  We derive raising and lowering operators for orthogonal polynomials on the
unit circle and find second order differential and $q$-difference equations for
these polynomials. A general functional equation is found which allows one to
relate the zeros of the orthogonal polynomials to the stationary values of an
explicit quasi-energy and implies recurrences on the orthogonal polynomial
coefficients. We also evaluate the discriminants and quantized discriminants of
polynomials orthogonal on the unit circle.
