6533b82bfe1ef96bd128d806
RESEARCH PRODUCT
On the discreet spectrum of fractional quantum hydrogen atom in two dimensions
V. A. Stephanovichsubject
fractional Schrödinger equationFOS: Physical sciencesPosition and momentum space01 natural sciences010305 fluids & plasmasSchrödinger equationMomentumsymbols.namesakeKepler problem0103 physical sciencesPrincipal quantum number010306 general physicsCondensed Matter - Statistical MechanicsMathematical PhysicsMathematical physicsPhysicsQuantum PhysicsStatistical Mechanics (cond-mat.stat-mech)fractional statisticsSpectrum (functional analysis)Mathematical Physics (math-ph)Hydrogen atomCondensed Matter PhysicsAtomic and Molecular Physics and OpticsVariational methodsymbolsQuantum Physics (quant-ph)hydrogenic problemsdescription
We consider a fractional generalization of two-dimensional (2D) quantum-mechanical Kepler problem corresponding to 2D hydrogen atom. Our main finding is that the solution for discreet spectrum exists only for $\mu>1$ (more specifically $1 < \mu \leq 2$, where $\mu=2$ corresponds to "ordinary" 2D hydrogenic problem), where $\mu$ is the L\'evy index. We show also that in fractional 2D hydrogen atom, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number $n$ but also on orbital $m$. To solve the spectral problem, we pass to the momentum representation, where we apply the variational method. This permits to obtain approximate analytical expressions for eigenvalues and eigenfunctions with very good accuracy. Latter fact has been checked by numerical solution of the problem. We also found the new integral representation (in terms of complete elliptic integrals) of Schr\"odinger equation for fractional hydrogen atom in momentum space. We point to the realistic physical systems like bulk semiconductors as well as their heterostructures, where obtained results can be used.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2019-01-01 | Physica Scripta |