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

Abstract

The wave functions of the ground ({psi}{sub 0}) and the first excited ({psi}{sub 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 {psi}{sub 0} and {psi}{sub k} = {psi}{sub k}{psi}{sub 0} are determined as the eigenfunctions of the N-particle Schroedinger equation from a system of coupled equations for {psi}{sub 0}, {psi}{sub 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) {psi}{sub 0} and {psi}{sub k} are derived from a model potential of interaction between He{sup 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{sub 0} of the potential barrier is a free parameter. Except for V{sub 0}, the model does not have any free parameters or functions. The calculated values of the structuremore » factor, the ground-state energy E{sub 0}, and the quasiparticle spectrum E(k) of He II are in agreement with the experimental values for V{sub 0} {approx} 100 K. The second-order correction to the logarithm of {psi}{sub 0} significantly affects the value of E{sub 0} and provides the asymptotics E(k {sup {yields}} 0) = ck, while the second-order correction to {psi}{sub k} slightly affects the E(k). The second-order corrections to {psi}{sub 0} and {psi}{sub 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 {psi}{sub 0} and {psi}{sub k} well describe the microstructure of He II. Thus, the series for {psi}{sub 0} and {psi}{sub k} can be truncated in spite of the fact that the expansion parameter is not very small ({approx}1/2)« less

Authors:
 [1]
  1. National Academy of Sciences of Ukraine, Bogolyubov Institute of Theoretical Physics (Ukraine), E-mail: mtomchenko@bitp.kiev.ua
Publication Date:
OSTI Identifier:
21067753
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 102; Journal Issue: 1; Other Information: DOI: 10.1134/S106377610601016X; Copyright (c) 2006 Nauka/Interperiodica; Article Copyright (c) 2006 Pleiades Publishing, Inc; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; ASYMPTOTIC SOLUTIONS; BOSE-EINSTEIN GAS; CORRECTIONS; EIGENFUNCTIONS; EXCITED STATES; GROUND STATES; HELIUM II; MICROSTRUCTURE; PARTICLES; POTENTIALS; SCHROEDINGER EQUATION; STRUCTURE FACTORS; WAVE FUNCTIONS

Citation Formats

Tomchenko, M. D. Calculation of the wave functions of the ground and weakly excited states of helium II. United States: N. p., 2006. Web. doi:10.1134/S106377610601016X.
Tomchenko, M. D. Calculation of the wave functions of the ground and weakly excited states of helium II. United States. doi:10.1134/S106377610601016X.
Tomchenko, M. D. Sun . "Calculation of the wave functions of the ground and weakly excited states of helium II". United States. doi:10.1134/S106377610601016X.
@article{osti_21067753,
title = {Calculation of the wave functions of the ground and weakly excited states of helium II},
author = {Tomchenko, M. D.},
abstractNote = {The wave functions of the ground ({psi}{sub 0}) and the first excited ({psi}{sub 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 {psi}{sub 0} and {psi}{sub k} = {psi}{sub k}{psi}{sub 0} are determined as the eigenfunctions of the N-particle Schroedinger equation from a system of coupled equations for {psi}{sub 0}, {psi}{sub 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) {psi}{sub 0} and {psi}{sub k} are derived from a model potential of interaction between He{sup 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{sub 0} of the potential barrier is a free parameter. Except for V{sub 0}, the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E{sub 0}, and the quasiparticle spectrum E(k) of He II are in agreement with the experimental values for V{sub 0} {approx} 100 K. The second-order correction to the logarithm of {psi}{sub 0} significantly affects the value of E{sub 0} and provides the asymptotics E(k {sup {yields}} 0) = ck, while the second-order correction to {psi}{sub k} slightly affects the E(k). The second-order corrections to {psi}{sub 0} and {psi}{sub 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 {psi}{sub 0} and {psi}{sub k} well describe the microstructure of He II. Thus, the series for {psi}{sub 0} and {psi}{sub k} can be truncated in spite of the fact that the expansion parameter is not very small ({approx}1/2)},
doi = {10.1134/S106377610601016X},
journal = {Journal of Experimental and Theoretical Physics},
number = 1,
volume = 102,
place = {United States},
year = {Sun Jan 15 00:00:00 EST 2006},
month = {Sun Jan 15 00:00:00 EST 2006}
}