The theory of a many boson system with the deformed Heisenberg algebra

We propose to consider nonlinear fluctuations in the theory of liquid ⁴He deforming the commutation relations between the generalized coordinates and momenta. The generalized coordinates are the coefficients of the density fluctuations of the Bose particles. The deformation parameter takes into acco...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2015
Hauptverfasser:	Vakarchuk, I.O., Panochko, G.I.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 2015
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/153552
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	The theory of a many boson system with the deformed Heisenberg algebra / I.O. Vakarchuk, G.I. Panochko // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33002: 1–14. — Бібліогр.: 48 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-153552
record_format	dspace
spelling	irk-123456789-1535522019-06-15T01:27:32Z The theory of a many boson system with the deformed Heisenberg algebra Vakarchuk, I.O. Panochko, G.I. We propose to consider nonlinear fluctuations in the theory of liquid ⁴He deforming the commutation relations between the generalized coordinates and momenta. The generalized coordinates are the coefficients of the density fluctuations of the Bose particles. The deformation parameter takes into account effects of three- and four-particle correlations in the behavior of the system. This parameter is defined from the experimental values of the elementary excitation spectrum and the structure factor extrapolated to T=0 K. The numerical estimation of the ground state energy and the Bose condensate fraction is made. The elementary exitation spectrum and the potential of interaction between the helium atoms are recovered. Запропоновано врахувати нелiнiйнi флуктуацiї в теорiї рiдкого ⁴He, деформуючи комутацiйнi спiввiдношення мiж узагальненими координатами та iмпульсами. В якостi узагальнених координат обрано коефiцiєнти флуктуацiї густини бозе-частинок. Параметр деформацiї, що враховує вплив три- та чотиричастинкових кореляцiй на поведiнку бозе-систем, обрано виходячи з експериментальних значень для спектра елементарних збуджень та екстрапольованих експериментальних даних структурного фактора до температури T = 0 K. З модельним параметром деформацiї проведено чисельну оцiнку енергiї основного стану та кiлькостi Бозе-конденсату, вiдтворено спектр елементарних збуджень та потенцiал взаємодiї мiж атомами ⁴He. 2015 Article The theory of a many boson system with the deformed Heisenberg algebra / I.O. Vakarchuk, G.I. Panochko // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33002: 1–14. — Бібліогр.: 48 назв. — англ. 1607-324X arXiv:1503.04746 DOI:10.5488/CMP.18.33002 PACS: 05.30.Jp, 02.40.Gh, 67.25.-k, 67.25.dt, 03.65.-w http://dspace.nbuv.gov.ua/handle/123456789/153552 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We propose to consider nonlinear fluctuations in the theory of liquid ⁴He deforming the commutation relations between the generalized coordinates and momenta. The generalized coordinates are the coefficients of the density fluctuations of the Bose particles. The deformation parameter takes into account effects of three- and four-particle correlations in the behavior of the system. This parameter is defined from the experimental values of the elementary excitation spectrum and the structure factor extrapolated to T=0 K. The numerical estimation of the ground state energy and the Bose condensate fraction is made. The elementary exitation spectrum and the potential of interaction between the helium atoms are recovered.
format	Article
author	Vakarchuk, I.O. Panochko, G.I.
spellingShingle	Vakarchuk, I.O. Panochko, G.I. The theory of a many boson system with the deformed Heisenberg algebra Condensed Matter Physics
author_facet	Vakarchuk, I.O. Panochko, G.I.
author_sort	Vakarchuk, I.O.
title	The theory of a many boson system with the deformed Heisenberg algebra
title_short	The theory of a many boson system with the deformed Heisenberg algebra
title_full	The theory of a many boson system with the deformed Heisenberg algebra
title_fullStr	The theory of a many boson system with the deformed Heisenberg algebra
title_full_unstemmed	The theory of a many boson system with the deformed Heisenberg algebra
title_sort	theory of a many boson system with the deformed heisenberg algebra
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/153552
citation_txt	The theory of a many boson system with the deformed Heisenberg algebra / I.O. Vakarchuk, G.I. Panochko // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33002: 1–14. — Бібліогр.: 48 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT vakarchukio thetheoryofamanybosonsystemwiththedeformedheisenbergalgebra AT panochkogi thetheoryofamanybosonsystemwiththedeformedheisenbergalgebra AT vakarchukio theoryofamanybosonsystemwiththedeformedheisenbergalgebra AT panochkogi theoryofamanybosonsystemwiththedeformedheisenbergalgebra
first_indexed	2025-07-14T05:00:58Z
last_indexed	2025-07-14T05:00:58Z
_version_	1837597205392785408
