  The position and momentum probability densities of a multidimensional quantum
system are fully characterized by means of the radial expectation values
$\langle r^\alpha \rangle$ and $\left\langle p^\alpha \right\rangle$,
respectively. These quantities, which describe and/or are closely related to
various fundamental properties of realistic systems, have not been calculated
in an analytical and effective manner up until now except for a number of
three-dimensional hydrogenic states. In this work we explicitly show these
expectation values for all discrete stationary $D$-dimensional hydrogenic
states in terms of the dimensionality $D$, the strength of the Coulomb
potential (i.e., the nuclear charge) and the $D$ state's hyperquantum numbers.
Emphasis is placed on the momentum expectation values (mostly unknown,
specially the ones with odd order) which are obtained in a closed compact form.
Applications are made to circular, $S$-wave, high-energy (Rydberg) and
high-dimensional (pseudo-classical) states of three- and multidimensional
hydrogenic atoms. This has been possible because of the analytical algebraic
and asymptotical properties of the special functions (orthogonal polynomials,
hyperspherical harmonics) which control the states' wavefunctions. Finally,
some Heisenberg-like uncertainty inequalities satisfied by these dispersion
quantities are also given and discussed.
