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arXiv:quant-ph/0105153 · Journal of Physics A Mathematical and General

Semiclassical approximations in phase space with coherent states

Michel Baranger, Marcus A. M. de Aguiar, F Keck, H. J. Korsch, B Schellhaaß

Published Sep 3, 2001• 176 citations
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Abstract

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.

DOI
10.1088/0305-4470/34/36/309

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arXiv:quant-ph/0607136•Journal of Physics A Mathematical and General•Oct 10, 2006•16 citations
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.
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.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in \cite{Bar01} for the two corresponding path integral forms suggested by Klauder and Skagerstan in \cite{Klau85}. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.
The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.
The overcompleteness of the coherent states basis leads to a multiplicity of representations of Feynman's path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Two such semiclassical formulas were derived in [1] for the two corresponding path integral forms suggested by Klauder and Skagerstan in [2]. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in other. In this paper we construct a third representation of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself.
We derive a semiclassical approximation for the coherent state propagator $〈{z}^{\ensuremath{''}}|{e}^{\ensuremath{-}iHt/\ensuremath{\Elzxh}}|{z}^{\ensuremath{'}}〉$ using a path integral formulation in which the intermediate coherent states can have arbitrary widths. Our semiclassical formula involves complex trajectories of the smoothed Hamiltonian $\mathcal{H}(q,p,b)=〈z|\ifmmode \hat{H}\else \^{H}\fi{}|z〉$ where b, the width of the coherent state ∣z〉,|z〉,∣z〉, is a free function that can be chosen conveniently. The generality of this formalism enables us to derive a semiclassical approximation which contains, as particular cases, other similar approximations known in the literature, providing a natural link between them. We present numerical results showing that the semiclassical propagation can be very sensitive to the choice of b and we suggest an energy dependent value b=bE{b=b}_{E}b=bE​ that results in considerable improvement over other choices. This value for the width will be generally different from the widths ${\ensuremath{\sigma}}^{\ensuremath{'}}$ or ${\ensuremath{\sigma}}^{\ensuremath{''}}$ of the initial and final states $|{z}^{\ensuremath{'}}〉$ and $|{z}^{\ensuremath{''}}〉.$
We construct a representation of the coherent state path integral using the Weyl symbol of the Hamiltonian operator. This representation is very different from the usual path integral forms suggested by Klauder and Skagerstam (1985 Coherent States: Applications in Physics and Mathematical Physics (Singapore: World Scientific)), which involve the normal or the antinormal ordering of the Hamiltonian. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. We show that the semiclassical limit of the coherent state propagator in the Weyl representation involves classical trajectories that are independent of the width of coherent states. This propagator is also free from the phase corrections found in Baranger et al ( 2001 J. Phys. A: Math. Gen. 34 7227 ) for the two Klauder forms and provides an explicit connection between the Wigner and the Husimi representations of the evolution operator.