On the evaluation of the quantum corrections to periodic TDHF orbits

James J. Griffin, Peter C. Lichtner, Maria Dworzecka +1 more

It is suggested that the TDHF method be viewed, not as an approximation to but as a model of the exact Schroedinger system; that is, as a gedanken many-body experiment whose analysis with digital computers provides data worthy in itself of theoretical study. From such a viewpoint attention is focused on the structural analogies of the TDHF system with the exact theory rather than upon its quantitative equivalence, and the TDHF many-body system is studied as a challenge of its own which, although much simpler than the realistic problem, may still offer complexity enough to educate theorists in the present state of knowledge. In this spirit, the TDHF description of continuum reactions can be restructured from an initial-value problem into a form analogous to the S-matrix version of the Schroedinger theory. The resulting TD-S-HF theory involves only self-consistent single determinantal solutions of the TDHF equations and invokes time averaging to obtain a consistent interpretation of the TDHF analogs of quantities which are constant in the exact theory, such as the S-matrix and the asymptotic reaction channel characteristics. Periodic solutions then play the role of stationary eigenstates in the construction of suitable asymptotic reaction channels. If these periodic channel states occur only at discrete energies, then the resulting channels are mutually orthogonal (on the time average) and the theory exhibits a structure fully analogous to the exact theory. In certain special cases where the periodic solutions are known to occur as an energy continuum, the requirement that the periodicity of the channel solutions be gauge invariant provides a natural requantization condition which (suggestively) turns out to be identical with the Bohr-Sommerfeld quantization rule. 11 references.

Felix M. H. Villars

This report contains two main parts.The first part summarizes the well known construction based on TDHF of a collective Hamiltonian .!J((p, q) from an A-particle Hamiltonian fl.This construction is stated as a prescription, that may also be applied to any one particle operator 'Y, and its time derivative [iH, 'Y].The operators Q} (p, q) and ciJ (p, q) so obtained are required to obey ci}(p, q)=[i.!J((p, q), Q}(p, q)].It is shown that this consistency condition on collective dynamics can indeed be enforced, and leads to the same conditions as the TDHF-variational principle.The second part presents an alternative to TDHF, which is quantum mechanical.from the outset.Eigenfunctions of the collective "position" operator Q are constructed by projection.In the adiabatic limit, the results obtained by variationally optimizing the trial function for the system are in substantial agreement with quantized TDHF.Zero-point energy corrections are automatically included.

BERTRAND G. GIRAUD, Jean LeTourneux

Jean-Louis Basdevant, Jean Dalibard

Wang De-Hua

Using the periodic orbit theory,the quantum spectra in the two dimensional rectangular billiard are calculated and analyzed.The results show that each peak in the Fourier transformed spectrum of this system corresponds to the contribution of one classical periodic orbit;and with the increase of the length of the orbit,the number of the classical periodic orbit increases,and the peaks in the corresponding quantum spectra have also increased.This study suggests the good correspondence between the quantum spectra and the classical orbits in the rectangular billiards.Therefore,the correctness of the periodic orbit theory has been verified,which further deepen people's awareness of the relations between the classical and quantum mechanics.