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A traceless Hermitian matrix defined on the unit sphere S^2 with control parameters is called a semi-quantum system, where S^2 is viewed as a set of classical variables associated with rotation and the Hermitian matrix as a quantum variable (or an operator). This Hermitian matrix is called a Hamiltonian, the size of which is related to the number of energy levels. Two- or three-level semi-quantum system means that the Hamiltonian in question is put in 2×2 or 3×3 matrix form. The Hamiltonian is assumed to be invariant under the action of a discrete subgroup of the rotation group SO(3). The D_3 and O groups are known to be discrete subgroups of SO(3), which are isomorphic with symmetric groups S_3 and S_4, respectively. A choice of discrete groups depends on molecules (or many-particle systems) which one is interested in. Accompanying with a change in control parameters, the energy levels (or eigenvalues of the Hamiltonian) undergoes a change, which occurs through level crossing, or through the degeneration of eigenvalues.
If all eigenvalues are distinct throughout S^2, the eigenspace of an eigenvalue is assigned on each point of S^2 to form a complex line bundle and the first Chern number is attached. The first Chern numbers of respective complex line bundles will change accompanying with the change in control parameters. The amount of the change in Chern numbers is closely related with the symmetry of the semi-quantum system in question, on which this talk's interest centers. Explicit examples are given to show change in Chern numbers.


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