  We consider the problem of the evolution of the fine structure coefficient
alpha under the assumption that the scalar field coupling to the Maxwell term
satisfies the condition mt>>1 for coherent dark matter oscillations. In this
case we find that the coupling scale f in the leading order coupling
-(phi/4f)F^2 affects the cosmological evolution of alpha according to
ln(alpha/alpha_0) xi(m_{Pl}/f)ln(tanh(t/2 tau)/tanh(t_0/2 tau)). A fit to the
QSO observations by Murphy et al. yields f/xi= 2.12^{+0.58}_{-0.37} 10^5m_{Pl}.
Here m_{Pl} is the reduced Planck mass, and xi^2=rho_phi/rho_m parametrizes the
contribution of phi to the matter density in the universe.
