Quantum-information entropies for highly excited states of single-particle systems with power-type potentials

Abstract

The asymptotics of the Boltzmann-Shannon information entropy as well as the Renyi entropy for the quantum probability density of a single-particle system with a confining (i.e., bounded below) power-type potential ${V(x)=x}^{2k}$ with $k\ensuremath{\in}N$ and $x\ensuremath{\in}R,$ is investigated in the position and momentum spaces within the semiclassical (WKB) approximation. It is found that for highly excited states both physical entropies, as well as their sum, have a logarithmic dependence on its quantum number not only when $k=1$ (harmonic oscillator), but also for any fixed k. As a by-product, the extremal case $\stackrel{\ensuremath{\rightarrow}}{k}\ensuremath{\infty}$ (the infinite well potential) is also rigorously analyzed. It is shown that not only the position-space entropy has the same constant value for all quantum states, which is a known result, but also that the momentum-space entropy is constant for highly excited states.