fulltext	Condensed Matter Physics, 2015, Vol. 18, No 3, 33002: 1–14 DOI: 10.5488/CMP.18.33002 http://www.icmp.lviv.ua/journal Theory of a many-boson system with deformed Heisenberg algebra I.O. Vakarchuk 1, G.I. Panochko2 1 Department for Theoretical Physics, Ivan Franko National University of Lviv, 12 Dragomanov St., 79005 Lviv, Ukraine 2 College of Natural Sciences, Ivan Franko National University of Lviv, 107 Tarnavsky St., 79010 Lviv, Ukraine Received March 14, 2015, in final form May 20, 2015 We propose to consider nonlinear fluctuations in the theory of liquid 4He deforming the commutation relations between the generalized coordinates and momenta. Generalized coordinates are coefficients of density fluctu- ations of Bose particles. The deformation parameter takes into account the effects of three- and four-particle correlations in the behavior of a system. This parameter is defined from the experimental values of the ele- mentary excitation spectrum and the structure factor extrapolated to T = 0 K. The numerical estimation of the ground state energy and the Bose condensate fraction is made. The elementary excitation spectrum and the potential of interaction between the helium atoms are recovered. Key words: deformed Heisenberg algebra, collective variables, elementary excitation spectrum of liquid 4He, Bose-condensate PACS: 05.30.Jp, 02.40.Gh, 67.25.-k, 67.25.dt, 03.65.-w 1. Introduction The method of collective variables is an effective approach to the study of an N -particle Bose system. In this method, independent variables are the Fourier coefficients of the density fluctuations ρk of the particle (k is the wave vector). This method was originally proposed by Bohm [1–4]. It was suggested that from the whole set of an infinite number of variables ρk, only DN variables are taken, where D is the space dimensionality. To achieve this, the domain of the wave-vector absolute values was restricted by some kc. An obvious imperfection of this approach is an ambiguous choice of variables. Bogoliubov and Zubarev [5] proposed an approach where the set of values k is not limited. However, the transition from Carte- sian coordinates to collective variables ρk can be provided by the weighting function. The hermitian Hamiltonian of the Bose liquid in the ρk representation is written as a sum of Hamiltonians of an infinite number of non-interacting harmonic oscillators describing the oscillations of the Bose-liquid density plus a contribution from anharmonicities of these oscillations. Justifications of the feasibility of the method of collective variables are given in many articles using different approaches. Usually, the anharmonic con- tribution was considered using the perturbation operator [6–20]. The respective results can be brought to the numerical calculations both for model systems and for strongly non-ideal systems like liquid 4He. A different approach for the study of the Bose system arose following the introduction of quan- tum spaces of minimal length. The simplest deformation is that of Kempf being quadratic in general- ized momentum [21, 22]. The deformed commutation relations can be examined in multidimensional case. Such kind of deformation was applied to a wide variety of quantum mechanical problems, among which we distinguish the eigenvalue problem for D-dimensional isotropic harmonic oscillator [23], three- dimensional Dirac oscillator [24], (2+1)-dimensional Dirac equation in a constant magnetic field [25] that were solved exactly. A composite system problem in the deformed space was also considered [26, 27]. © I.O. Vakarchuk , G.I. Panochko, 2015 33002-1 http://dx.doi.org/10.5488/CMP.18.33002 http://www.icmp.lviv.ua/journal I.O. Vakarchuk,G.I. Panochko Some aspects of field theories were investigated in the deformed space, for example: the electromag- netic field [29, 30], a photoelectric phenomenon [31], radiation and absorption of photons for deformed field [32], the Casimir effect for the deformed field [33]. An urgent problem is to find a possible gener- alization of arbitrary one-dimensional Heisenberg algebra with minimal length (or/and momentum) for multidimensional case [28]. Kempf ’s deformation can be applied not only to the operators of positions and momenta but also to collective variables. The idea to use a deformation of Poisson brackets to ex- plore a Bose system firstly appeared in the article [34]. In paper [35], the Bose particles were represented as a set of q-deformed harmonic oscillators. In paper [36], there was used a representation of deformed creation and annihilation operators with a generalized four-parameter q-algebra. We note that the de- formed creation and annihilation operators associate with the non-linear f -oscillator operators. In paper [37], two-parameter deformed Bose gas model was proposed to find the correlation functions of the par- ticle. In a series of papers, the thermodynamics of ideal Bose and Fermi systems [38–40] was studied. In paper [41], the phenomenon of the Bose-Einstein condensation of the relativistic ideal Bose gas with deformed commutation relations for positions and momenta operators is investigated. In our paper, we associate the difficulties connected with anharmonic contribution with deformation of commutation relations between the positions and momenta. Thus, we replace the multimode Hamilto- nian of the Bose liquid in ρk representation by a sum of single-mode Hamiltonians. We assume that this model successfully describes