Anomalous quantum mechanics

Abstract: 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.