user:Nlaskin

Path integral over the

Lévy-like quantum mechanical paths allows one to develop a

generalization of quantum mechanicsN. Laskin, Phys. Rev. E 62, 3135, 2000; namely, if the path

integral over Brownian trajectories leads to the well known

Schrödinger equation, then the path integral over Lévy trajectories

leads to the fractional Schrödinger equation. The

fractional Schrödinger equation includes a space derivative

of order ᾳ instead of the second order (ᾳ=2) space derivative

in the standard Schrödinger equation. Thus, the fractional

Schrödinger equation is a fractional differential equation

in accordance with modern terminology.

This is the main point of the term fractional

Schrödinger equation or of the more general term

fractional quantum mechanics. As mentioned

above, at ᾳ=2 the Lévy motion becomes Brownian motion.

Thus, fractional quantum mechanics includes the standard quantum mechanics as a particular Gaussian

case at ᾳ=2. The quantum mechanical path integral over

the Lévy paths at ᾳ=2 becomes the well known Feynman

path integral.