the properties of the Bose system when we make an appropriate choice of the deformation parameter. In this approach, we treat the collective variables as generalized coordinates. Thus, we attempt to take into account the anharmonic contribution described in [42] with deformed Pois- son brackets quadratic in generalized momenta. In this paper, we use the deformation function which is quadratic in generalized coordinates. The correctness and efficiency of the proposed approach is shown for the analysis of Bose-systems with developed anharmonisms of the density fluctuations. Such a treat- ment of anharmonisms lowers the ground state energy if the deformation parameter is negative. It is interesting to note that the energy levels are bounded for such values of the deformation parameter. We make numerical estimations of the analytical results. 2. Hamiltonian of Bose-liquid in the deformed space of collective vari- ables Let us consider a system of N spinless Bose particles of mass m and by the coordinates r1, . . . ,rN which move in the D-dimension space of the volume V . The Hamiltonian of the system reads Ĥ = N ∑ j=1 p̂ j 2 2m + ∑ 1Éi< jÉN Φ(\|ri −r j \|), (2.1) where the first term is the kinetic energy operator, p̂ j = −iħ∇ j is the momentum operator of the j th particle; the second term is the potential energy consisting of a sum of the particle interaction potentials Φ(\|ri −r j \|). We introduce the collective coordinates representation which takes the form: ρk = 1 p N N ∑ j=1 e −ikr j , k, 0. (2.2) The Hamiltonian of the Bose liquid (2.1) can be rewritten in the following form [5, 6]: Ĥ = ∑ k,0 ħ2k2 2m ( − ∂2 ∂ρk∂ρ−k + 1 4 ρkρ−k − 1 2 ) + N (N −1) 2V ν0 + N 2V ∑ k,0 νk (ρkρ−k −1)+∆Ĥ , (2.3) where νk = ∫ e −ikR Φ(R)dR (2.4) 33002-2 Theory of many-boson system with deformed algebra is the Fourier image of the interaction potential. The operator ∆Ĥ contains all anharmonic terms and in addition contains the term from the specific linear anharmonic by ρk and quadratic by ∂/∂ρk: ∆Ĥ = ∑ k,0 ∑ k′,0 ħ2(kk′) 2m p N ρk+k′ ∂2 ∂ρk∂ρk′ + ∑ nÊ3 (−)n 4n(n−1)( p N )n−2 ∑ k1,0 . . . ∑ kn,0 k1+...+kn=0 ħ2 2m (k2 1 +·· ·+k2 n)ρk1 ...ρkn . (2.5) The introduced in (2.2) variables ρk are complex and can be represented as a linear combination of real variables: ρk = ρc k − iρs k , ρc k = 1 p N N ∑ j=1 coskr j , ρs k = 1 p N N ∑ j=1 sin kr j . Since they are the complex conjugate value ρ∗ k = ρ−k, that is ρ c k = ρc −k , ρs k =−ρ−k, and the variables with a particular index value k equal to (−k). Thus, independent variables are ρk when the wave vector k takes up values to the half-space of the wave vectors. Taking into account all the remarks we have made, let us consider the harmonic part of the Hamiltonian of the Bose-liquid (2.3) as an infinite set of non-interacting harmonic oscillators with oscillation frequency ωk : Ĥ = ∑ µ=c ,s ∑ k,0 ′ ( P̂ 2 k,µ 2mk + mkω 2 k Q̂2 k,µ 2 ) + N (N −1) 2V ν0 − ∑ k,0 (ħ2k2 4m + N 2V νk ) , (2.6) where the generalized momentum operator P̂k,µ is conjugate to the generalized coordinate Q̂k,µ. In the coordinate ρk-representations Q̂k,c = ρc k , Q̂k,s = ρs k . (2.7) Explicit form of the generalized momentum operator is as follows: P̂k,µ =−iħ ∂ ∂ρk,µ . (2.8) The ∑′ k,0 means that the sum takes the values k just from the haft-space of the domain. Comparing Hamiltonians (2.6) and (2.3) we have mk = 2m k2 , ωk = ħk2 2m αk , αk = √ 1+ 2N V νk /ħ2k2 2m . (2.9) We note that the operator P̂k,µ has the dimension of action ħ, and Q̂k,µ is dimensionless. The Hamiltonian of the system (2.3) contains harmonic terms. We suggest that the anharmonicity ∆Ĥ can be taken into account by deformation of the commutation relations between generalized coordinates and momenta: Q̂k,µP̂k,µ− P̂k,µQ̂k,µ = iħ ( 1+βkQ̂2 k,µ ) , (2.10) where the dimensionless deformation parameter βk depends on the absolute value of the wave-vector. The operators Q̂k,µ and P̂k′,µ′ with different index commute. The deformation parameter for 4He can take negative values. We do not expect a full description of the properties of the Bose liquid Hamiltonian (2.6) with the condition (2.10) but assume that this model describes the behavior of the many-boson systems. 3. Energy levels and wave functions of a Bose liquid We find solutions of the stationary Schrödinger equation from harmonic oscillators in the deformed space with Hamiltonian (2.6). We suppose that the deformation is positive. For this purpose, we use the 33002-3 I.O. Vakarchuk,G.I. Panochko canonically conjugated operators q̂k,µ, p̂k,µ with the standard Heisenberg algebra: P̂k,µ = p̂k,µ , Q̂k,µ = tan (q̂k,µ √ βk ) √ βk . (3.1) To solve the eigenvalue problem we use a representation of the deformed operators P̂k,µ and Q̂k,µ (2.10) which express them in terms of canonically conjugate operators q̂k,µ, p̂k,µ. The representation we have chosen reads: q̂k,µp̂k,µ− p̂k,µ q̂k,µ = iħ . (3.2) Taking into account the representation (3.1) we rewrite Hamiltonian (2.6) in the form: Ĥ = ∑ µ=c ,s ∑ k,0 ′ [ p̂ 2 k,µ 2mk + mkω 2 k 2 tan2 (q̂k,µ √ βk ) βk ] + N (N −1) 2V ν0 − ∑ k,0 (ħ2k2 4m + N 2V νk ) . (3.3) The eigenvalues and the wave functions of the harmonic oscillator with deformed Heisenberg algebra were calculated in [6]. Taking into consideration our notations (2.9) we find: E...,nk,c , ...; ...,nk,s , ... = ∑ µ=c ,s ∑ k,0 ′ħ2k2 2m αk   ( nk,µ+ 1 2 ) √ 1+ ( βk 2αk )2 + βk 2αk ( n2 k,µ+nk,µ+ 1 2 )  + N (N −1) 2V ν0 − ∑ k,0 (ħ2k2 4m + N 2V νk ) , (3.4) here, quantum numbers nk,µ = 0,1,2, . . ., µ = c, s. In the case of positive deformation parameter, the energy spectrum is infinite. The wave functions in the coordinate representation q̂k,µ = qk,µ, p̂k,µ = −iħ∂/∂qk,µ can be written as follows: ψ...,nk,c , ...; ...,nk,s , ...