An planck-asymptotic analysis of the error in the thawed Gaussian approximation and in the corresponding initial value representation of the quantum propagator
An algebraic approach based on the multimode two-photon Lie algebra and its corresponding Lie group is followed to derive a formal solution to the time-dependent Schrödinger equation. This solution is written as an expansion series whose leading term corresponds to the thawed Gaussian approximation (TGA). Our scheme provides the most general expression reported so far for this approximation. By using the coherent state representation of the formal solution, the correction term to the TGA is analysed in the zero ℏ asymptotic limit. The error is generally found not to vanish in this semiclassical limit. The same approach is followed to study the remainder to the TGA initial value representation (IVR) of the quantum propagator. This correction is found not to vanish either in the zero ℏ limit. Hence, the TGA IVR would not be the correct semiclassical asymptotic form of the quantum propagator. The origin of this behaviour is shown to be in the existence of contributions from unphysical saddle points in the semiclassical limit. These would unveil an incorrect analytic structure of the TGA IVR propagator in that limit.
An algebraic approach based on the multimode two-photon Lie algebra and its corresponding Lie group is followed to derive a formal solution to the time-dependent Schrödinger equation. This solution is written as an expansion series whose leading term corresponds to the thawed Gaussian approximation (TGA). Our scheme provides the most general expression reported so far for this approximation. By using the coherent state representation of the formal solution, the correction term to the TGA is analysed in the zero ℏ asymptotic limit. The error is generally found not to vanish in this semiclassical limit. The same approach is followed to study the remainder to the TGA initial value representation (IVR) of the quantum propagator. This correction is found not to vanish either in the zero ℏ limit. Hence, the TGA IVR would not be the correct semiclassical asymptotic form of the quantum propagator. The origin of this behaviour is shown to be in the existence of contributions from unphysical saddle points in the semiclassical limit. These would unveil an incorrect analytic structure of the TGA IVR propagator in that limit.
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.
We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.
The Herman−Kluk (HK) or coherent state initial value representation (IVR) of semiclassical (SC) theory expresses a time-evolving wave function as a sum (integral) over the initial conditions of classically evolved coherent states (aka minimum uncertainty wave packets). An exact expansion of the wave function can also be made in terms of these same coherent states, thinking of them simply as a (time-dependent) basis set. This paper shows the precise set of approximations that are necessary to degenerate the exact expansion into the SC-IVR result. As a byproduct, we also obtain a generalization of the HK “prefactor”.
Abstract Recent experimental progress in monitoring the dynamics of ultracold gases in optical lattices necessitates a quantitative theoretical description for a significant number of bosons. In the present paper, we investigate if time-dependent semiclassical initial value methodology, with propagators expressed as integrals over phase space and using classical trajectories, is suitable to describe interacting bosons, concentrating on a single mode. Despite the nonlinear contribution from the self-interaction, the corresponding classical dynamics allows for a largely analytical treatment of the semiclassical propagator. We find that application of the Herman–Kluk (HK) propagator conserves unitarity in the semiclassical limit, but a decay of the norm is seen for low particle numbers n . The frozen Gaussian approximation (FGA) (HK with unit prefactor) is explicitly shown to violate unitarity in the present system for non-vanishing interaction strength, even in the semiclassical limit. Furthermore, we show by evaluating the phase space integral in steepest descent approximation, that the HK propagator reproduces the exact spectrum correctly in the semiclassical limit ( ). An error is, however, incurred in next-to-next-to-leading order (small parameter ), as seen upon numerical evaluation of the integral and confirmed analytically by considering finite n corrections to the steepest descent calculations. The FGA, in contrast, is only accurate to lowest order, and an erroneous next-to-leading order term in the energy spectrum was found analytically. Finally, as an example application, we study the dynamics of wave packets by computing the time evolution of the Wigner function. While the often-used truncated Wigner approximation cannot capture any interferences present in the exact quantum mechanical solution (known analytically), we find that the HK approach, despite also using classical information only, reproduces the salient features of the exact solution correctly.
In order to rigorously derive the amplitude factor of the semiclassical approximation to the quantum propagator, we extend an existing method originally devised to evaluate Gaussian path-integral expressions. Using a result which relates the determinant of symmetric block-tridiagonal matrices to the determinants of their blocks, two difference equations are obtained. The first one allows to establish the connection of the amplitude factor to Jacobi’s accessory equations in the continuous-time limit, while the second one leads to an additional factor which, however, contributes to the final result only in exceptional cases. In order to demonstrate the wide applicability of these difference equations, we treat explicitly the case where the time-sliced Lagrangian is written in generalized coordinates, for which a general derivation has so far been unavailable.
The coherent path integral is generalized such that the identity operator represented in a complete (actually overcomplete) set of the coherent states with the ``time-variable'' exponents are inserted between two consecutive short-time propagators. Since such a complete set of any given exponent can constitute the identity operator, the exponent may be varied from time to time without loss of generality as long as it is set common to all the Gaussians. However, a finite truncation of the coherent state expansion should result in different values of the propagator depending on the choice of the exponents. Furthermore, approximation methodology to treat with the exact propagator can also depend on this choice, and thereby many different semiclassical propagators may emerge from these combinations. Indeed, we show that the well-known semiclassical propagators such as those of Van Vleck, Herman-Kluk, Heller's thawed Gaussian, and many others can be derived in a systematic manner, which enables one to comprehend these semiclassical propagators from a unified point of view. We are particularly interested in our generalized form of the Herman-Kluk propagator, since the relative accuracy of this propagator has been well established by Kay, and since, nevertheless, its derivation was not necessarily clear. Thus our generalized Herman-Kluk propagator replaces the classical Hamiltonian with a Gaussian averaged quantum Hamiltonian, generating non-Newtonian trajectories. We perform a numerical test to assess the quality of such a family of generalized Herman-Kluk propagators and find that the original Herman-Kluk gives an accurate result. The reason why this has come about is also discussed.