(but not all) are rather similar for the two interactions. Since the best milieu for the existence of scissors mode excitations (orbital magnetic dipole excitations) are strongly deformed systems, one would expect the $Q \cdot Q$ Hamiltonian to yield strong scissors modes. But is this also true for the realistic interaction ? We will address this question. Another motivation for introducing the $Q \cdot Q$ Hamiltonian is that it is easy to establish a connection via energy weighted sum rule techniques between isovector orbital $B(M1)$’s and isoscalar and isovector $B(E2)$’s.

We shall be performing the calculations, not only in the $0\hbar\omega$ space (small space) but also in a space which allows $2\hbar\omega$ admixtures (large space). For the $Q \cdot Q$ Hamiltonian in the small space the energy matrix is proportional to $\chi$. Hence the energy eigenvalues depend linearly on $\chi$, but the eigenfunctions (and $B(M1)$’s and $B(E2)$’s) are independent of the interaction strength. In a large space calculation there is one more parameter: the energy splitting induced by $\frac{p^2}{2m} + \frac{1}{2}m\omega^2r^2$ i.e. $2\hbar\omega$. Thus the wave function and the corresponding $B(M1)$’s and $B(E2)$’s will also depend on $\chi$.

We have chosen values of $\chi$ appropriate for the large space calculation. 