(. . . , qk,µ, . . .) = ∏ k,0 ′ ∏ µ=c ,s ψnk,µ (qk,µ) . (3.5) The ground-state wave function (nk,µ = 0) reads: ψ0(qk,µ) =β1/4 k √ Γ(ν+1) Γ(1/2)Γ(ν+1/2) cos ν q, for nk,µ Ê 1, we have ψn (qk,µ) = β1/4 √ Γ(ν+n+1)Γ(n+2ν) n!Γ(1/2)Γ(ν+n+1/2)Γ(2ν+2n) × ( − d dq +νtan q ) · · · ( − d dq + (ν+n−1)tan q ) cos ν+n q , (3.6) here, ν = 1 2 + αk βk √ 1+ ( βk 2αk )2 , (3.7) q = qk,µ √ βk , n = nk,µ . The wave functions are orthonormalized: π/(2 p βk ) ∫ −π/(2 p βk ) ψn′ (qk,µ)ψn(qk,µ)dqk,µ = δn′,n . (3.8) 33002-4 Theory of many-boson system with deformed algebra The explicit form of the wave function of the first excited state is as follows: ψ1(qk,µ) =β1/4 k √ 2Γ(ν+2) Γ(1/2)Γ(ν+1/2) cos ν q sin q. The energy spectrum is quadratic in quantum numbers nk,µ, similarly to the theory of anharmonic oscil- lator in the case it takes into account the terms proportional to ∼ħ2 [6, 43]. Now we consider the case βk < 0. Then [Q̂k,µ, P̂k,µ]= iħ ( 1−\|βk \|Q̂2 k,µ ) , (3.9) and we impose the requirement ( 1−\|βk \| 〈 Q̂2 k,µ 〉) > 0, (3.10) here, the brackets 〈(. . . )〉 denote an average: 〈(. . . )〉 = ∫ ψ0(qk,µ)(. . . )ψ∗ 0 (qk,µ)dqk,µ . (3.11) The uncertainty relation obtained for the deformed algebra (3.9) leads to the fact that minimal uncer- tainty for the momentum operator is equal to zero. The new canonically conjugated variables (3.1) and a structure of Hamiltonian (3.3) with the changes βk →\|βk \|, tan (q̂k,µ √ βk ) → tanh (q̂k,µ √ \|βk \|) will be the same. Using the factorization method [6] we can write the wave functions of the transformed Hamilto- nian for (k,µ) modes: ψ0(qk,µ) = \|βk \|1/4 √ Γ(ν+1/2) Γ(1/2)Γ(ν) 1 coshν q , when n Ê 1 : ψn(qk,µ) = \|βk \|1/4 √ Γ(ν−n+1/2)Γ(2ν−2n+1) n!Γ(1/2)Γ(ν−n)Γ(2ν−n+1) × ( − d dq +νtanh q ) · · · ( − d dq + (ν−n+1)tanh q ) cosh −ν+n q , (3.12) here, ν = −1 2 + αk \|βk \| √ 1+ ( \|βk \| 2αk )2 , (3.13) q = qk,µ √ \|βk \|, n = nk,µ . In case of deformed algebra (3.9), the quantum numbers are limited by the number n < ν and the energy levels are bounded from above. When ν Ê n, the energy spectrum is continuous. The wave function is also normalized. We note that the integration variable qk,µ runs from ∞ up to −∞ ∞ ∫ −∞ ψn (qk,µ)ψn (qk,µ)dqk,µ = 1. (3.14) For example, we get for n = 1: ψ1(qk,µ) = \|βk \|1/4 √ 2Γ(ν+1/2) Γ(1/2)Γ(ν−1) sinh q coshν q . In the non-deformed case (βk = 0), we obtain the energy levels of the Bose liquid in the main approxima- tion [6]. When in equations (3.6) and (3.12), for the wave functions, the quantity βk → 0 and ν→∞, we obtain: cos ν q = βk→0 ( 1− q2 k,µ βk 2 +·· · )αk /βk = βk→0 e −q2 k,µ αk /2 , 33002-5 I.O. Vakarchuk,G.I. Panochko cosh −ν q = \|βk \|→0 ( 1+ q2 k,µ \|βk \| 2 +·· · )−αk /\|βk \| = \|βk \|→0 e −q2 k,µ αk /2 . Taking into account the asymptotic formula for the gamma function Γ(ν+ a) ∼ p 2πe−ννν+a−1/2, when ν→∞, we obtain the wave functions of the harmonic oscillator: ψnk,µ (qk,µ) = (αk π )1/4 1 √ nk,µ!2nk,µ ( − d dη +η )nk,µ e −η2/2 , (3.15) here, η= qk,µ p αk , αk = mkωk /ħ. 4. Ground-state energy If the quantum numbers in equation (3.4) are equal to zero, nk,µ = 0, we obtain the ground-state energy E0 which can be written after transformations in the form: E0 = N (N −1) 2V ν0 − ∑ k,0 ħ2k2 8m (αk −1) 2 + ∑ k,0 ħ2k2 4m αk [ √ 1+ ( βk 2αk )2 + βk 2αk −1 ] . (4.1) The first two terms recover the ground-state energy in the Bogoliubov approximation [44]. The last term in equation (4.1) takes into account the anharmonicity and always leads to the negative values for the parameter βk . The lowering of the ground-state energy of the liquid 4He can by achieved by a direct consideration of the anharmonic operator ∆Ĥ in the Hamiltonian (2.3) from perturbation theory [19, 20]. The Fourier coefficient ν0 in the limit k → 0 can be expressed via the speed of the first sound for liquid 4He (when T = 0 K, the speed is c = 238.2m/s). The definition of the speed of the first sound when T = 0 K, N ∂2E0 ∂N 2 = mc2 , allows us to obtain the equation: mc2 = N V ν0 − 1 4N ∑ k,0 ħ2k2 4m (α2 k −1)2 α3 k 1 (1+ (βk /2αk )2)3/2 . (4.2) Having taken into account the obtained equation (4.2) we rewrite the ground-state energy in the thermo- dynamic limit: E0 = N mc2 2 − 1 4 ∑ k,0 ħ2k2 2m (αk −1) 2 + 1 2 ∑ k,0 ħ2k2 2m αk [ √ 1+ ( βk 2αk )2 + βk 2αk −1 ] + 1 16 ∑ k,0 ħ2k2 2m 1 αk ( αk − 1 αk )2 [ 1+ ( βk 2αk )2]−3/2 . (4.3) The found expressions can be used for the models with exact or perturbative solutions. To verify the va- lidity of the expression (4.3), we might compare it with the results given by an exactly solvable model or a model which allows perturbative consideration. Such a comparison brings about an interesting question concerning the choice of deformation parameter βk . 