfactory accuracy for the cases studied.

The formulation of a general strategy for implementing semiclassical quantization using the adiabatic-switching method, presented in the preceding paper for two-dimensional systems, is generalized to multidimensional systems. The method is shown to produce accurate semiclassical eigenvalues for the realistic polynomial potential describing

{\mathrm{SO}}_{2}

as well as for model potentials for other multidimensional systems. The results are compared with semiclassical results obtained with use of other methods.

The time-dependent WKB approximation for a coherent state is expanded to third order around a guiding real trajectory, allowing for the novel treatment of nonlinearity in its semiclassical dynamics, which is generally present in all physical systems far from the classical regime. The result is a closed-form solution consisting of a linear combination of Airy functions and their derivatives multiplied by an exponential. The expression's ability to capture nonlinearity is demonstrated by examining the autocorrelation of initial coherent states in anharmonic systems with few to many contributing periodic orbits. Its accuracy is compared to the quadratic expansion and found to be superior in regimes of ℏ where the curvature begins to be significant, as expected. Moreover, the expression is shown to be a real-trajectory uniformization over two coalescing saddle points that are emblematic of significant curvature. This extends real-trajectory time-dependent wave-packet semiclassical methods to highly anharmonic systems for the first time and establishes their regime of validity.

A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.