  The redistribution of energy levels between energy bands is studied for a
family of simple effective Hamiltonians depending on one control parameter and
possessing axial symmetry and energy-reflection symmetry. Further study is made
on the topological phase transition in the corresponding semi-quantum and
completely classical models, and finally the joint spectrum of the two
commuting observables $(H=E,J_z)$ (also called the lattice of quantum states)
is superposed on the image of the energy-momentum map for the classical model.
Through these comparative analyses, mutual correspondence is demonstrated to
exist among the redistribution of energy levels between energy bands for the
quantum Hamiltonian, the modification of Chern numbers of eigenline bundles for
the corresponding semi-quantum Hamiltonian, and the presence of Hamiltonian
monodromy for the complete classical analog. In particular, as far as the band
rearrangement is concerned, a fine agreement is found between the
redistribution of the energy levels described in terms of joint spectrum of
energy and momentum in the full quantum model and the evolution of
singularities of the energy-momentum map of the complete classical model. The
topological phase transition observed in the present semi-quantum and the
complete classical models are analogous to topological phase transitions of
matter.
