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American Journal of Physics

Further properties of hydrogenic wave functions

J D Hey

Published Aug 1, 1993• 28 citations
Publisher

Abstract

The generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms. The two Pasternack relations, which connect the expectation values 〈rk〉 for values of k differing by an integer, are examined, as well as a generalization of the first relation, and the advantages which they hold for computational purposes discussed. An earlier calculation by the present author, of the electron impact ionization cross section for hydrogen atoms in the 2S metastable state, is compared with other experimental data from the literature. The calculation of the wavelengths of transitions between hydrogen-like levels in highly excited ions possessing a few additional, tightly bound, electrons is described and compared with experimental data.

DOI
10.1119/1.17150

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The generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms. The two Pasternack relations, which connect the expectation values 〈rk〉 for values of k differing by an integer, are examined, as well as a generalization of the first relation, and the advantages which they hold for computational purposes discussed. An earlier calculation by the present author, of the electron impact ionization cross section for hydrogen atoms in the 2S metastable state, is compared with other experimental data from the literature. The calculation of the wavelengths of transitions between hydrogen-like levels in highly excited ions possessing a few additional, tightly bound, electrons is described and compared with experimental data.
The wave functions in momentum space for discrete states of hydrogenlike atoms are discussed in terms of both the early contributions to the subject and their present usefulness in atomic physics. The derivation of some of their important properties is outlined, in particular the expectation values of various powers of momentum and kinetic energy, and the kinetic energy distribution. As an application, the cross section for ionization of hydrogen in the metastable 2S state by electron impact is calculated with the aid of the binary-encounter approximation, which requires the use of momentum-space wave functions.
The radial expectation values of the probability density of a quantum system in position and momentum spaces allow one to describe numerous physical quantities of the system as well as to find generalized Heisenberg-like uncertainty relations and to bound entropic uncertainty measures. It is known that the position and momentum expectation values of the main prototype of the D-dimensional Coulomb systems, the D-dimensional hydrogenic system, can be expressed in terms of some generalized hypergeometric functions of the type p+1Fp(z) evaluated at unity with p = 2 and p = 3, respectively. In this work we determine the position and momentum expectation values in the limit of large D for all hydrogenic states from ground to very excited (Rydberg) ones in terms of the spatial dimensionality and the hyperquantum numbers of the state under consideration. This is done by means of two different approaches to calculate the leading term of the special functions 3F21 and 5F41 involved in the large D limit of the position and momentum quantities. Then, these quantities are used to obtain the generalized Heisenberg-like and logarithmic uncertainty relations, and some upper and lower bounds to the entropic uncertainty measures (Shannon, Rényi, Tsallis) of the D-dimensional hydrogenic system.
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.
Abstract The position and momentum probability densities of a multidimensional quantum system are fully characterized by means of the radial expectation values ⟨ r α ⟩ and p α , 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.
In this chapter, we focus on three-dimensional problems that have rotational symmetry, with a concentrated effort on describing the properties of the hydrogen atom. This is an exactly solvable problem that is extremely important. To get there, we first have to describe how we work in spherical coordinate systems, and we need to develop the general rules for the factorization method. We develop the concept of a vector operator, of radial momentum, and the rules needed to solve problems by the Schrödinger factorization method. We first examine the isotropic harmonic oscillator, and then we establish the energy scales of hydrogen and solve first for the energies and eigenstates and then the wavefunctions in position and momentum space. We derive the Kramers–Pasternak relations for expectation values of powers of r and then focus on important experiments, which include the Pickering–Fowler lines, the discovery of deuterium, visualizing the atom in momentum space with e-2e spectroscopy, Rydberg atoms, the Lamb shift, and hyperfine structure and its applications to radio astronomy.
will be added. One can explain the nature of the term by examining the relativistic case. Below the formulas for the radial and angular parts of the wave function were evaluated. The term for the energetic levels of the hydrogen atom for specific quantum numbers was given. We calculated the values of several specific radial functions for different quantum numbers, as well as the values of the radial function at the origin of coordinates and wide apart. A number of mean values for different powers of r was calculated. For the continuous spectrum the radial eigenfunctions were calculated.