Abstract
The super-rotor model is discussed in more detail. The respective states of SO(5) symmetry are discussed by relating the model to well-known models of nuclear theories. Furthermore, branching rules for the correlation of these states to well-known states with an ordinary angular momentum quantum number are discussed. These rules follow from the fact that the symmetry group SO(5) incorporates SO(3), the group of three-dimensional rotations, as a proper subgroup. Furthermore, a new technique to find equivalent rotations of permutation/inversion operations is discussed. With this generalization of the well-known Longuett–Higgins method, also the limits, which were discussed in Chap. 6 are overcome. It is now possible to map the elements of the permutation group of five identical particles to rotations in five dimensions. With this, Pauli’s principle can also be applied to the super-rotor states.
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Notes
- 1.
Both subgroups are different and so are the branchings. This suggests that the decoupling of the super-rotation into internal and overall rotation is somewhat different to the treatment in terms of one single rotating object. We discuss this feature in Chap. 10. For now, the SO(3) subgroup is used to express the SO(5) states in a known basis of rigid rotor functions.
- 2.
The dimension 3N is indeed only odd, if the number of nuclei is odd and is even if N is even, respectively.
- 3.
One could think that the permutation group of five particles is embedded in the five dimensional rotation group due to the number of particles. This is not the case, the group \({\varvec{S}}_4\) is, e.g., isomorphic to a subgroup of SO(3). The number five appears coincidentally in both groups.
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Schmiedt, H. (2017). Super-Rotor States and Their Symmetry. In: Molecular Symmetry, Super-Rotation, and Semiclassical Motion. Springer Series on Atomic, Optical, and Plasma Physics, vol 97. Springer, Cham. https://doi.org/10.1007/978-3-319-66071-4_8
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DOI: https://doi.org/10.1007/978-3-319-66071-4_8
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