  We construct a classical action for a system of $N$ point-like sources which
carry SU(2) non-Abelian charges coupled to non-Abelian Chern-Simons gauge
fields, and develop a quantum mechanics for them. Adopting the coherent state
quantization and solving the Gauss' constraint in an appropriately chosen
gauge, we obtain a quantum mechanical Hamiltonian given in terms of the
Knizhnik-Zamolodchikov connection. Then we study the non-Abelian Aharonov-Bohm
effect, employing the obtained Hamiltonian for two-particle sector. An explicit
evaluation of the differential cross section for the non-Abelian Aharonov-Bohm
scattering is given.
