Quantum and classical analyses of Hamiltonian 
constructed from Casimir operators of Lie algebras

The algebraic method for molecules is known as a technique 
to formulate and to solve a model of molecular vibration. 
This method enables us to construct a so-called algebraic 
Hamiltonian from a Lie algebra and to calculate its spectrum, 
which can be compared with experimentally measured spectrum. 
Up to the present, it has been known that an algebraic 
Hamiltonian based on the su(2) algebra contains the Morse 
oscillator. But the contents of systems based on more general 
algebras like su(3) and su(4) are not yet uncovered.

In this master thesis, properties of the dynamical systems 
based on su(2), su(3) and su(4) are examined from both the 
viewpoint of quantum and classical mechanics. Since the 
algebraic Hamiltonian has some symmetry by its definition, 
separation of variables is applicable in the context of 
quantum mechanics. On the other hand, in the context of
classical mechanics, the Marsden-Weinstein method, which 
reduces the degrees of freedom of dynamical system with 
symmetry, is applicable. As the result, in both contexts, 
it is shown that the Morse oscillator is induced from the 
original algebraic systems. It is also shown that the Morse 
oscillators induced from the algebraic systems have different 
inner product structure for probability interpretation or 
different metric structure from the ordinary Morse oscillator.