Marcel Vénéroni

Derivation of an adiabatic time-dependent Hartree-Fock formalism from a variational principle

Abstract A derivation of the adiabatic time-dependent Hartree-Fock formalism is given, which is based on a variational principle analogous to Hamiltons principle in classical mechanics. The method leads to a Hamiltonian for collective motion which separates into a potential and a kinetic energy and gives mass and potential parameters in terms of the nucleon-nucleon interaction. The adiabatic approximation assumes slow motion but not small amplitudes and can therefore describe anharmonic effects. The RPA is a limiting case where both amplitudes and velocities are small. The variational approach provides a consistent way of extracting coordinates and momenta from the density matrix and of obtaining equations of motion when particular trial forms for this density matrix are chosen. One such choice leads to Thouless-Valatin formula. An other choice leads to irrotational hydrodynamics.

Extension through time-smoothing of the time-dependent mean field theory

Abstract The mean field theory is considered from the viewpoint of the “contraction of the description” for complicated systems of fermions. Keeping only some small number of variables, considered as “simple” or collective, may or may not produce irreversibility. The single-particle density ϱ still contains non-collective aspects, presumably associated with high frequencies. The mechanism which generates these high frequencies from an initial slow motion is illustrated by considering the Landau-Zener-Stueckelberg model. Since it does not appear consistent to keep such rapid oscillations, time-smoothing is proposed as a possible procedure for erasing them. An evolution equation for the density ϱ is thus obtained by “naively” time-averaging the TDHF equation. The resulting evolution, however, is time-reversal invariant, and it may be unstable with respect to the initial conditions. To cure these defects, a different, iterative, type of time-smoothing is introduced, which prevents from the outset the development of high frequencies. This adds to the TDHF equation a term containing the RPA kernel and the smoothing parameter. Irreversibility occurs in the sense of an increase of entropy. Depending on the nature of the two-body interaction and on the initial conditions, the “mechanical energy” associated with the slow motion may either decrease with time, dissipating into fast motions, or increase. The occupation numbers evolve according to a balance equation similar to that of Pauli and they show a tendency towards equalization. The iterative time-smoothing, which involves a loss-of-memory mechanism, may also be viewed as an approximate way to account for the destructive coupling between the correlations and the rapid single-particle motions. Indeed it is recovered through the simulation of the residual interaction by a random noise.

Self-consistent calculation of the fission barrier of 240Pu

Abstract Completely self-consistent calculations using the Skyrme force have been carried out for the fission energy curve of 240 Pu. We use a deformed oscillator basis including 13 major shells and convergence has been checked by extending the size of the basis to 15 shells. We obtain a double humped barrier with energies E A = 9 MeV for the first barrier, E B = 13 MeV for the second barrier and E II = 4 MeV for the isomeric state. Corrections to our calculation, such as inclusion of non-axial and symmetric shapes and zero-point rotational motion, are likely to improve quantitative agreement with experimental data.

Time-Dependent Variational Principle for the Expectation Value of an Observable: Mean-Field Applications☆

In non equilibrium statistical mechanics one is often interested in making some prediction at a time t1 from the knowledge of some properties at an earlier time t0. In quantum statistical mechanics all the knowledge about the state (whether pure or not) of the system of interest is included, at a given time, in the density operator. Let us assume that this system has been prepared in such a way that the density operator D(t0), characterizing the initial state, is known. At some later time t1 one intends to perform a measurement of some observable A. The theoretical problem is the prediction of the average value of A at the time t1. In principle the calculation of this expectation value

A Hartree-Fock calculation for the stability of super-heavy nuclei

A({{t}_{1}};D,{{t}_{0}})\equiv TrAD({{t}_{1}},{{t}_{0}})

COULOMB ENERGY DIFFERENCES IN MIRROR NUCLEI IN THE HARTREE--FOCK APPROXIMATION.

(1.1) requires only the knowledge of the Hamiltonian H of the system, since the exact evolution of the density operator D(t, t0) is given by the Liouville-von Neumann equation (2.4).

A method for calculating adiabatic mass parameters: Application to isoscalar quadrupole modes in light nuclei

Abstract A general procedure is first reviewed for constructing variational principles (v.p.) suited to the optimization of a quantity of interest. A geometric interpretation of the method is given. Conditions under which the solution may be obtained as a maximum rather than as a saddle-point are examined. Applications are then worked out, providing v.p. adapted to the evaluation of expectation values, fluctuations, or correlations of quantum observables, whether the system is at thermal equilibrium or whether it has evolved. In particular, a v.p. suited to the optimization of Tr Ae−sH is built, generalizing the usual free energy v.p. to observables A ≠ 1. To evaluate both the expectation value of an observable Q and its fluctuations (as well as correlations between Q1 and Q2), the characteristic function is determined variationally by letting A = exp[−ξQ], and then expanded in powers of ξ. Within a given class of trial states, the best approximate state depends in general on the question raised. Another class of v.p. concerns dynamical problems. The general method allows one to recover time-dependent v.p. for a state and an observable which was previously proposed and which deals with the following question: Given the state of the system at the time t0, what is the expectation value 〈A〉 at a later time t1? A more general v.p. applies to situations in which the initial state is too complicated to be handled exactly. Both the approximate initial conditions and the approximate evolution are then determined so as to optimize 〈A〉. Finally, analogous v.p. are constructed in classical statistical mechanics and Hamiltonian dynamics. The recent formulation of classical mechanics in terms of covariant Poisson brackets due to Marsden et. al. comes out naturally in this context from the v.p. for the evaluation of a classical expectation value.

Variational derivation of nuclear hydrodynamics

Spherical Hartree-Fock calculations have been carried out in the super-heavy region with the Skyrme interaction. Shell effects are discussed and stability against alpha and beta-decay are investigated. A strong shell effect is found to occur at N = 228.