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R\'enyi entropies of the highly-excited states of multidimensional harmonic oscillators by use of strong Laguerre asymptotics
- Publication Year :
- 2016
-
Abstract
- The R\'enyi entropies $R_{p}[\rho]$, $p>0,\neq 1$ of the highly-excited quantum states of the $D$-dimensional isotropic harmonic oscillator are analytically determined by use of the strong asymptotics of the orthogonal polynomials which control the wavefunctions of these states, the Laguerre polynomials. This Rydberg energetic region is where the transition from classical to quantum correspondence takes place. We first realize that these entropies are closely connected to the entropic moments of the quantum-mechanical probability $\rho_n(\vec{r})$ density of the Rydberg wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-norms of the associated Laguerre polynomials. Then, we determine the asymptotics $n\to\infty$ of these norms by use of modern techniques of approximation theory based on the strong Laguerre asymptotics. Finally, we determine the dominant term of the R\'enyi entropies of the Rydberg states explicitly in terms of the hyperquantum numbers ($n,l$), the parameter order $p$ and the universe dimensionality $D$ for all possible cases $D\ge 1$. We find that (a) the R\'enyi entropy power decreases monotonically as the order $p$ is increasing and (b) the disequilibrium (closely related to the second order R\'enyi entropy), which quantifies the separation of the electron distribution from equiprobability, has a quasi-Gaussian behavior in terms of $D$.<br />Comment: Accepted in The European Physical Journal B (EPJ B)
- Subjects :
- Physics
Monotonic function
Condensed Matter Physics
01 natural sciences
010305 fluids & plasmas
Electronic, Optical and Magnetic Materials
symbols.namesake
Quantum state
Excited state
0103 physical sciences
Orthogonal polynomials
Laguerre polynomials
Rydberg formula
symbols
010306 general physics
Wave function
Harmonic oscillator
Mathematical Physics
Mathematical physics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....c71e3a016fe6c406abb55d0db8d64ec6