The Shannon entropy of high-dimensional hydrogenic and harmonic systems

There exists an increasing interest on the dimensionality dependence of the entropic properties for the stationary states of the multidimensional quantum systems in order to contribute to its emergent informational representation, which extends and complements the standard energetic representation. Nowadays, this is specially so for high-dimensional systems as they have been recently shown to be very useful in both scientific and technological fields. In this work, the Shannon entropy of the discrete stationary states of the high-dimensional harmonic (ie, oscillator-like) and hydrogenic systems is analytically determined in terms of the dimensionality, the potential strength, and the state's hyperquantum numbers. We have used an information-theoretic methodology based on the asymptotics of some entropy-like integral functionals of the orthogonal polynomials and hyperspherical harmonics which control the wave functions of the quantum states, when the polynomial parameter is very large; this is basically because such a parameter is a linear function of the system's dimensionality. Finally, it is shown that the Shannon entropy of the D-dimensional harmonic and hydrogenic systems has a logarithmic growth rate of the type D log D when D → ∞.