# 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 »

- Authors:

- 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}

}