Semiclassical approximation in the coherent states representation

Abstract

This paper presents a semiclassical theory for the computation of matrix elements of the type 〈u‖v〉 when either one or both ‖u〉 and ‖v〉 are coherent states (in different representations). Our results can be considered as an extension of Miller’s semiclassical theory [Adv. Chem. Phys. 25, 69 (1974)]. Such an extension has been presented also by Heller [J. Chem. Phys. 66, 5777 (1977)]. We were able to simplify considerably some of Heller’s results by exploiting the canonical properties of the classical coherent variables. This enabled us to relate the elements 〈u‖v〉 to certain generalized, complex generator functions in a manner that is very similar to the relations that appear in the original Miller’s theory. The advantages that are inherent in the coherent states representation are illustrated in a few elementary examples. We were able to derive an excellent approximation to the eigenstates of the harmonic oscillator which is valid even for the ground state. Furthermore, we have demonstrated that it is possible to describe semiclassically dynamical processes that are classically forbidden by real time trajectories in a certain generalized phase space.

Semiclassical theory of weak values

“…the eigenstates of the momentum operator, and coherent states) can be obtained straightforwardly with the help of the semiclassical algebra [22,23], as long as the single-term approximation (cf. (4)) holds for the corresponding semiclassical generating function Z.…”

“…I use the coherent states [25] that are characterized with the help of a complex symplectic transformation [26,23] …”

Functional integrals and inequivalent representations in Quantum Field Theory

“…[56,57], apart from the second piece 5 on the RHS of Eq.(80). Among other things, the aforementioned correction gives the correct zero point energy contribution for the Bogoliubov model, studied in Section 5.2.…”

The semiclassical coherent state propagator for systems with spin

“…For canonical coherent states, Weissman [5,6] re-derived the results of Klauder using the semiclassical theory of Miller [7]. In Weissman's work, and also in the original Klauder's papers, the fluctuations around the critical trajectory have not been accurately performed, and a 'phase' factor was missed.…”