  We investigate the constraints that OH microwave transitions in megamasers
and molecular absorbers at cosmological distances may place on the evolution of
the fine structure constant alpha=e^2/(hbar c). The centimeter OH transitions
are a combination of hyperfine splitting and lambda-doubling that can constrain
the cosmic evolution of alpha from a single species, avoiding systematic errors
in alpha measurements from multiple species which may have relative velocity
offsets. The most promising method compares the 18 and 6 cm OH lines, includes
a calibration of systematic errors, and offers multiple determinations of alpha
in a single object. Comparisons of OH lines to the HI 21 cm line and CO
rotational transitions also show promise.