5. Elementary excitation spectrum Let us find an expression for an elementary excitation given by the wave vector q. Suppose that only the quantum number nq,c = 1 and the other quantum numbers nk,µ = 0 when k , q, µ , c. From equation (3.4) we obtain: E...,0,nq,c=1,0, ...; ...,0, ... = E...,0, ...; ...,0,nq,s=1,0, ...; ...,0, ... = E0 +Eq , 33002-6 Theory of many-boson system with deformed algebra where the elementary excitation spectrum Eq = ħ2q2 2m αq [ √ 1+ ( βq 2αq )2 + βq αq ] . (5.1) If βk = 0, the latter equation leads to the Bogoliubov’s excitation spectrum [44]: E B q = ħ2q2 2m αq . (5.2) Equation (5.1) is an exact solution of the Schrödinger equation with Hamiltonian (3.3). Here, we do not obtain the fading phenomenon of the elementary excitation because in Hamiltonian (3.3) there are no terms that describe the dissipation and the decay of the elementary excitation. The decay of the elemen- tary excitations in the liquid 4He can be explained by the fact that the spectrum of Bose-liquid ends at k ≃ 3.6 Å−1 [46]. 6. Structure factor The structure factor of the system can be defined as an average of the square of the density fluctua- tions: Sk = 〈\|ρk\|2〉 . (6.1) Now, we calculate the average (6.1) taking into consideration the fact that the collective variables are generalized ones and using the ground state wave functions (3.6). Since the collective variables are gen- eralized coordinates Q̂k,µ from equation (2.7), we find the average (6.1) on the wave functions of the ground-state (3.6): Sk = ∑ µ=c ,s 〈Q̂2 k,µ〉 = 1 βk ∑ µ=c ,s π/(2 p βk ) ∫ −π/(2 p βk ) ψ2 0(qk,µ) tan 2 ( √ βk qk,µ ) dqk,µ (6.2) = 2 βk Γ(ν+1) Γ(ν+1/2)Γ(1/2) π/2 ∫ −π/2 cos 2ν q tan 2 qdq = 4 βk Γ(ν+1) Γ(ν+1/2)Γ(1/2) π/2 ∫ −π/2 sin 2 q cos 2(ν−1) qdq. This integral is reduced to the beta function: Sk = 1 βk Γ(ν−1/2) Γ(ν+1/2) = 1 βk (ν−1/2) . The parameter ν should be taken from the relation 3.7 and after some transformations we obtain: Sk = 1 αk √ 1+ (βk /2αk )2 . (6.3) When βk = 0, we recover the well-know result by Bogoliubov and Zubarev [5], Sk = 1/αk . (6.4) The elementary excitation spectrum (5.1) reads: Eq = ħ2q2 2mSq + ħ2q2 2m βq . (6.5) 33002-7 I.O. Vakarchuk,G.I. Panochko The structure factor can be written as follows: Sk −1= 2V N δE0 δνk , here, F is the free energy of the system, which is equal to the ground-state energy E0 at T = 0 K. Having performed elementary calculations with equation (4.1) we obtain the result (6.3). Note that the struc- ture factor Sk is an analytic function of the deformation parameter βk . This fact can be seen when we calculated Sk and make a parameter βk →−\|βk \| in the wave functions (3.12): Sk = ∑ µ=c ,s 1 \|βk \| ∞ ∫ −∞ ψ2 0(qk,µ) tanh 2 ( √ \|βk \|qk,µ ) dqk,µ (6.6) = 2 \|βk \| Γ(ν+1/2) Γ(1/2)Γ(ν) ∞ ∫ −∞ tanh2 q cosh2ν q dq = 4 \|βk \| Γ(ν+1/2) Γ(1/2)Γ(ν) ∞ ∫ 0 sinh2 q cosh2ν+2 q dq. This integral is reduced to the beta function: Sk = Γ(ν+1/2) \|βk \|Γ(ν+3/2) = 1 \|βk \|(ν+1/2) , here, the parameter ν is taken from the equation (3.13). Having performed some simplifications we obtain the structure factor from the equation (6.3). We note that condition (3.10) leading to the restrictions on the deformation parameter can be represented in the following form: \|βk \|Sk 2 < 1. (6.7) 7. Potential and kinetic energy Having used the structure factor we rewrite the ground-state energy of the many-body system (4.1): E0 = N mc2 2 + 1 16 ∑ k,0 ħ2k2 2mSk (S2 k −1) 2 − 1 4 ∑ k,0 ħ2k2 2m ( 1− 1 Sk )2 +∆E0 , (7.1) here, ∆E0 = 1 16 ∑ k,0 ħ2k2 2mSk [ 2(S2 k −1)+ ( βk Sk 2 )2]( βk Sk 2 )2 + 1 4 ∑ k,0 ħ2k2 2m [ βk + ( βk 2 )2] . (7.2) Themean value for the potential energy can be calculated for the ground state using equations (4.2), (6.3). After some transformations, we arrive at the relation for the mean value of the potential energy: 〈Φ〉 = N (N −1) 2V ν0 + N 2V ∑ k,0 νk (〈\|ρk\|2〉−1) = N (N −1) 2V ν0 + ∑ k,0 ħ2k2 8m (α2 k −1) { 1 αk √ 1+ (βk /2αk )2 −1 } . (7.3) Let us rewrite the average potential energy in terms of Sk and speed of the first sound: 〈Φ〉 = N mc2 2 + 1 16 ∑ k,0 ħ2k2 2mSk (S2 k −1) 2 − 1 4 ∑ k,0 ħ2k2 2m (Sk −1) ( 1− 1 S2 k ) +∆〈Φ〉, (7.4) here, ∆〈Φ〉 = 1 32 ∑ k,0 ħ2k2 2m β2 k (Sk −1) 2 (Sk +2)+ 1 16 ∑ k,0 ħ2k2 2mSk ( βk Sk 2 )4 . (7.5) 33002-8 Theory of many-boson system with deformed algebra The mean value of the kinetic energy is the average value of the operator (2.6) in case νk = 0: 〈K 〉 = ∑ µ=c ,s ∑ k,0 ′ħ2k2 4m [ 1 ħ2 〈P̂ 2 k,µ〉+〈Q̂2 k,µ〉−1 ] . (7.6) We calculate the first term in the square brackets on the ground state wave function from the equa- tion (3.6): 〈P̂ 2 k,µ〉 = −ħ2 Γ(ν+1) Γ(1/2)Γ(ν+1/2) √ βk π/(2 p βk ) ∫ −π/(2 p βk ) cos ν qk,µ ∂2 ∂q2 k,µ cos ν qk,µdqk,µ (7.7) = ħ2 Γ(ν+1) Γ(1/2)Γ(ν+1/2) √ βk π/(2 p βk ) ∫ −π/(2 p βk ) ∣ ∣ ∣ ∣ ∂ ∂qk,µ cos ν qk,µ ∣ ∣ ∣ ∣ 2 dqk,µ (7.8) = 2βkħ2 ν2 Γ(ν+1) Γ(1/2)Γ(ν+1/2) π/2 ∫ 0 sin 2 q cos 2ν−2 qdq. After simple transformations we get: 〈P̂ 2 k,µ〉 = ħ2βkν 2 ν−1/2 = ħ2 2Sk ( 1+ βk Sk 2 )2 , (7.9) here, the expression for ν is taken from equation (3.7) and the structure factor from equation (6.3). The second term in equation (7.6) is the structure factor by definition. Finally, we can write: 〈K 〉 = ∑ k,0 ħ2k2 8m { αk [ √ 1+ (βk /2αk )2 +βk /2αk ]2 √ 1+ (βk /2αk )2 + 1 αk √ 1+ (βk /2αk )2 −2 } . (7.10) Taking into consideration equation (6.3) we obtain: 〈K 〉 = 1 4 ∑ k,0 ħ2k2 2m (1−Sk )2 Sk +∆〈K 〉, (7.11) ∆〈K 〉 = 1 4 ∑ k,0 ħ2k2 2m βk ( 1+ βk Sk 4 ) . (7.12) This result for the kinetic energy can be obtained as a derivative