Usor:Tchougreeff/Determinans Slateri

Determinans Slateri/slaterianum est functio ad statūs multi-electronicōs (multōrum electronum vel multārum particulārum - fermionum) representandum adhibita.

Definitio: Sint ${\textstyle \varphi _{1}\left(x\right),...,\varphi _{N}\left(x\right)}$ functiones (spira-orbitalia) describentēs statūs unius fermionis (electronis). Tum functio in forma determinantis:$$\Psi \left(x_{1},...,x_{N}\right)={\frac {1}{\sqrt {N!}}}{\begin{vmatrix}\varphi _{1}\left(x_{1}\right)&...&\varphi _{1}\left(x_{1}\right)\\\cdots &&\cdots \\\varphi _{N}\left(x_{1}\right)&...&\varphi _{N}\left(x_{N}\right)\end{vmatrix}}(1)$$(columnae numeris particulārum, lineas numeris functionum designatae) erit functio $N$ fermionum describens quae determinans Slateri nuncupatur. Hujus functionis signum mutatur quando duae functiones vel duae coordinatae permutantur. Notandum est functiones $\varphi _{i}\left(x\right)$ non orthogonales (lineare dependentes) solum suārum partibus orthogonalibus (lineare independentibus) determinanti (1) contrĭbŭunt (pars functionis $\varphi _{i}\left(x\right)$ aliquid functioni $\varphi _{j}\left(x\right)$ proportionalis nullum determinanti (1) contrĭbŭit).

Pro determinantibus Slateri ex orbitalibus orthogonalibus constructis talis theorema plenitudinis/plenitatis Löwdini pertinens/Löwdiniana certa est. Sit basis fuctionum $\varphi _{i}\left(x\right)$ variabilis $x$ completa secundum metricam $L^{2}$ in spatiei functionum variabilis $x$, basis totium determinatium Slateri $N$ fermionum erit completa in spatiei functionum varabilium $x_{1},...,x_{N}$ secundum eadem metricam in spatiei $L^{2}(x_{1},...,x_{N})$.