  In this paper we construct a non-commutative version of the Hopf bundle by
making use of Jaynes-Commings model and so-called Quantum Diagonalization
Method. The bundle has a kind of Dirac strings. However, they appear in only
states containing the ground one (${\cal F}\times \{\ket{0}\} \cup
\{\ket{0}\}\times {\cal F} \subset {\cal F}\times {\cal F}$) and don't appear
in remaining excited states. This means that classical singularities are not
universal in the process of non-commutativization.
  Based on this construction we moreover give a non-commutative version of both
the Veronese mapping which is the mapping from $\fukuso P^{1}$ to $\fukuso
P^{n}$ with mapping degree $n$ and the spin representation of the group SU(2).
  We also present some challenging problems concerning how classical
(beautiful) properties can be extended to the non-commutative case.
