Schrödinger-Poisson model for very-high-pressure cold helium

Abstract

An alternative numerical solution of the Hartree-Fock integro-differential equations is reported that consists of reformulating the self-consistent ansatz as a nonlinear set of coupled Schrödinger and Poisson equations. In the simple case of helium, this yields an amazingly simple dynamical system whose statistical properties are constrained by the virial theorem. This approach leads to an equation of state for helium at zero temperature and very high densities, covering the whole range of astrophysical interest, up to the pressure ionization phase transition that is predicted around 44 Mbar by the present model.

Nonlinear Schrödinger–Poisson theory for quantum-dot Helium

“…Therefore we recover an important property that has already been emphasized in the N = 2 Coulomb case, both for atomic Helium 12 and hydrogen ion H − . 13 Namely that the SP nonlinear differential description yields surprisingly accurate values for the ground state energy when compared to their corresponding mean-field Hartree-Fock ones.…”

“…For GaAs quantum-dot Parahelium defined by the confinement hω = 3.37 meV, the dielectric constant of the bulk material κ = 12.4 and the effective mass M = 0.067 me (where me is the electron mass), 5 we have N = 2.53 and Ẽ = 1.831 by use of (respectively) Eqs ( 13) & ( 14). This last SP ground-state energy per particle value appears here as the intersection of its virial (continuous line) and its Koopman (dashed-dotted line) definitions as respectively provided by Eqs ( 10) & (11)(12). PGM's exact numerical value Ẽ = 1 2 (12.28) meV /3.37 meV = 1.822 given in Ref.…”

“…defined by Eqs (7)(8)(9)(10)(11)(12) which states that the trap harmonicity ω, or equivalently its quantum-dot dimensionless length k = h/M ω/(h 2 /M e 2 ), must be identical for the two modes u 1, 3 that enter the calculation of the scalar product (18). Practically, in the numerical simulations of Eqs (18-20), we will consider Eq.…”

Asymptotic decay estimates for the repulsive Schrödinger–Poisson system

“…The study of these properties is relevant for the fabrication of semiconductor superlattices [1][2][3]. This has motivated in the last years, based on the classical modelling of an electron gas by means of the Schr odinger-Poisson system [4,5], the appearance of di erent adaptations of this model in the literature for electron transport under particular contexts [6][7][8]. Particularly, the analysis of the quantum transport in an homogeneous semiconductor can be simpliÿed thanks to the 372 Ã O.…”

Insulator-to-Conducting Transition in Dense Fluid Helium

“…Solid He at low temperature is a wide band-gap insulator [3], with an excitonic level of 21.58 eV. Many calculations have predicted the insulator to conductor transition pressure for solid He with the most recent estimates [4][5][6][7] spanning the range 4:4-25 TPa (i.e., 10-21 g=cm 3 ) much out of reach of static experiments. At elevated temperatures, insulator to conductor transitions have been observed in many wide band-gap insulators, for example, in dense H 2 , O 2 , and N 2 [8][9][10] under multishock quasi-isentropic compression, and in Al 2 O 3 , LiF, H 2 O, D 2 , SiO 2 , and diamond [11][12][13] under single shock compression.…”