New Method for Constructing the Oscillator Functions of a Quantum System of Identical Particles in Symmetrized Coordinates

The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and affected by the external field of a target, is formulated in the new symmetrized coordinates. A new method and symbolic algorithm for generating (A − 1)-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles, is elaborated and implemented using the MAPLE computer algebra system. Examples of generating the symmetrized coordinate representation for composite systems of several identical particles in one-dimensional Euclidean space are given and their symmetry properties are analyzed. The systems composed from three to six particles in one dimensional Euclidean space were analyzed a correspondence between the representations of the symmetry groups D3 and Td for A = 3 and A = 4 and symmetric or antisymmetric oscillator functions was found. It is shown that the transformations of (A− 1)-dimensional oscillator functions from the symmetrized coordinates to the Jacobi coordinates, reducible to permutations of coordinates and (A − 1)-dimensional finite rotation, are implemented by means of the (A − 1)-dimensional oscillator Wigner functions. The examples of construction of the symmetric or antisymmetric oscillator functions in closed analytical form by means of mathematical induction and the algorithm are given. The approach is aimed at solving the problem of tunnelling the clusters, consisting of several identical particles, through repulsive potential barriers of a target.

method and algorithm, identical particles, symmetric or antisymmetric oscillator functions, symmetrized coordinates