On the Hylleraas Coordinates
| dc.creator | Pan, Xiao-Yin | |
| dc.creator | Sahni, Viraht | |
| dc.creator | Massa, Lou | |
| dc.date | 2003-10-24 | |
| dc.date | 2004-12-15 | |
| dc.date.accessioned | 2026-07-07T05:50:19Z | |
| dc.date.available | 2026-07-07T05:50:19Z | |
| dc.description | The Hylleraas coordinates $s=r_{1}+r_{2}$, $t=r_{1}-r_{2}$, $u=|{\bf r}_{1}-{\bf r}_{2}|$ are the natural coordinates for the determination of properties of the Helium atom, the positive ions of its isoelectronic sequence, and the negative Hydrogen ion. In this paper, we derive a new expression for integrals representing properties such as the energy, normalization and expectation of arbitrary operators, as written in the $(s,t,u)$ coordinates. The expression derived is valid for both \emph{finite} and \emph{infinite} space. The integrals for the various properties are comprised in each case of two components $A$ and $B$. The contribution of these components to the volume of integration and the normalization of a wave function for finite space, and in variational calculations of the ground state energy of the Helium atom confined in a finite volume is demonstrated by example. We prove that when the integration space is \emph{infinite}, the expression for the energy and other properties employed by Hylleraas corresponds \emph{only} to that of integral $A$. We further prove that for the approximate variational wave functions employed by Hylleraas and other authors, the contribution of the term $B$ vanishes. This contribution also vanishes for the exact wave function. It is interesting to note that the component $B$ to the integral is not mentioned in the literature. A principle purpose of the paper, therefore, is to point out the existence of this term. | |
| dc.description | 9 pages, 4 figures, 1 table | |
| dc.identifier | https://arxiv.org/abs/physics/0310128 | |
| dc.identifier | http://arxiv.org/abs/physics/0310128 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/85677 | |
| dc.subject | Atomic Physics | |
| dc.subject | Chemical Physics | |
| dc.title | On the Hylleraas Coordinates | |
| dc.type | text |