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 \gg 1 for coherent dark matter oscillations . In this case we find that the coupling scale f 
in the leading order coupling - ( \phi / 4 f ) F _ { \mu \nu } F ^ { \mu \nu } 
affects the cosmological evolution of \alpha according to \ln ( \alpha / \alpha _ { 0 } ) \propto \xi ( m _ { Pl } / f ) \times \ln ( \tanh ( t / 2 \tau ) / \tanh ( t _ { 0 } % / 2 \tau ) ) .
A fit to the QSO observations by Murphy et al .
yields f = \xi \times 2.12 ^ { +0.58 } _ { -0.37 } \times 10 ^ { 5 } m _ { Pl } .
Here m _ { Pl } = ( 8 \pi G _ { N } ) ^ { -1 / 2 } is the reduced Planck mass , and \xi ^ { 2 } = \varrho _ { \phi } / \varrho _ { m } parametrizes the contribution of \phi to the matter density in the universe .