The position- and momentum-space entropies of the isotropic harmonic oscillator and the hydrogen atom in D dimensions are shown to be related to some entropy integrals which involve classical orthogonal polynomials. These integrals are exactly calculated for Chebyshev polynomials and only in an approximate way for Gegenbauer polynomials. The physical entropies are explicitly obtained in the ground state and in a few low-lying excited states. Finally, the dimensionality dependence of the ground-state entropies of the two above-mentioned quantum-mechanical systems is analyzed (numerically) and the values of the entropies in a large class of excited states of the D-dimensional (D=1,2,3) harmonic oscillator and hydrogen atom are tabulated and discussed.

The information entropies of the two-dimensional harmonic oscillator, ${\text{V(x,y)=1/2λ(x}}^2{\mathrm{+y}}^2\mathrm{),}$ and the one-dimensional hydrogen atom, $\text{V(x)=−1/|x|,}$ can be expressed by means of some entropy integrals of Laguerre polynomials whose values have not yet been analytically determined. Here, we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of Laguerre polynomials. Then, this result (which is also important by itself in a context of both approximation theory and potential theory) is employed to analyze the information entropies of the aforementioned quantum-mechanical potentials for the very excited states in both position and momentum spaces. It is observed, in particular, that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state. Finally, the rate of convergence of the entropies is numerically examined.

The spreading of the quantum-mechanical probability distribution density of $D$ -dimensional hydrogenic orbitals is quantitatively determined by means of the local information-theoretic quantity of Fisher in both position and momentum spaces. The Fisher information is found in closed form in terms of the quantum numbers of the orbital.

The information entropy of the harmonic oscillator potential V(x)=1/2λx² in both position and momentum spaces can be expressed in terms of the so‐called ‘‘entropy of Hermite polynomials,’’ i.e., the quantity S_n(H):= −∫_−∞^+∞H²_n(x)log H²_n(x) e−x²dx. These polynomials are instances of the polynomials orthogonal with respect to the Freud weights w(x)=exp(−‖x‖^m), m≳0. Here, a very precise and general result of the entropy of Freud polynomials recently established by Aptekarev et al. [J. Math. Phys. 35, 4423–4428 (1994)], specialized to the Hermite kernel (case m=2), leads to an important refined asymptotic expression for the information entropies of very excited states (i.e., for large n) in both position and momentum spaces, to be denoted by S_ρ and S_γ, respectively. Briefly, it is shown that, for large values of n, S_ρ+1/2logλ≂log(π√2n/e)+o(1) and S_γ−1/2log λ≂log(π√2n/e)+o(1), so that S_ρ+S_γ≂log(2π²n/e²)+o(1) in agreement with the generalized indetermination relation of Byalinicki‐Birula and Mycielski [Commun. Math. Phys. 44, 129–132 (1975)]. Finally, the rate of convergence of these two information entropies is numerically analyzed. In addition, using a Rakhmanov result, we describe a totally new proof of the leading term of the entropy of Freud polynomials which, naturally, is just a weak version of the aforementioned general result.

The Boltzmann–Shannon information entropy of quantum‐mechanical systems in central potentials can be expressed in terms of the entropy S_n of the classical orthogonal polynomials. Here, an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained. It is shown that this entropy is intimately related to the relative entropy I (ρ₀,ρ) of the equilibrium measure ρ₀(x) and the weight function ρ(x) of the polynomials. To do so, the theory of strong asymptotics of orthogonal polynomials on compact sets is used.