from the ground state energy with respect to the mass: 〈K 〉 =−m ∂E0 ∂m . (7.13) The sum of expressions (7.4) and (7.11) is the total energy (7.1). Now we find the potential of the interaction between the Bose particles: Φ(r )= 1 (2π)3 ∫ e ikrνk dk, (7.14) here, νk is the Fourier coefficient of the interaction potential between Bose particles represented as a function of the structure factor (6.3): νk = V N ħ2k2 4m ( 1 S2 k −1− β2 k 4 ) . (7.15) Thus, the interaction potential between the helium atoms reads: Φ(r )= 1 2π2ρr ħ2 2m ∞ ∫ 0 k3 sin kr ( 1 S2 k −1 ) dk +∆Φ(r ), (7.16) 33002-9 I.O. Vakarchuk,G.I. Panochko here, the contribution is ∆Φ(r ) caused by the deformation of the commutation relations: ∆Φ(r )=− 1 8π2ρr ħ2 2m ∞ ∫ 0 k3β2 k sinkr dk. (7.17) 8. Momentum distribution To find the average number of atoms Nk with the momentum ħk, k , 0 it is sufficient to calculate the variational derivative from the free energy of the system with respect to the free-particle spectrum ħ2k2/2m. We note that the free energy of the system coincides with the ground state energy when T = 0 K [equation (4.3) by nk,µ = 0]. After simple calculations we get: Nk = 1 4 { αk [ √ 1+ (βk /2αk )2 +βk /2αk ]2 √ 1+ (βk /2αk )2 + 1 αk √ 1+ (βk /2αk )2 −2 } , (8.1) or after some transformation Nk = 1 4Sk [ (1−Sk ) 2 +βk Sk + (βk Sk 2 )2] . (8.2) The expression for the kinetic energy can be represented as a function of average numbers of particles: 〈K 〉 = ∑ k,0 ħ2k2 2m Nk . (8.3) The average number of atoms Nk with the momentum ħk can be obtained from the expression for the mean value of the kinetic energy (8.3). To do so, we take into account the relation (7.10). After simple calculations, we obtain the expression for the average numbers of atoms Nk which coincides with the expression (8.1). We estimate the relative number of atoms in the case of their momenta being equal to zero (Bose condensate): N0 N = 1− 1 N ∑ k,0 Nk = 1− 1 4N ∑ k,0 (1−Sk )2 Sk + ∆N0 N , (8.4) ∆N0 N =− 1 4N ∑ k,0 βk ( 1+ βk Sk 4 ) . (8.5) At intermediate calculation, we assume that βk is of positive value. For numerical calculations of the deformation parameter it is negative. 9. Deformation parameter. Numerical calculations For the numerical evaluation of the deformation parameter we proceed from the expression of the elementary excitation spectrum (6.5). The value of the structure factor and excitation spectrum are taken from experimental papers [45, 46]: βk = Ek ħ2k2/2m − 1 Sk . (9.1) We have the values of the deformation parameter for a limited range of wave vectors because the elemen- tary excitation spectrum of the Bose liquid has the ultimate point of completion, and the experimental data of the structure factor are given up to 7.3 Å−1. However, as k →∞ the elementary excitation spec- trum should be equal to the free-particle spectrum Ek → ħ2k2/2m and the structure factor Sk → 1. In figure 1, the deformation parameter βk based on the experimental data for Sk and Ek is shown. We 33002-10 Theory of many-boson system with deformed algebra 1 2 3 4 5 6 -1,5 -1,0 -0,5 0,0 k k,Å-1 Figure 1. Deformation parameter (9.1). shall model the deformation parameter by the function where a free parameter will be taken from the extrapolated data for the structure factor of the Bose liquid at T = 0 K [47]: βk =−Sk \|Sk −1\|3. (9.2) The sign of the deformation parameter determines the behavior of the elementary excitation spectrum we have received (6.5). Therefore, βk < 0 because the experimental data for the elementary excitation spectrum of the liquid helium (triangles in figure 3, right panel) is lower than that of the theoretically calculated Feynman’s spectrum (circles in figure 3, right panel). The graph of the function (see equation 9.2) is shown in figure 2 (left panel). Note that this choice of the deformation parameter gives a correct behavior in the long-wavelength domain. Function (9.2) can be used to find the physical quantities in the limit of T → 0. The form of the curve for βk (see figure 2) should be such that the elementary excitation spectrum (solid line in figure 3) is reproduced for all the values of the wave vector. The point k = 1 Å−1 corresponds to the typical maximum on the curve of the elementary excitation spectrum (see figure 3, right panel: triangles and circles). Thus, βk (see figure 2) should have a clearly expressed minimum in this point. In figure 2 (right panel) the dependence reflecting limitations imposed by the deformation parameter (6.7) is shown. 0 1 2 3 4 -0,12 -0,10 -0,08 -0,06 -0,04 -0,02 0,00 k k,Å-1 0 1 2 3 4 -0,06 -0,05 -0,04 -0,03 -0,02 -0,01 0,00 k,Å-1 fk Figure 2. (Left panel): Model of the deformation parameter (9.2); (Right panel): The function fk =βk Sk /2 as a condition to limit the deformation parameter (9.2). One can offer the model functions satisfying these considerations provided that contributions to the 33002-11 I.O. Vakarchuk,G.I. Panochko basic physical quantities of the system reproduce the results in the post-RPA approximation in the theory of liquid helium-4. With the deformation parameter from (9.2), numerical calculations of the obtained quantities per particle are made. In the thermodynamic limits (V →∞, N →∞, N /V = const), ∑ k → V (2π)3 ∫ dk, the contributions of the physical quantities connected with the deformed