  The internal disorder of a D-dimensional hydrogenic system, which is strongly
associated to the non-uniformity of the quantum-mechanical density of its
physical states, is investigated by means of the shape complexity in the two
reciprocal spaces. This quantity, which is the product of the disequilibrium or
averaging density and the Shannon entropic power, is mathematically expressed
for both ground and excited stationary states in terms of certain entropic
functionals of Laguerre and Gegenbauer (or ultraspherical) polynomials. We
emphasize the ground and circular states, where the complexity is explicitly
calculated and discussed by means of the quantum numbers and dimensionality.
Finally, the position and momentum shape complexities are numerically discussed
for various physical states and dimensionalities, and the dimensional and
Rydberg energy limits as well as their associated uncertainty products are
explicitly given. As a byproduct, it is shown that the shape complexity of the
system in a stationary state does not depend on the strength of the Coulomb
potential involved.
