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Physical Review A

Position and momentum information entropies of the D -dimensional harmonic oscillator and hydrogen atom

Rafael J. Yáñez, Walter Van Assche, J. S. Dehesa

Published Oct 1, 1994• 284 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.50.3065

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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 position and momentum information entropies of D -dimensional quantum systems with central potentials, such as the isotropic harmonic oscillator and the hydrogen atom, depend on the entropies of the (hyper)spherical harmonics. In turn, these entropies are expressed in terms of the entropies of the Gegenbauer (ultraspherical) polynomials C (λ) n ( x ), the parameter λ being either an integer or a half-integer number. Up till now, however, the exact analytical expression of the entropy of Gegenbauer polynomials of arbitrary degree n has only been obtained for the particular values of the parameter λ = 0, 1, 2. Here we present a novel approach to the evaluation of the information entropy of Gegenbauer polynomials, which makes use of trigonometric representations for these polynomials and complex integration techniques. Using this method, we are able to find the analytical expression of the entropy for arbitrary values of both n and .
The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.
The problem of calculating the information entropy in both position and momentum spaces for the nth stationary state of the one-dimensional quantum harmonic oscillator reduces to the evaluation of the logarithmic potential Vn(t)=−∫−∞∞(Hn(x))2 ln |x−t|e−x2 dx at the zeros of the Hermite polynomial Hn(x). Here, a closed analytical expression for Vn(t) is obtained, which in turn yields an exact analytical expression for the entropies when the exact location of the zeros of Hn(x) is known. An inequality for the values of Vn(t) at the zeros of Hn(x) is conjectured, which leads to a new, nonvariational, upper bound for the entropies. Finally, the exact formula for Vn(t) is written in an alternative way, which allows the entropies to be expressed in terms of the even-order spectral moments of the Hermite polynomials. The asymptotic (n≫1) limit of this alternative expression for the entropies is discussed, and the conjectured upper bound for the entropies is proved to be asymptotically valid.
We analyse the entropy of Hermite polynomials and orthogonal polynomials for the Freud weights w(x)=exp(-|x|^m) on the real line and show how these entropies are related to information entropy of the one-dimensional harmonic oscillator. The physical interest in such entropies comes from a stronger version of the Heisenberg uncertainty principle, due to Bialynicki-Birula and Mycielski, which is expressed as a lower bound for the sum of the information entropies of a quantum-mechanical system in the position space and in the momentum space.
In this paper, we study Renyi and Tsallis information entropies for a Hamiltonian system with position-dependent mass confined in harmonic oscillator potential. Gegenbauer polynomials are used to obtain the position eigenfunction of such a system, and the modified Bessel function of the second kind is used to determine the equivalent momentum eigenfunction. By means of probability densities of both representations, we analytically and numerically evaluate the Heisenberg-like uncertainty of this system. Because the Renyi and Tsallis information entropies in position representation are described by integral functionals of the Gegenbauer polynomials, these quantities are much more difficult to calculate. To get around this problem, we evaluate these information entropies at the system s asymptotical limit, which corresponds to the behavior of an undeformed harmonic oscillator. Nevertheless, no approximation method is used to obtain the Tsallis and Renyi information entropies in momentum representation. In both representations, we find that these information entropies approach the Shannon entropy when the entropic parameter alpha-> 1 and are closed to the results of similar models of the literature. Finally, we evaluate the related entropic uncertainty relations numerically to validate the latter obersevations.