Energy Spectrum of Elementary Excitations in Liquid Helium

Abstract

The elementary excitations in liquid ${\mathrm{He}}^4$ are studied using the Brillouin-Wigner perturbation theory. For the four-particle distribution function occurring in the third-order energy correction, a convolution approximation is introduced which generates the corresponding convolution form for the three-particle distribution function in the sequential relation. The proposed convolution form proves useful in estimating matrix elements. To carry out the numerical evaluation of the elementary excitation spectrum, we use Massey's theoretical liquid-structure function with slight modifications. In the low-momentum region it is shown that the ratios of the second- and third-order energy corrections to the Feynman excitation spectrum are proportional to the square of the wave vector; the result for the excitation energy with the leading correction is $\epsilon \mathrm{}\left(k\right)=\frac{(1-0.334\mathrm{}{k}^2{\AA }^2){\hslash }^2{k}^2}{2\mathrm{m}S\left(k\right)\mathrm{}}, k\ll {\rho }^{\frac{1}{3}}.$