319 lines
6.6 KiB
Plaintext
319 lines
6.6 KiB
Plaintext
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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\begin_preamble
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\usepackage{unicode-math}
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% Toto je trik, jimž se z fontspec získá familyname pro následující
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% definujeme novou rodinu, jež se volá pomocí \MyCyr pro běžné použití, avšak pro účely \DeclareSymbolFont je nutno získat název pomocí getfamilyname definovaného výše
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}
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\spacing single
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\use_hyperref true
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\pdf_title "Accelerating lattice mode calculations with T-matrix method"
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\pdf_author "Marek Nečada"
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\index Index
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\shortcut idx
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\begin_body
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\begin_layout Standard
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\begin_inset FormulaMacro
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\newcommand{\svecp}[1]{#1}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\svect}[1]{#1}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\vect}[1]{\mathbf{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\FoR}[1]{\mathfrak{#1}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\ud}{\mathrm{d}}
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\end_inset
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\begin_inset FormulaMacro
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\newcommand{\WignerD}{\mathcal{D}}
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\end_inset
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\end_layout
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\begin_layout Title
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Testování numerické správnosti QPMS
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\end_layout
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\begin_layout Author
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Marek Nečada
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\end_layout
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\begin_layout Abstract
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Všeliké poznámky vztahující se k psaní testů knihovny QPMS.
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\end_layout
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\begin_layout Section
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Operátor přesunu
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\end_layout
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\begin_layout Standard
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Rozmohl se mi takový nešvar, že souměrné soustavy (například vůči zrcadlení
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\begin_inset Formula $y\leftrightarrow-y$
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\end_inset
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) dávají nesouměrné výsledky např.
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pro účinný průřez.
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Zdá se, že k chybě dochází v některém z kroků výpočtu operátoru přesunu
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\begin_inset Formula $S(b\leftarrow a)$
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\end_inset
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.
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Ověřme jeho výpočtem v různě otočených či převrácených soustavách souřadnic.
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Buďtež tedy
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\begin_inset Formula $\FoR F$
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\end_inset
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,
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\begin_inset Formula $\FoR G$
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\end_inset
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dvě různé soustavy souřadnic v euklidovském navzájem otočené či převrácené,
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a pakliže vyjádření vektoru
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\begin_inset Formula $\vect v$
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\end_inset
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v soustavě
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\begin_inset Formula $\FoR F$
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\end_inset
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je
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\begin_inset Formula $\vect v_{j}^{\FoR F}$
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\end_inset
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, pak jeho vyjádření v soustavě
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\begin_inset Formula $\FoR G$
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\end_inset
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budiž
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\begin_inset Formula $\vect v_{j}^{\FoR G}=\Psi_{ji}\vect v_{i}^{\FoR F}$
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\end_inset
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, kde
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\begin_inset Formula $\Psi\in O(3)$
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\end_inset
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.
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Odpovídající transformace kulového vektoru (např.
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vyzařovaných kulových el.
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vln) jest
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\begin_inset Formula
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\[
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\svect A_{l'm'}^{\FoR G}=\WignerD_{l'm'\leftarrow lm}^{(\Psi)}\svect A_{lm}^{\FoR F}.
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\]
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\end_inset
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Pakliže se jedná o kombinovaný kulový vektor-pseudovektor (jako třeba vyzařovaný
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ch kulových elektrických a magnetických vln), dostáváme (OVĚŘ)
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\begin_inset Formula
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\[
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\svecp A_{t'l'm'}^{\FoR G}=\WignerD_{t'l'm'\leftarrow tlm}^{(\Psi)}\svecp A_{tlm}^{\FoR F}=\left(\det\Psi\right)^{\left(t'-t\right)}\WignerD_{l'm'\leftarrow lm}^{(\Psi)}\svecp A_{tlm}^{\FoR F}.
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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Vezměmež elementární případ dílčího rozptylu částice v bodě
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\begin_inset Formula $b$
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\end_inset
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záření vyzařovaného částicí v bodě
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\begin_inset Formula $a$
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\end_inset
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.
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Nezávisle na soustavě:
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\begin_inset Formula
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\[
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P^{(b)}=S_{b\leftarrow a}T^{(a)}P^{(a)}
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\]
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\end_inset
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\end_layout
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\begin_layout Standard
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V soustavě
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\begin_inset Formula $\FoR F$
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\end_inset
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:
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\begin_inset Formula
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\[
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P^{(b)\FoR F}=S_{(b\leftarrow a)^{\FoR F}}^{\FoR F}T^{(a)\FoR F}P^{(a)\FoR F}.
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\]
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\end_inset
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V soustavě
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\begin_inset Formula $\FoR G$
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\end_inset
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(pro jednoduchost píši
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\begin_inset Formula $\WignerD\equiv\WignerD^{(\Psi)}$
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\end_inset
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atd.):
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\begin_inset Formula
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\begin{eqnarray*}
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P^{(b)\FoR G} & = & \WignerD P^{(b)\FoR F}=\WignerD S_{(b\leftarrow a)^{\FoR F}}^{\FoR F}T^{(a)\FoR F}P^{(a)\FoR F}\\
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& = & \underbrace{\WignerD S_{(b-a)^{\FoR F}}^{\FoR F}\WignerD^{-1}}_{S_{(b-a)^{\FoR F}}^{\FoR G}=S_{\Psi^{-1}(b-a)^{\FoR G}}^{\FoR G}???}\underbrace{\WignerD T^{(a)\FoR F}\WignerD^{-1}}_{T^{(a)\FoR G}}\underbrace{\WignerD P^{(a)\FoR F}}_{P^{(a)\FoR G}}.
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\end{eqnarray*}
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\end_inset
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\end_layout
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\begin_layout Standard
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Nemá první svorka býti
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\begin_inset Formula $S_{\left(b-a\right)^{\FoR G}}^{\FoR G}$
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\end_inset
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?!
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\end_layout
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\begin_layout Standard
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Test správnosti tedy může vypadat následovně:
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\end_layout
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\begin_layout Enumerate
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Vytvoř náhodně vektor přesunu
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\begin_inset Formula $\vect v$
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\end_inset
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(což bude naše
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\begin_inset Formula $(b-a)^{\FoR G}$
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\end_inset
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) a transformaci
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\begin_inset Formula $\Psi\in O(3)$
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\end_inset
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.
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\end_layout
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\end_body
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\end_document
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