THE ONE-DIMENSIONAL FRACTIONAL QUANTUM OPERATOR OF MOMENTUM
P. Miškinis
Department of Physics, Vilnius Gediminas Technical University, Saulėtekio 11, LT-10223 Vilnius, Lithuania
E-mail: paulius.miskinis@fm.vtu.lt

Received 7 April 2006

In the case of the quantum generalization of Lévy processes, expressions for the Hermitian operator of momentum and its eigenfunctions are proposed. The normalization constant has been determined and its relation to the translation operator is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality α.

Keywords: Lévy processes, quantum mechanics, fractional calculus

VIENMATIS TRUPMENINIS KVANTINIS JUDESIO KIEKIO OPERATORIUS
P. Miškinis
Vilniaus Gedimino technikos universitetas, Vilnius, Lietuva

Kvantinio Lévy (Levi) vyksmo atveju pasiūlyta ermitinio vienmačio judesio kiekio operatoriaus išraiška
$\begin{equation*} \hat{p}=\,\hbar l^{\alpha-1}_0 / 2 \cdot \big [(-\mathrm{i}\partial^{}_{+})^{\alpha}+(\mathrm{i} \tilde{\partial}^{}_{-})^{\alpha}\big ] \,, \end{equation*}$
rastos jo tikrinės funkcijos bei tikrinės vertės$\begin{equation*} \psi(x,t)=A\,\mathrm{e}^{\mathrm{i}\kappa\,x-\mathrm{i} \, E/\,\hbar \, t}\,,\, \quad \kappa= \big [p/(\hbar l^{\alpha-1}_0) \big ]^{1/\alpha}\,. \end{equation*}$
$\begin{equation*} \psi(x,t)=A\,\mathrm{e}^{\mathrm{i}\kappa\,x-\mathrm{i} \, E/\,\hbar \, t}\,,\, \quad \kappa= \big [\,p/\,(\hbar l^{\alpha-1}_0) \big ]^{1/\alpha}\,. \end{equation*}$
Išreikštu pavidalu rasta normavimo konstanta
$\begin{equation*} A = \sqrt {\left| \,\kappa \,\right|/(2\pi\! \left|\,p \,\right|)} \, \rightarrow 1/\sqrt{2 \pi \,\hbar }\,,\quad \mathrm{kai}\quad {\alpha \to 1} \, , \end{equation*}$
ir poslinkio operatoriaus išraiška. Sąryšis tarp judesio kiekio ir banginio skaičiaus
$\begin{equation*} p = \,\hbar l_0^{\alpha - 1} \kappa ^\alpha \end{equation*}$
apibendrintas vyksmams, turintiems trupmeninį Lévy indeksą.