commutation relations are es- timated. We do not expect a complete agreement between the properties of the Bose liquid within the proposedmethod and the perturbative results. However, we obtained the results that are consistent with those of the perturbation theory. The ground state energy with the deformation taken into consideration: E0 N = E B 0 N + ∆E0 N , ∆E0 N =−1.89 K. (9.3) The numerical value of the first term corresponds to the ground state energy in the Bogoliubov approxi- mation, see [47] (with ρ = 0.0219 Å−3 and speed of the first sound c = 238.2 m/s): E B 0 /N =−5.31 K. Thus, the full energy per particle in the deformation case is E0/N =−7.2 K. The experimental result of the en- ergy is E0/N =−7.13 K. Note that the linear corrections to the ground state energy over the deformation parameter give a leading contribution. This choice of the deformation parameter (9.2) offers an insignif- icant amendment to the value of the Bose condensate fraction obtained in the zeroth approximation: N0 N = N B 0 N + ∆N0 N , ∆N0 N = 0.11. (9.4) In the zeroth approximation, N B 0 /N gives a wrong result: N B 0 /N =−0.31. 0 2 4 6 8 0 50 100 150 200 250 (r), K r, Å 0 1 2 3 4 5 0 7 13 20 26 33 39 E(q), K q,Å-1 Figure 3. (Left panel): The potential of the interaction between helium atoms. Circles correspond to the non-deformed case equation (7.16) with ∆Φ = 0. Solid line is the potential with models of the deforma- tion parameter equation (7.16) with ∆Φ , 0; (Right panel): The elementary excitation spectrum. Circles denote Feynman’s spectrum; solid line is the spectrum of the deformed case; triangles correspond to the experimental data for the spectrum [46]. In our method, the anharmonic contributions from higher correlations in the Bose-liquid are taken into account within the single-mode approximation. The Bose condensate fraction taking into account the correction ∆N0/N in the deformed case [see equation (8.5)] is calculated in the approximation of the one sum over the wave vector. In this approximation, the value N0/N is negative. Nonlinear terms related to the deformation are quadratic in the expression for the interaction po- tential. This contribution vanishes, whereas the linear correction from the deformation parameter to the elementary excitation spectrum gives a significant contribution at q ≃ 2 Å−1. This correction does not 33002-12 Theory of many-boson system with deformed algebra contribute in the long-wavelength limit (figure 3). The behavior of the elementary excitation spectrum in the long-wavelength limit was solved in [48] of the method of two-time temperature Green’s functions. At least a two-parametric deformation should be sought for to consider anharmonic terms ∆H in Hamilto- nian (2.3) in a proper way. This issue will be a subject of our future studies. Acknowledgements The authors appreciate the help in the preparation of this article of Ph.D. Andrij Rovenchak and Ph.D. Mykola Stetsko. References 1. Bohm D., Pines D., Phys. Rev., 1951, 82, 625; doi:10.1103/PhysRev.82.625. 2. Bohm D., Pines D., Phys. Rev., 1952, 85, 338; doi:10.1103/PhysRev.85.338. 3. Bohm D., Pines D., Phys. Rev., 1953, 92, 609; doi:10.1103/PhysRev.92.609. 4. Bohm D., General Theory of Collective Coordinates, Mir, Moscow, 1964, 150 p. (in Russian). 5. Bogoliubov N.N., Zubarev D.N., Sov. Phys. JETP, 1955, 83, 1 [Zh. Eksp. Fiz. Nizk. Temp., 1955, 28, 129 (in Russian)]. 6. Vakarchuk I.O., Quantum Mechanics, Ivan Franko National University of Lviv, Lviv, 2012, 789–791; (in Ukrainian). 7. Hiroike K., Prog. Theor. Phys., 1959, 21, 327; doi:10.1143/PTP.21.327. 8. Hiroike K., Prog. Theor. Phys., 1975, 54, 308; doi:10.1143/PTP.54.308. 9. Sunakawa S., Yumasaki S., Kebukawa T., Prog. Theor. Phys., 1969, 41, 919, doi:10.1143/PTP.41.919. 10. Sunakawa S., Yumasaki S., Kebukawa T., Prog. Theor. Phys., 1975, 54, 348 doi:10.1143/PTP.54.348. 11. Vakarchuk I.A., Yukhnovskii I.R., Theor. Math. Phys., 1974, 18, 63; doi:10.1007/BF01036928. 12. Vakarchuk I.A., Yukhnovskii I.R., Theor. Math. Phys., 1979, 40, 100; doi:10.1007/BF01019246. 13. Vakarchuk I.A., Yukhnovskii I.R., Theor. Math. Phys., 1980, 42, 73; doi:10.1007/BF01019263. 14. Grest G.S., Rajagopal A.K, Phys. Rev. A., 1974, 10, 1395; doi:10.1103/PhysRevA.10.1395. 15. Grest G.S., Rajagopal A.K, Phys. Rev. A., 1974, 10, 1837; doi:10.1103/PhysRevA.10.1837. 16. Sasaki S., Matsuda K., Prog. Theor. Phys., 1976, 56, 375; doi:10.1143/PTP.56.375. 17. Tserkovnikov Yu.A., Theor. Math. Phys., 1976, 26, 50; doi:10.1007/BF01038256. 18. Vakarchuk I.A., Glushak P.A., Theor. Math. Phys., 1988, 75, 399; doi:10.1007/BF01017174. 19. Vakarchuk I.A., Theor. Math. Phys., 1989, 80, 983; doi:10.1007/BF01016193. 20. Vakarchuk I.A., Theor. Math. Phys., 1990, 82, 308; doi:10.1007/BF01029225. 21. Kempf A., J. Math. Phys., 1994, 35, 4483; doi:10.1063/1.530798. 22. Kempf A., Mangano G., Mann R.B., Phys. Rev. D, 1995, 52, 1108; doi:10.1103/PhysRevD.52.1108. 23. Quesne C., Tkachuk V.M., J. Phys. A: Math. Gen., 2004, 37, 10095; doi:10.1088/0305-4470/37/43/006. 24. Quesne C., Tkachuk V.M., J. Phys. A: Math. Gen., 2005, 38, 1747; doi:10.1088/0305-4470/38/8/011. 25. Pedram P., Amirfakhrian M., Shababi H., Preprint arXiv:1412.1425v2, 2015. 26. Buisseret F., Phys. Rev. A, 2010, 82, 062102; doi:10.1103/PhysRevA.82.062102. 