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Jonas Hahn
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59 lines
1.6 KiB
Typst
59 lines
1.6 KiB
Typst
// Main VL template
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#import "../preamble.typ": *
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// Fix theorems to be shown the right way in this document
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#import "@preview/ctheorems:1.1.3": *
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#show: thmrules
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// Main settings call
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#show: conf.with(
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// May add more flags here in the future
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num: 5,
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type: 0, // 0 normal, 1 exercise
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date: datetime.today().display(),
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//date: datetime(
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// year: 2025,
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// month: 5,
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// day: 1,
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//).display(),
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)
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= Uebersicht
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Wiederholung
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$
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E_("pot") = N/2 sum _(i != j) ( ... ) \
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E_(i j) := "WW-Energie zwischen Ion" i and j \
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E_(i) := "WW-Energie des Ion" i \, space E_(i) := sum _(i != j) E_(i j) \
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E_("pot") = N/2 E_(i) "Summe der Ionenpole nach dem Summationsprinzip".
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$
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Zur Loesung einfacher Molekuele wird die LCAO-Methode genutzt um das Molekuelorbital durch eine Linearkombination zu modellieren.
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Helium kann durch Kombination aus $phi_(A) "und" phio_(B) $ dargestellt werden.
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Das H-atom wird dargestellt als Kombination aus $phi_(A) and phi_(B) $.
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Zeitunabhaengiger Hamilton-Operator
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$
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- planck.reduce ^2 / (2 m ) arrow(nabla) _(e) ^2 - (e ^2 ) / (4 pi epsilon_0 ) (1/r_(A) + 1/r_(B) - 1/R ) phi = E phi.
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$
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Ansatz
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$
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psi (arrow(r), R) = c_(A) phi_(A) (arrow(r)_(A) ) + c_(B) phi_(B) (arrow(r)_(B) ) \
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phi_(i) (arrow(r)) = phi_(i) (arrow(r )_(i) ) = 1/(sqrt(pi a_0 ^3 )) e ^( - r_(i) /a_0 ) \, space a_0 = "const."
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$
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Die gesamte Wellenfunktion ist normiert
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$
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integral abs(psi)^2 dif ^3 r = 1.
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$
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Das Ueberlappungsintegral ist gegeben durch
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$
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S_(A B) = R_0 integral phi_(A) ^(star ) phi_(B) dif ^3 r.
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$
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Das Helium Ion ist symmetrisch, also
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$
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H_(A A) = H_(B B) \
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c_(A) = +- c_(B).
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$
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