  The fractional operators together with exponential quantum in coordinate and
momentum space corresponding to the power of observables are introduced. Based
on an exponential relation between energy and momentum, the fractional
Schr\"odinger equations for the free particle and the one in potential fields
in heterogeneous complex media are found. The fractional equation of motion and
the fractional virial theorem for anomalous quantum mechanics are then
developed. Applying the fractional virial theorem, we derive an anomalous
hydrogen atom whose transition energy values are much higher than that of Bohr
hydrogen atom. The anomalous Heisenberg picture being equivalent to the
fractional Schr\"odinger picture is also discussed.
