Energy Spectrum of the Excitations in Liquid Helium
A wave function previously used to represent an excitation (phonon or roton) in liquid helium, inserted into a variational principle for the energy, gave an energy-momentum curve having the qualitative shape suggested by Landau; but the value computed for the minimum energy $\ensuremath{\Delta}$ of a roton was 19.1\ifmmode^\circ\else\textdegree\fi{}K, while thermodynamic data require $\ensuremath{\Delta}=9.6\ifmmode^\circ\else\textdegree\fi{}$K. A new wave function is proposed here. The new value computed for $\ensuremath{\Delta}$ is 11.5\ifmmode^\circ\else\textdegree\fi{}K. Qualitatively, the wave function suggests that the roton is a kind of quantum-mechanical analog of a microscopic vortex ring, of diameter about equal to the atomic spacing. A forward motion of single atoms through the center of the ring is accompanied by a dipole distribution of returning flow far from the ring.In the computation both the two-atom and three-atom correlation functions appear. The former is known from x-rays, while for the latter the Kirkwood approximation of a product of three two-atom correlation functions is used. A method is developed to estimate and correct for most of the error caused by this approximation, so that the residual uncertainty due to this source is negligible.