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Journal of Physics A Mathematical and General

A large-N phase integral approximation of Coulomb-type systems using SO(2,1) coherent states

Christopher C. Gerry, James B. Togeas

Published Dec 21, 1986• 11 citations
Publisher

Abstract

A phase integral approximation, based on the notion that large N is a semiclassical limit, is presented for spherically symmetric systems of the Coulomb type. Since the dynamical group SO(2,1) is the spectrum generating group for the radial point Coulomb problem and the Casimir operators of O(N) and SO(2,1) are coupled, the authors write a path integral expression of the Green function in terms of SO(2,1) coherent states. The resulting phase integral approximation is applied to the Yukawa potential and the Coulomb plus linear r confining potential.

DOI
10.1088/0305-4470/19/18/025

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A phase integral approximation, based on the notion that large N is a semiclassical limit, is presented for spherically symmetric systems of the Coulomb type. Since the dynamical group SO(2,1) is the spectrum generating group for the radial point Coulomb problem and the Casimir operators of O(N) and SO(2,1) are coupled, the authors write a path integral expression of the Green function in terms of SO(2,1) coherent states. The resulting phase integral approximation is applied to the Yukawa potential and the Coulomb plus linear r confining potential.
From our recently developed representation of the path integral in terms of coherent states of the dynamical group SU(1,1), we obtain the semiclassical limit as a large-NNN limit. Specifically, the parameter k=12(l+N2)k=\frac{1}{2}(l+\frac{N}{2})k=21​(l+2N​) which labels the relevant SU(1,1) representations plays the role of ${\ensuremath{\hbar}}^{\ensuremath{-}1}$. A phase-integral quantization rule is given and used to calculate the energy eigenvalues for various isotropic anharmonic oscillators. We obtain results comparable to the first-order JWKB calculations even at N=1N=1N=1.
We derive a regularized semiclassical radial propagator for the Coulomb potential, a case for which standard approaches run into well-known difficulties associated with a non-Cartesian radial coordinate and a potential singularity. Following Kleinert [Path Integrals in Quantum Mechanics, Statistics and Polymer Physics (World Scientific, Singapore, 1990)], we first perform a quantum-mechanical regularization of the propagator. The semiclassical limit is then obtained by stationary phase approximation of the resulting integrals. The semiclassical propagator so derived has the standard Van Vleck--Gutzwiller form for the radial Coulomb problem with a potential correction (Langer modification) term included. The regularized semiclassical propagator is applied to compute the autocorrelation function for a Gaussian Rydberg wave packet.
Abstract We present a rigorous path integral treatment of a dynamical system in the axially symmetric potential $V(r,\theta ) = V(r) + \tfrac{1} {{r^2 }}V(\theta ) $ . It is shown that the Green’s function can be calculated in spherical coordinate system for $V(\theta ) = \frac{{\hbar ^2 }} {{2\mu }}\frac{{\gamma + \beta \sin ^2 \theta + \alpha \sin ^4 \theta }} {{\sin ^2 \theta \cos ^2 \theta }} $ . As an illustration, we have chosen the example of a spherical harmonic oscillator and also the Coulomb potential for the radial dependence of this noncentral potential. The ring-shaped oscillator and the Hartmann ring-shaped potential are considered as particular cases. When α = β = γ = 0, the discrete energy spectrum, the normalized wave function of the spherical oscillator and the Coulomb potential of a hydrogen-like ion, for a state of orbital quantum number l ≥ 0, are recovered.
Using the generalized coherent states it is shown that the path integral formulas for SU(2) and SU(1,1) (in the discrete series) are WKB exact, if it is started from the trace of e−iTĤ, where Ĥ is given by a linear combination of generators. In this case, the WKB approximation is achieved by taking a large ‘‘spin’’ limit: J,K→∞, under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result. It is further pointed out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression sometimes leads to a wrong result. Therefore great care must be taken when some geometrical action would be adopted, even if it is so beautiful as the starting ingredient of path integral. Discussions on generalized coherent states are also presented both from geometrical and simple oscillator (Schwinger boson) points of view.
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.