原子核原子分子物理学中的群论方法

The representation theory of group and algebra has been one of the most useful mathematical tools studying the modern physics. In the present project, first of all, the analytic expressions for irreducible representations of some simple Lie algebras were studied by making use of the irreducible tensor basis method, together with Racah coefficients, branching rules, and reduction factors with respect to the corresponding group chains; On the base of the above results, a MATHEMATICA software package was designed. Second, two kinds of nonlinear algebras, polynomial deformations of Lie algebra O(3) and algebras of square-root type, were studied. The relationship of square-root algebras and ordinary classical Lie algebras were revealed for the first time, and various representations including irreducible representations, indecomposable representations, (multi-) boson and (multi-variable) differential realizations of the square-root algebras were discussed in detail. Finally, we investigated (nonlinear) dynamical symmetries and supersymmetries of simple solvable quantum mechanical systems, and some applications of Lie algebras and Lie superalgebras to the algebraic models of atomic and nuclear structure. The significative results we obtained in this project are very useful for studying nuclear, atomic and molecular physics.

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