Efective calculation of harmonic oscillator transformation brackets

The wave function of a self-bound system in the absence of external fields must be invariant in respect of spatial translations as well as antisymmetric in respect of all permutations of the fermions. One of the possibilities to find a solution to the problem of translational invariance of any wave function is based on direct construction of the many-fermion wave function, which is independent of the center of mass coordinate [1]. In this case, the harmonic-oscillator (HO) basis set in terms of intrinsic (Jacobi) coordinates is necessary. Only the set of HO functions can be chosen in such a manner that the transformation from one set of Jacobi coordinates to other results in a corresponding expansion with a finite number of terms. In this approach, the essential feature is the Talmi-Moshinsky transformation [2] and corresponding HO brackets. Since HO brackets are constantly employed in various model calculations of the nuclear and hadron structure, it is desirable to have a simple and efficient method to calculate them [3]Fizikos katedraVytauto Didžiojo universiteta