Helium and Other Two Electron Systems

Y.-C. Lin, Chih‐Yuan Lin, Y. K. Ho

Recently, there has been considerable interest in investigating quantum entanglement in two-electron systems, such as two-electron model atoms and quantum dots. In this work, the quantum entanglement of the helium atom and helium-like ions are explored using the two-electron wave functions constructed by the $B$-spline basis. As a measure of the spatial entanglement, the linear entropy of the reduced density matrix is investigated for the ground state and the singlet excited $1sn{s}^{1}S$ states, with $n=2$--10, of the helium atom. The variation of the spatial entanglement with the nuclear charges $Z=2$--15 for the helium-like ions is reported. Results are compared with the existing predictions.

Igor Pavlovich Popov

The purpose of the work is to check the validity of generalizing the an-gular momentum ћ of a Cooper pair to any many-particle quantum system, including positronium. Methodology. For the purposes of this work, a semiclassical consideration is quite sufficient. To increase the reliability for helium, a quantum solution is additionally given. It is believed that the total minimum angular momentum of a two-particle quan-tum system – positronium is equal to ћ. When generalizing the angular momentum ћ of a Cooper pair and an electron-positron pair to a pair of helium electrons, its radius turns out to be almost four times less than the established value. And this means only one thing – the generalization of the angular momentum ћ of a Cooper pair to a pair of helium electrons is illegal. Therefore, each helium electron has a minimum angular mo-mentum equal to ћ. And the angular momentum of a pair of helium electrons is equal to the sum of the angular momentum of each electron. The double standards lie in the fact that, on the one hand, a quantum system – an electron-positron pair – is endowed with a quantum of angular momentum ћ, and on the other hand, a quantum system – a pair of helium electrons – is not endowed with this quantum (it is endowed with two quanta). And since double standards are unacceptable, an elementary dilemma arises – either the radius of helium should be quadrupled, or the radius of positronium should be quadrupled. The task is simplified by the fact that recently the value of the radius of positronium was completely unambiguously corrected, which turned out to be exactly four times larger than was commonly believed. Thus, the helium atom retains its size. It has been established that endowing a Cooper pair of electrons with a quantum of angular momentum ћ is just as illegitimate as in the case of positronium. The angular momentum of a many-particle quantum system, due to its additivity, must be divided between the particles of the system. It has been established that the quantum of angular momentum ћ can be attributed to only one quantum particle, and cannot be attributed to a quantum collection of particles.

S. M. Blinder

Olaf Stenzel

Y.-C. Lin, Y. K. Ho

Quantum entanglement for two electrons in the excited states of helium-like atom or ions is investigated using two-electron wave functions constructed in the B-spline basis. As a measure of spatial (electron – electron orbital) entanglement, the von Neumann entropy and linear entropy of the reduced density matrix are calculated for the 1s2s 1,3S excited states for systems with some selected Z values from Z = 2 to 100. Results for the helium atom are compared with other available calculations. We have also investigated the entropies for these excited states when the nucleus charge is reduced from Z = 2 continuously to Z = 1. At such a critical charge, all the singly excited states of this system become unbound, and the linear entropies and the von Neumann entropies for the excited states approach 1/2 and 1, respectively; the limits for the entropies when one electron is bound to the nucleus, and the other is free.