Elementary Excitations in Liquid Helium

An analysis of some recent experimental and theoretical investigations of the elementary excitation spectrum of liquid He II is carried out. The spectrum of density fluctuations, which is measured in an inelastic neutron scattering experiment, is shown to consist of two distinct parts: direct excitation of single quasi-particles from the condensed zero-momentum state; and excitation of more complex configurations arising from the interaction of two, three, or more quasi-particles. With the aid of general sum-rule arguments it is demonstrated that: (1) in the long-wavelength limit, a single quasi-particle excitation exhausts the $f$-sum rule; the resulting excitation spectrum is identical to the phonon spectrum proposed by Feynman and found experimentally by Henshaw and Woods; (2) this asymptotic behavior of the density fluctuation spectrum may be used to normalize the experimental results of Henshaw and Woods; one thereby obtains a somewhat altered liquid structure-factor curve, detailed information on the efficiency of quasi-particle excitation from the condensed state of an incident slow neutron, and an estimate of the depletion of the zero-momentum state as a consequence of particle interaction; (3) the backflow introduced by Feynman and Cohen corresponds to taking into account the coupling between a Feynman excitation and higher configurations involving several elementary excitations.