  In this paper, quantum mechanics on a circle with finite number of
{\alpha}-uniformly distributed points is discussed. The angle operator and
translation operator are defined. Using discrete angle representation, two
types of discrete angular momentum operators and Hermitian Hamiltonian on a
circle with d {\alpha}-distributed discrete angles are constructed. The energy
levels are computed for a free particle on a circle where the wave function is
defined in the d {\alpha}-distributed discrete angles.
