A time-dependent variational procedure is proposed that possesses the same constants of the motion as the exact many-body Schr\"odinger dynamics. The class of trial wave functions is larger than the manifold of Slater determinants that supports the time-dependent Hartree-Fock dynamics. These wave functions can be regarded as superpositions of the eigenfunctions of the conserved observable of interest and the variational equations display the usual parametric structure, with properly admixed energy gradients and the symplectic metric tensor. In an application to the Lipkin-Meshkov-Glick model, significant improvements over the usual mean-field or determinantal dynamics can be achieved.NUCLEAR STRUCTURE Mean field symmetry breaking; symmetry restoration; nondeterminantal wave function; time-dependent variational principle; parametric equations of motion; mean energy metric tensor; canonical coordinates; quasispin models; comparison with time-dependent Hartree-Fock dynamics.
The time-dependent Hartree-Fock solutions of the two-level Lipkin-Meshkov-Glick model are studied by transforming the time-dependent Hartree-Fock equations into Hamilton's canonical form and analyzing the qualitative structure of the Hartree-Fock energy surface in the phase space. It is shown that as the interaction strength increases these time-dependent Hartree-Fock solutions undergo a qualitative change associated with the ground state phase transition previously studied in terms of coherent states. For twobody interactions stronger than the critical value, two types of time-dependent Hartree-Fock solutions (the "librations" and "rotations" in Hamilton's mechanics) exist simultaneously, while for weaker interactions only the rotations persist. It is also shown that the coherent states with the maximum total pseudospin value are determinants, so that time-dependent Hartree-Fock analysis is equivalent to the coherent state method.NUCLEAR STRUCTURE Lipkin-Meshkov-Glick model, TDHF, transformation of TDHF equations to Hamilton's canonical form, coherent states, ground state phase transition.
A generalization of many-fermion Feynman-Goldstone diagrams is derived based on a time-dependent unperturbed Hamiltonian along with a time-dependent reference state. In lowest order, the time-dependent Hartree-Fock equations follow immediately from the same arguments which lead to static Hartree-Fock when the basis is time independent. Systematic corrections to the time-dependent Hartree-Fock mean-field approximation are obtained through higher order diagrams.[NUCLEAR STRUCTURE Generalization of Feynman-Goldstone diagrams to a time-dependent reference state. Diagrammatic derivation of the time-dependent Hartree-Fock equations and their systematic corrections.]
This article examines the time-dependent Hartree-Fock (TDHF) approximation of single-particle dynamics in systems of interacting fermions. We find the TDHF approximation to be accurate when there are sufficiently many particles and the initial many-particle state is a Slater determinant, or any Gibbs equilibrium state for noninteracting fermions. Assuming a bounded two-particle interaction, we obtain a bound on the error of the TDHF approximation, valid for short times. We further show that the error of the the TDHF approximation vanishes at all times in the mean field limit.
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.