An adiabatic time-dependent Hartree-Fock theory of collective motion in finite systems

We show how to derive the parameters of a phenomenological collective model from a microscopic theory. The microscopic theory is Hartree-Fock, and we start from the time-dependent Hartree-Fock equation. To this we add the adiabatic approximation, which results in a collective kinetic energy quadratic in the velocities, with coefficients depending on the coordinates, as in the phenomenological models. The crucial step is the decom-position of the single-particle density matrix ϱ in the form exp(iχ)ϱ0 exp(−iχ), where ϱO represents a time-even Slater determinant and plays the role of coordinate. Then χ plays the role of momentum, and the adiabatic assumption is that χ is small. The energy is expanded in powers of χ, the zeroth-order being the collective potential energy. The analogy with classical mechanics is stressed and studied. The same adiabatic equations of motion are derived in three different ways (directly, from the Lagrangian, from the Hamiltonian), thus proving the consistency of the theory. One equation of motion (Eq. I) expresses the relation between velocity and momentum; the other (Eq. II) is the dynamical equation. Equation II is not necessary for writing the energy or for the subsequent quantization which leads to a Schrodinger equation, but it must be used if one wants to check the validity of various approximation schemes, particularly when one attempts to reduce the problem to a few degrees of freedom. The role of the adiabatic hypothesis, its definition, and range of validity, are analyzed in great detail. It assumes slow motion, but not small amplitude, and is therefore suitable for large-amplitude collective motion. The RPA is obtained as the limiting case where the amplitude is also small. The translational mass is correctly given, and the moment of inertia under rotation is that of Thouless and Valatin. For a quadrupole two-body force, the Baranger-Kumar formalism is recovered. The self-consistency brings additional terms to the Inglis cranking formula. Comparison is also made with generator coordinate methods.

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