Journal of Experimental and Theoretical Physics

Calculation of the wave functions of the ground and weakly excited states of helium II

Maksim Tomchenko

Published Jan 1, 2006• 2 citations

Abstract

The wave functions of the ground (Ψ 0 ) and the first excited (Ψ k ) states of He II in the second-order approximation, i.e., up to the first two corrections to the corresponding solutions for a weakly nonideal Bose gas, are determined by the collective variable method, which was proposed by Bogolyubov and Zubarev and developed in the studies by Yukhnovskii and Vakarchuk. The functions Ψ 0 and Ψ k = ψ k Ψ 0 are determined as the eigenfunctions of the N -particle Schrödinger equation from a system of coupled equations for Ψ 0 , Ψ k , and the quasiparticle spectrum E ( k ) of helium II. The results consist in the following: (1) these equations are solved numerically for a higher order approximation compared with those investigated earlier (the first-order approximation), and (2) Ψ 0 and ψ k are derived from a model potential of interaction between He 4 atoms (rather than from the structure factor as earlier) in which the potential barrier is joined with the attractive potential found from experiment. The height V 0 of the potential barrier is a free parameter. Except for V 0 , the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E 0 , and the quasiparticle spectrum E ( k ) of He II are in agreement with the experimental values for V 0 ≈ 100 K. The second-order correction to the logarithm of Ψ 0 significantly affects the value of E 0 and provides the asymptotics E ( k → 0) = ck , while the second-order correction to ψ k slightly affects the E ( k ). The second-order corrections to Ψ 0 and ψ k have a smaller effect on the results compared with the first-order corrections, whereby the theory is in agreement with experiment; therefore, one may assume that the truncated Ψ 0 and ψ k well describe the microstructure of He II. Thus, the series for Ψ 0 and Ψ k can be truncated in spite of the fact that the expansion parameter is not very small (∼1/2).

DOI: 10.1134/s106377610601016x

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