27. Quesne C., Tkachuk V.M., Phys. Rev. A, 2010, 81, 012106; doi:10.1103/PhysRevA.81.012106. 28. Nowicki A., Tkachuk V.M., J. Phys. A: Math. Theor., 2014, 47, 025207; doi:10.1088/1751-8113/47/2/025207. 29. Camacho A., Int. J. Mod. Phys. D, 2003, 12, 1687; doi:10.1142/S0218271803004341. 30. Camacho A., Gen. Relat. Gravit., 2003, 35, 1153, doi:10.1023/A:1024437522212. 31. Vakarchuk I.O., Diakiv Iu.M., J. Phys. Stud., 2014, 18, 2004–1 (in Ukrainian). 32. Vakarchuk I.O., Condens. Matter Phys., 2008, 11, 409; doi:10.5488/CMP.11.3.409. 33. Vakarchuk I.O., J. Phys. A: Math. Theor., 2008, 41, 185402; doi:10.1088/1751-8113/41/18/185402. 34. Monteiro M.R., Rodrigues L.M.C.S., Wulck S., Phys. Rev. Lett., 1996, 76, 1098; doi:10.1103/PhysRevLett.76.1098. 35. Monteiro M.R., Rodrigues L.M.C.S., Physica A, 1998, 259, 245; doi:10.1016/S0378-4371(97)00633-X. 36. Mizrahi S.S., Camargo Lima J.P., Dodonov V.V., J. Phys. A: Math. Gen., 2004, 37, 11, 3707; doi:10.1088/0305-4470/37/11/012. 37. Gavrilik A.M., Mishchenko Yu.M., Nucl. Phys. B, 2015, 891, 466; doi:10.1016/j.nuclphysb.2014.12.017. 38. Lavagno A., Swamy P.N., Int. J. Mod. Phys. B, 2009, 23, 235; doi:10.1142/S0217979209049723. 39. Lavagno A., Swamy P.N., Found. Phys., 2010, 40, No. 7, 814; doi:10.1007/s10701-009-9363-0. 40. Fityo T.V., Phys. Lett. A, 2008, 372, No. 37, 5872; doi:10.1016/j.physleta.2008.07.047. 41. Xiu-Ming Z., Chi T., Chinese Phys. Lett., 2015, 32, 010303; doi:10.1088/0256-307X/32/1/010303. 42. Vakarchuk I.O., Ukr. J. Phys., 2010, 55, No. 1, 36–43. 33002-13 http://dx.doi.org/10.1103/PhysRev.82.625 http://dx.doi.org/10.1103/PhysRev.85.338 http://dx.doi.org/10.1103/PhysRev.92.609 http://dx.doi.org/10.1143/PTP.21.327 http://dx.doi.org/10.1143/PTP.54.308 http://dx.doi.org/10.1143/PTP.41.919 http://dx.doi.org/10.1143/PTP.54.348 http://dx.doi.org/10.1007/BF01036928 http://dx.doi.org/10.1007/BF01019246 http://dx.doi.org/10.1007/BF01019263 http://dx.doi.org/10.1103/PhysRevA.10.1395 http://dx.doi.org/10.1103/PhysRevA.10.1837 http://dx.doi.org/10.1143/PTP.56.375 http://dx.doi.org/10.1007/BF01038256 http://dx.doi.org/10.1007/BF01017174 http://dx.doi.org/10.1007/BF01016193 http://dx.doi.org/10.1007/BF01029225 http://dx.doi.org/10.1063/1.530798 http://dx.doi.org/10.1103/PhysRevD.52.1108 http://dx.doi.org/10.1088/0305-4470/37/43/006 http://dx.doi.org/10.1088/0305-4470/38/8/011 http://arxiv.org/abs/1412.1425v2 http://dx.doi.org/10.1103/PhysRevA.82.062102 http://dx.doi.org/10.1103/PhysRevA.81.012106 http://dx.doi.org/10.1088/1751-8113/47/2/025207 http://dx.doi.org/10.1142/S0218271803004341 http://dx.doi.org/10.1023/A:1024437522212 http://dx.doi.org/10.5488/CMP.11.3.409 http://dx.doi.org/10.1088/1751-8113/41/18/185402 http://dx.doi.org/10.1103/PhysRevLett.76.1098 http://dx.doi.org/10.1016/S0378-4371(97)00633-X http://dx.doi.org/10.1088/0305-4470/37/11/012 http://dx.doi.org/10.1016/j.nuclphysb.2014.12.017 http://dx.doi.org/10.1142/S0217979209049723 http://dx.doi.org/10.1007/s10701-009-9363-0 http://dx.doi.org/10.1016/j.physleta.2008.07.047 http://dx.doi.org/10.1088/0256-307X/32/1/010303 I.O. Vakarchuk,G.I. Panochko 43. Landau L.D., Lifshiz E.M., Quantum Mechanics, Nauka, Moscow, 1974 (in Russian). 44. Bogoliubov N., J. Phys. (USSR), 1947, 9, 23. 45. Donnelly R.J., Donnelly J. A., Hills R.N., J. Low Temp. Phys., 1981, 44, 471; doi:10.1007/BF00117839. 46. Cowley R.A., Woods A.D.B., Can. J. Phys., 1971, 49, 177; doi:10.1139/p71-021. 47. Vakarchuk I.O., Introduction into the Many-Body Problem, Ivan Franko National University of Lviv, Lviv, 1999 (in Ukrainian). 48. Rovenchak A., Z. Naturforsch. A, 2015, 70, 73; doi:10.1515/zna-2014-0211. Теорiя багатобозонної системи з деформованою алгеброю Гайзенберга I.О. Вакарчук1, Г.I. Паночко2 1 Кафедра теоретичної фiзики, Львiвський нацiональний унiверситет iменi Iвана Франка, вул. Драгоманова 12, Львiв, 79005, Україна 2 Львiвський нацiональний унiверситет iменi Iвана Франка, Природничий коледж, вул. Тарнавського 107, Львiв, 79010, Україна Запропоновано врахувати нелiнiйнi флуктуацiї в теорiї рiдкого 4He, деформуючи комутацiйнi спiввiдно- шення мiж узагальненими координатами та iмпульсами. В якостi узагальнених координат обрано коефi- цiєнти флуктуацiї густини бозе-частинок. Параметр деформацiї, що враховує вплив три- та чотиричастин- кових кореляцiй на поведiнку бозе-систем, обрано виходячи з експериментальних значень для спектра елементарних збуджень та екстрапольованих експериментальних даних структурного фактора до тем- ператури T = 0 K. З модельним параметром деформацiї проведено чисельну оцiнку енергiї основного стану та кiлькостi Бозе-конденсату, вiдтворено спектр елементарних збуджень та потенцiал взаємодiї мiж атомами 4He. Ключовi слова: деформована алгебра Гайзенберга, спектр елементарних збуджень рiдкого 4He, Бозе-конденсат 33002-14 http://dx.doi.org/10.1007/BF00117839 http://dx.doi.org/10.1139/p71-021 http://dx.doi.org/10.1515/zna-2014-0211 Introduction Hamiltonian of Bose-liquid in the deformed space of collective variables Energy levels and wave functions of a Bose liquid Ground-state energy Elementary excitation spectrum Structure factor Potential and kinetic energy Momentum distribution Deformation parameter. Numerical calculations

The theory of a many boson system with the deformed Heisenberg algebra

Institution

Ähnliche Einträge