A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator
U(tf,ti). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of
U is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.
Some limitations of the conventional time-dependent Hartree-Fock method for describing complex reactions are noted, and one particular ubiquitous defect is discussed in detail: the post-breakup spurious cross channel correlations which arise whenever several asymptotic reaction channels must be simultaneously described by a single determinant. A reformulated time-dependent---
S-matrix Hartree-Fock theory is proposed, which obviates this difficulty. Axiomatic requirements minimal to assure that the time-dependent---
S-matrix Hartree-Fock theory represents an unambiguous and physically interpretable asymptotic reaction theory are utilized to prescribe conditions upon the definition of acceptable asymptotic channels. That definition, in turn, defines the physical range of the time-dependent---
S-matrix Hartree-Fock theory to encompass the collisions of mathematically well-defined "time-dependent Hartree-Fock droplets." The physical properties of these objects then circumscribe the content of the Hartree-Fock single determinantal description. If their periodic vibrations occur for continuous ranges of energy then the resulting "classical" time-dependent Hartree-Fock droplets are seen to be intrinsically dissipative, and the single determinantal description of their collisions reduces to a "trajectory" theory which can describe the masses and relative motions of the fragments but can provide no information about specific asymptotic excited states beyond their constants of motion, or the average properties of the limit, if it exists, of their equilibrization process. If, on the other hand, the periodic vibrations of the time-dependent Hartree-Fock droplets are discrete in energy, then the time-dependent---
S-matrix Hartree-Fock theory can describe asymptotically the time-average properties of the whole spectrum of such periodic vibrations of these "quantized" time-dependent Hartree-Fock droplets which are asymptotically stationary on a time-averaged basis. Such quantized time-dependent Hartree-Fock droplet spectra promise the closest analog to the rich array of asymptotic channel eigenstates in the exact Schr\"odinger theory which single time-dependent Hartree-Fock self-consistent determinants might describe. Whether the time-dependent Hartree-Fock droplets are classical or quantized is determined by the mathematical properties of the periodic solutions of the time-dependent Hartree-Fock equation. If the droplets are in fact classical, then the question remains open whether an explicit requantization, by assumption, could consistently restore the close structural analogy with the exact theory. We argue that if the statistical interpretation of the single determinantal wave function, which is central to the present restructuring of the conventional time-dependent Hartree-Fock description, were to be found inadmissible, then no basis would remain for considering the determinant as a "wave function" in the Schr\"odinger sense. Finally, we note that the conceptual basis of this time-averaged
S-matrix theory need not be restricted to the time-dependent Hartree-Fock theory, but might apply as well to other nonlinear Schr\"odinger-type models which purport to provide an approximate wave function to describe physical reaction processes.NUCLEAR REACTIONS Structural analysis of scattering theory for time-dependent Hartree-Fock single determinantal wave functions; scattering matrix defined as a time average; channels labeled by time-averaged properties of periodic solutions; time-averaged channel orthogonality; classical and quantal time-dependent Hartree-Fock droplets.
A general hierarchy of approximations to the many-body Schr\"odinger equation is presented which reduces to the time-dependent mean-field approximation in lowest order and provides systematic corrections in subsequent orders. The theory is applied to two interacting systems described by the Lipkin model Hamiltonian. Comparison of the results of the lowest two orders of approximation with the exact solution demonstrates the practicality of the method and its potential for generalizing nuclear dynamics beyond the mean field theory.NUCLEAR STRUCTURE Time-dependent approximation to nuclear dynamics. Corrections to time-dependent mean-field theory. Solution of Lipkin model for two interacting systems.