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Register a new userUltra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. II. Spin-singlet (excited) and spin-triplet (lowest) states of Helium sequence
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As a continuation of Part I (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z leq 20$, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (4-5 significant digits (s.d.)) of two excited states of He-like ions: the spin-singlet (first) excited state $2^1 S$ and for lowest spin-triplet $1^3 S$ state. For both states it provides absolute accuracy for energy $sim 10^{-3}$,a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy $sim 10^{-2}$. Bressanini-Reynolds observation about the special form of nodal surface of $2^1 S$ state for Helium is confirmed and extended to ions with $Z > 2$. Critical charges $Z=Z_B$, where ultra-compact trial functions loose their square-integrability, are estimated: $Z_B(1^1 S)approx Z_B(2^1 S)sim 0.905$ and $Z_B(1^3 S)sim 0.902$. For both states the Majorana formula - the energy as the second degree polynomial in $Z$ - provides accurately the 4-5 significant digits for $Z leq 20$.
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Several ultra-compact accurate wave functions in the form of generalized Hylleraas-Kinoshita functions and Guevara-Harris-Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), or, equivalently, the Non-Relativistic QED (NRQED), for the ground state energies (4-5 significant digits (s.d.)) of He-like and Li-like iso-electronic sequences in the static approximation with point-like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in $Z$ by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He-like two-electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy $sim 10^{-3}$,a.u. and the same relative accuracies $sim 10^{-2}-10^{-3}$ for both the cusp parameters and the six expectation values, is found. For the Li-like three-electron sequence the most accurate ultra-compact function taken as the variational trial function provides absolute accuracy for energy $sim 10^{-3}$,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for $Z leq 20$ and 3 s.d. for the two expectation values for $Z=3$.
A general formalism is used to express the long-range potential energies in inverse powers of the separation distance between two like atomic or molecular systems with $P$ symmetries. The long-range molecular interaction coefficients are calculated for the molecular symmetries $Delta$, $Pi$, and $Sigma$, arising from the following interactions: He($2 ^1P$)--He($2 ^1P$), He($2 ^1P$)--He($2 ^3P$), and He($2 ^3P$)--He($2 ^3P$). The electric quadrupole-quadrupole term, $C_{5}$, the van der Waals (dispersion) term $C_{6}$, and higher-order terms, $C_{8}$, and $C_{10}$, are calculated textit{ab initio} using accurate variational wave functions in Hylleraas coordinates with finite nuclear mass effects. A comparison is made with previously published results where available.
We show that the general quantum state of synthetically spin-orbit coupled ultra cold bosonic atom whose condensate was experimentally created recently ( Y. J. Lin {it et al.}, Nature, {bf 471}, 83, (2011)), shows entanglement between motional degrees of freedom ( momentum) and internal degrees of freedom (hyperfine spin). We demonstrate the violation of Bell-like inequality (CHSH) for such states that provides a unique opportunity to verify fundamental principle like quantum non-contextuality for commutating observables which are not spatially separated. We analyze in detail the Rabi oscillation executed by such atom-laser system and how that influneces quantities like entanglement entropy, violation of Bell like Inequality etc. We also discuss the implication of our result in testing the quantum non-contextuality and Bells Inequality vioaltion by macroscopic quantum object like Bose-Einstein Condensate of ultra cold atoms.
The 1s2-1s2l lines are the most intense He-like ions lines. They are used as spectroscopic diagnostics for solar active regions as well as for different laboratory plasmas. Nowadays, it exits very high spectral resolution instruments and, for intense X-ray sources, one can do spectroscopic diagnostics from line ratios. With XMM (RGS) and Chandra (LETGS, HETGS) spectral resolutions and for several atomic elements, it is particularly possible to separate a 3 blended line set, the so-called He-like triplet: Resonance (r), Intercombination (i) and Forbidden (f), which are dominated respectively by lines issued from the following levels : 1s2p 1P1, 1s2p 3P1,2 and 1s2s 3S1. We shall show that the measurement of two different ratios between these 3 lines (R = f/i and G = (f + i)/r) give quantitative informations on the nature of the emitting plasma (photo-ionized or collisional) and on its electronic density and temperature. A more refined analysis must also include satellite line contributions.
Due to the ever increasing power and cooling requirements of large-scale computing and data facilities, there is a worldwide search for low-power alternatives to CMOS. One approach under consideration is superconducting computing based on single-flux-quantum logic. Unfortunately, there is not yet a low-power, high-density superconducting memory technology that is fully compatible with superconducting logic. We are working toward developing cryogenic memory based on Josephson junctions that contain two or more ferromagnetic (F) layers. Such junctions have been demonstrated to be programmable by changing the relative direction of the F layer magnetizations. There are at least two different types of such junctions -- those that carry the innate spin-singlet supercurrent associated with the conventional superconducting electrodes, and those that convert spin-singlet to spin-triplet supercurrent in the middle of the device. In this paper we compare the performance and requirements of the two kinds of junctions. Whereas the spin-singlet junctions need only two ferromagnetic layers to function, the spin-triplet junctions require at least three. In the devices demonstrated to date, the spin-singlet junctions have considerably larger critical current densities than the spin-triplet devices. On the other hand, the spin-triplet devices have less stringent constraints on the thicknesses of the F layers, which might be beneficial in large-scale manufacturing.
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