Multidimensional hydrogenic states: Position and momentum expectation values

TL;DR

This paper derives explicit formulas for position and momentum expectation values of multidimensional hydrogenic states, including previously unknown momentum expectations, and discusses related uncertainty principles.

Contribution

It provides analytical expressions for all radial expectation values of D-dimensional hydrogenic states, extending beyond three dimensions and including momentum expectations with odd orders.

Findings

Explicit formulas for all radial expectation values in D dimensions.

Closed-form expressions for momentum expectation values, especially odd orders.

Discussion of Heisenberg-like uncertainty inequalities for these quantities.

Abstract

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.