Quantum systems with finite Hilbert space and Chebyshev polynomials

Finite quantum systems are considered and the Heisenberg–Weyl group of discrete displacements in the Z(d)×Z(d) phase space is studied. Matrix elements of various operators are calculated and the result is given in terms of Chebyshev polynomials and their derivatives. The SL(2,Z(d)) group of transformations in the phase space is studied. The general theory is applied in the context of spherical harmonics and provides a mathematical framework for the study of the angle–angular momentum quantum phase space.