Harmonic Potential Theorem: Extension to Spin-, Velocity-, and Density-Dependent Interactions

One of the few exact results for the description of the time evolution of an inhomogeneous, interacting many-particle system is given by the harmonic potential theorem (HPT). The relevance of this theorem is that it sets a tight constraint on time-dependent many-body approximations. In this contribution, we show that the original formulation of the HPT is valid also for the case of spin-, velocity-, and density-dependent interactions. This result is completely general and relevant, among the rest, for nuclear structure theory both in the case of ab initio and of more phenomenological approaches. As an example, we report on a numerical implementation by testing the small-amplitude limit of the time-dependent Hartree-Fock—also known as the random phase approximation—for the translational frequencies of a neutron system trapped in a harmonic potential.

Figure 1

Transition densities obtained from the RPA calculation (full line) and from the HPT in the form of Eq. (26) (dashed line), in the case of different neutron drops trapped in a harmonic potential with $\hslash \omega =10\mathrm{MeV}$.