Gruppentheoretische Untersuchungen zum Schalenmodell

Abstract

The harmonic oscillator shell model with LS-coupling is studied from a group-theoretical point of view. Leaving aside for the moment the spin and isospin degrees of freedom the most general transformation group, which leaves the hamiltonian invariant, is determined to be the unitary group in3 A dimensions ($\left( {\mathfrak{U}_{3A} } \right)$), whereA is the nucleon number. The following chain of subgroups is considered:

$$\mathfrak{U}_{3A} > \mathfrak{U}_3 \times \mathfrak{U}_A > \mathfrak{D}_3^{( + )} \times \mathfrak{U}_A > \mathfrak{D}_3^{( + )} \times \mathfrak{S}_A .$$

($\mathfrak{U}_3$ resp.$\mathfrak{U}_A$ unitary group in 3 resp.A dimensions, × means direct product,$\mathfrak{D}_3^{( + )}$ rotation group in 3 dimensions,$\mathfrak{S}_A$ group of all permutations of theA nucleons). The classification of the wavefunctions with respect to energy is equivalent to a classification according to irreducible representations of$\mathfrak{U}_{3A}$. Therefore we can classify the wavefunctions in each energy level with respect to irreducible representations of$\mathfrak{U}_3 , \mathfrak{D}_3^{( + )}$ and$\mathfrak{S}_A$ by studying the laws of decomposition of the irreducible representations of$\mathfrak{U}_{3A}$ into those of the subgroups. Such a classification is of physical importance in connection with the Elliot model. The last step going from$\mathfrak{D}_3^{( + )} \times \mathfrak{U}_A to \mathfrak{D}_3^{( + )} \times \mathfrak{S}_A$ leads in a natural way to the problem of center-of-mass motion and one obtaines a clear separation of the totality of wave-functions into “good” states and “spurious” states. The most appropriate mathematical tool to deal with this question is the theory of “inner plethysm”.