Calculation of four-particle 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. Translational invariance of a wave function means that it is dependent only on instrinsic degrees of freedom of the system. The traditional applications, such as the nuclear shell model, however, are based on a model Hamiltonian with individual one-particle variables. These wave functions are very attractive because they allow employing a simple procedure of antisymmetrization. Hence, the corresponding model wave functions, which are dependent on one-particle coordinates, are not translationally invariant and cannot represent the wave function of such a system in a proper way because the centre of mass of a free nucleus (or a free hadron) should be described by the exponential function corresponding to a freely moving point mass. This should not be a problem in the case when the expression for the realistic wave function in a harmonic-oscillator (HO) basis is known. 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 centre of mass coordinate [1–3]. In this case, the HO basis set in terms of intrinsic (Jacobi) coordinates is necessary. By a set of Jacobian coordinates for a system of N particles we have in mind the N − 1 independent vectors each representing the displacement of the centre of mass of two different subsystems. For N > 2, there exist more than one set of Jacobi coordinates that can be assigned to the N particle system. In general, when the transformation from one set of Jacobi coordinates to another is performed, one obtains an expression for the wave function containing an infinite number of terms. Only the set of HO functions can be chosen so that the transformation from one set of Jacobi coordinates to another results in a corresponding expansion with a finite number of terms. In this approach, the essential feature is the Talmi–Moshinsky transformation [4–7] and corresponding HO brackets. Since HO brackets are constantly employed in various model calculations of nuclear and hadron structure, it is desirable to have a simple and efficient method to calculate them. Many papers have been devoted to the study of the Talmi–Moshinsky transformation brackets, and various methods for the calculation of these brackets and several explicit expressions for them are described in [8–10]. In the latter papers three-particle Talmi–Moshinsky transformations are presented. The goal of the current study is to work out a simple and efficient method of the calculation of four-particle HO transformation brackets.