User:Nlaskin

Fractional Quantum Mechanics

Path integral over the Lévy-like quantum mechanical paths allows one to develop a generalization of quantum mechanics; 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.