  Fractional evolution equations lack generally accessible and well-converged
codes excepting anomalous diffusion. A particular equation of strong interest
to the growing intersection of applied mathematics and quantum information
science and technology is the fractional Schr\"odinger equation, which
describes sub-and super-dispersive behavior of quantum wavefunctions induced by
multiscale media. We derive a computationally efficient sixth-order split-step
numerical method to converge the eigenfunctions of the FSE to arbitrary
numerical precision for arbitrary fractional order derivative. We demonstrate
applications of this code to machine precision for classic quantum problems
such as the finite well and harmonic oscillator, which take surprising twists
due to the non-local nature of the fractional derivative. For example, the
evanescent wave tails in the finite well take a Mittag-Leffer-like form which
decay much slower than the well-known exponential from integer-order derivative
wave theories, enhancing penetration into the barrier and therefore quantum
tunneling rates. We call this effect \emph{fractionally enhanced quantum
tunneling}. This work includes an open source code for communities from quantum
experimentalists to applied mathematicians to easily and efficiently explore
the solutions of the fractional Schr\"odinger equation in a wide variety of
practical potentials for potential realization in quantum tunneling enhancement
and other quantum applications.
