Geometrical derivation of collective operators and paths in TDHF

Abstract

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.