Quantum mechanics on a circle with a finite number of α-uniformly distributed points

In this paper, quantum mechanics on a circle with finite number of α-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 α-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 α-distributed discrete angles.