  The wave functions for two dimensional Bloch electrons in a uniform magnetic
field at the mid-band points are studied with the help of the algebraic
structure of the quantum group $U_q(sl_2)$. A linear combination of its
generators gives the Hamiltonian. We obtain analytical and numerical solutions
for the wave functions by solving the Bethe Ansatz equations, proposed by
Wiegmann and Zabrodin on the basis of above observation. The semi-classical
case with the flux per plaquette $\phi=1/Q$ is analyzed in detail, by exploring
a structure of the Bethe Ansatz equations. We also reveal the multifractal
structure of the Bethe Ansatz solutions and corresponding wave functions when
$\phi$ is irrational, such as the golden or silver mean.
