Expectation values and fluctuations in a time-dependent mean-field approach

Abstract

Time-dependent expectation values of many-body nuclear observables and their quantum fluctuations are considered within the framework of functional integration. We propose two quantization procedures through saddle-point approximations to a functional integral representation of the real-time generating function. The first method, referred to as Derivation After Variation, leads to the well-known TDHF approximation obtained also in Alhassid, Muller, and Koonin, Phys. Rev. C 23 (1981), 487, with a different functional integration method. The resulting saddle-point approximation of the time-dependent quantum fluctuations is found to include the contribution of the collective phonons of the corresponding time-dependent R.P.A. in addition to the bare contribution of the TDHF mean-field. Direct comparison is made with the time-dependent mean-field approach of Balian and Veneroni, Phys. Lett. B 136 (1984), 301, obtained by application of the variational principle proposed in Balian and Veneroni, Phys. Rev. Lett. 47 (1981), 1353, 1765(E). The second method, referred to as Variation After Derivation, reproduces in a much simpler formulation the results of Troudet and Koonin, Phys. Rev. C 28 (1983), 1465, where the mean-field is found to be dependent upon the observable being measured. We propose this mean-field theory as a possible alternative to the previous mean-field approaches for the evaluation of many-body measurements that are known to be poorly described within the bare TDHF approximation, such as widths of mass distributions of final fragments in heavy-ion collisions. A comparative illustration of these various methods is proposed through the numerical evaluation of the time-dependent expectation values of a one-body observable for a simple shell-model hamiltonian.