  We discuss some of the mathematical properties of the fractional derivative
defined by means of Fourier transforms. We first consider its action on the set
of test functions $\Sc(\mathbb R)$, and then we extend it to its dual set,
$\Sc'(\mathbb R)$, the set of tempered distributions, provided they satisfy
some mild conditions. We discuss some examples, and we show how our definition
can be used in a quantum mechanical context.
