I read about Fractional Quantum Mechanics and it seemed interesting. But are there any justifications for this concept, such as some connection to reality, or other physical motivations, apart from the pure mathematical insight?

If there are none, why did anyone even bother to invent it?

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    $\begingroup$ Interesting question. Looking here it seems like you can derive generalizations of the usual formulas which look the same but with the Levy parameter in them (like the "fractional Bohr atom"), you get the usual answer by putting the parameter to 2. But, as you say....why? $\endgroup$ – twistor59 Nov 26 '12 at 12:37
  • $\begingroup$ Related: physics.stackexchange.com/q/4005/2451 and links therein. $\endgroup$ – Qmechanic Nov 27 '12 at 0:12

It seems the goal here is to be able to explain all kind of phenomena considering complex situations, in which nonlinearity could be infeasible to handle as it happens in non-quantic systems. According to this reference, the fractional Schrödinger equation

$$i\hbar\dfrac{\partial\Psi(\vec{x},t)}{\partial t}=-[D_{\alpha}(\hbar\nabla)^{\alpha}+V(\vec{x},t)]\Psi(\vec{x},t)$$

where $(\hbar\nabla)^{\alpha}$ is the quantum Riesz fractional derivative

$$(-\hbar ^2\Delta )^{\alpha /2}\Psi (\vec{x},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i\frac{\vec{p}\cdot\vec{x}}\hbar }|\mathbf{p}|^\alpha \varphi ( \vec{p},t)$$

still corresponds/represents quantic systems. For instance, Laskin shows that uncertainty (fractal) it does exist, because

$$\langle|\Delta x|^\mu\rangle^{1/{\mu}}\cdot\langle|\Delta p|^\mu\rangle^{1/{\mu}}>\dfrac{\hbar}{(2\alpha)^{1/{\mu}}}$$

for $\mu<\alpha$ and $1<\alpha\leq 2$.

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    $\begingroup$ quantic? You do mean quantum, right? I can see this equations, but I find no relating of physics with these mathematics up there. I found Qmechanic's link more enlightening. Check it out, its not bad. $\endgroup$ – resgh Dec 7 '12 at 11:35

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