Wave Functions of Condensed States

Alexander Yu. Cherny

Pair wave functions (PWF) which are eigenfunctions of the reduced density 2-matrix are considered for a homogeneous Bose liquid. With the Bogoliubov principle of the correlation weakening it is proved that the distribution of the "dissociated" pair states over momenta is exactly the product of the single-particle distribution functions. Thus, the "dissociated" pair states are naturally classified as condensate-condensate, condensate-supracondensate and supracondensate-supracondensate ones provided the Bose-Einstein condensate exists. The condensate-condensate as well as condensate-supracondensate PWF are expressed in terms of the averages of products of the creation and destruction Bose operators. This leads to the simple interpretation of the anomalous averages as the "scattering parts" of the condensate-condensate and condensate-supracondensate PWF. It is shown that in contrast to the Fermi liquid, the appearance of the anomalous averages for the Bose liquid does not necessarily mean that there exist bound states of pairs of particles. The PWF in the Hartree-Fock-Bogoliubov (HFB) approach are found. Given the density of the condensate is not zero, there are no bound pair states in the HFB scheme. The expansion of the pair correlation function in the set of PWF is very useful in order to take into account both short-range and long-range spatial correlations. Applications (possible and already realized) of the formalism developed are discussed.

Detlev Buchholz A

In this article, a novel characterization of Bose–Einstein condensates is proposed. Instead of relying on occupation numbers of a few dominant modes, which become macroscopic in the limit of infinite particle numbers, it focuses on the regular excitations whose numbers stay bounded in this limit. In this manner, subspaces of global, respectively, local regular wave functions are identified. Their orthogonal complements determine the wave functions of particles forming proper (infinite) condensates in the limit. In contrast to the concept of macroscopic occupation numbers, which does not sharply fix the wave functions of condensates in the limit states, the notion of proper condensates is unambiguously defined. It is outlined how this concept can be used in the analysis of condensates in models. The method is illustrated by the example of trapped non-interacting ground states and their multifarious thermodynamic limits, differing by the structure of condensates accompanying the Fock vacuum. The concept of proper condensates is also compared with the Onsager–Penrose criterion based on the analysis of eigenvalues of one-particle density matrices. It is shown that the concept of regular wave functions is also useful in that approach for the identification of wave functions forming proper condensates.

Jens C. Zorn, E. A. Zorn

Detlev Buchholz A

In this article a novel characterization of Bose-Einstein condensates is proposed. Instead of relying on occupation numbers of a few dominant modes, which become macroscopic in the limit of infinite particle numbers, it focuses on the regular excitations whose numbers stay bounded in this limit. In this manner, subspaces of global, respectively local regular wave functions are identified. Their orthogonal complements determine the wave functions of particles forming proper (infinite) condensates in the limit. In contrast to the concept of macroscopic occupation numbers, which does not sharply fix the wave functions of condensates in the limit states, the notion of proper condensates is unambiguously defined. It is outlined, how this concept can be used in the analysis of condensates in models. The method is illustrated by the example of trapped non-interacting ground states and their multifarious thermodynamic limits, differing by the structure of condensates accompanying the Fock vacuum. The concept of proper condensates is also compared with the Onsager-Penrose criterion, based on the analysis of eigenvalues of one-particle density matrices. It is shown that the concept of regular wave functions is useful there as well for the identification of wave functions forming proper condensates.

Rhett J. Allain