We consider a stability property in a Cauchy problem for a fractional Schrödinger equation with the principal operator $i\partial_{t}^{1/2}+\partial_{x}^{2}$ in a bounded interval $(0,L)$. To achieve our purpose, we first introduce a suitable transformation in order to convert the fractional operator $i\partial_{t}^{1/2}+\partial_{x}^{2}$ to a fourth order partial differential operator $\partial_{t}+\partial_{x}^{4}$. Next we apply the Carleman estimate to prove the conditional stability with data $z(t,0), \partial_{x}z(t,0), 0 < t < T$.