Calculation of the wave functions of the ground and weakly excited states of helium II
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SORDER, AND PHASE TRANSITIONS IN CONDENSED SYSTEMS
Calculation of the Wave Functions of the Ground and Weakly Excited States of Helium II M. D. Tomchenko Bogolyubov Institute of Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, 03143 Ukraine e-mail: [email protected] Received July 29, 2004
Abstract—The wave functions of the ground (Ψ0) and the first excited (Ψk) 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 Ψ0 and Ψk = ψkΨ0 are determined as the eigenfunctions of the N-particle Schrödinger equation from a system of coupled equations for Ψ0 , Ψk , 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) Ψ0 and ψk are derived from a model potential of interaction between He4 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 V0 of the potential barrier is a free parameter. Except for V0 , the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E0 , and the quasiparticle spectrum E(k) of He II are in agreement with the experimental values for V0 ≈ 100 K. The second-order correction to the logarithm of Ψ0 significantly affects the value of E0 and provides the asymptotics E(k 0) = ck, while the second-order correction to ψk slightly affects the E(k). The second-order corrections to Ψ0 and ψk 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 Ψ0 and ψk well describe the microstructure of He II. Thus, the series for Ψ0 and Ψk can be truncated in spite of the fact that the expansion parameter is not very small (~1/2). PACS numbers: 67.40.Db DOI: 10.1134/S106377610601016X
1. INTRODUCTION Since Landau’s pioneer works [1, 2], a variety of models have been proposed to describe the properties of helium II [3–30]. In my opinion, the most rigorous description of helium II from the viewpoint of microtheory is given by the following two approaches. I. A field-theoretic formalism, which develops the ideas of Bogolyubov [31], Belyaev [32], and Brueckner and Savada [5] (recently, interesting results have been obtained by Pashitskii [28]). II. A quantum-mechanical approach, in which the N-particle Schrödinger equation is solved for the ground and the first excited states of He II; this approach was developed in studies by Feynman [3, 4], Yukhnovskii and Vakarchuk [15–17], and Feenberg [9] and was used in the correlated basis function (CBF) [10, 11], hyp
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