Hyperoctahedral group - Revision history

en>Tomruen: /* Subgroups */

2014-12-18T17:03:06Z

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← Older revision		Revision as of 19:03, 18 December 2014
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	In [~~[atomic physics~~]], a ~~'''two-electron atom''' or '''helium-like ion'''~~ is ~~a [[quantum mechanical]] system consisting~~ of ~~one~~ [~~[Atomic nucleus\|nucleus]~~] with ~~a [[Electrical charge\|charge]] of ''Z[[Elementary charge\|e]]'' and just two electrons~~. ~~This~~ is the ~~first case~~ of ~~many-electron systems where the [[Pauli exclusion principle]] plays a central role~~.		[http://Www.Alexa.com/search?q=Colours&r=topsites_index&p=bigtop Colours] have a direct effect on human understanding.<br>Realizing the meaning of every colour may help you create a statement that is clear and to dress up. I am offering some tips on how to select the colour of your dresses for women in accordance to [http://tinyurl.com/nqbkz6z ugg boots] the special scenario to you. Let us start with black. Black is the colour of sophistication and strength.<br><br>

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	~~The first few two-electron atoms are:~~

	{\|
	\|-
	~~\| ''Z''=1: \|\| H<sup>–<~~/~~sup> \|\| [[hydrogen anion~~]]
	\|-
	~~\| ''Z''=2: \|\| He \|\| '''[~~[~~helium atom]]'''~~
	\|-
	~~\| ''Z''=3~~: ~~\|\| Li<sup>+<~~/~~sup> \|\| [[lithium]] ion~~
	\|-
	~~\| ''Z''=4~~: ~~\|\| Be<sup>2+<~~/~~sup> \|\| [[beryllium~~]~~] ion~~
	\|-
	~~\| ''Z''=5: \|\| B~~<~~sup~~>3+<~~/sup~~> ~~\|\| [[boron]] ion~~
	\|-
	\|}

	~~==Schrödinger equation==~~
	~~The Schrödinger equation for any two-electron system, such as~~ the ~~neutral [[Helium atom]] (He~~, '~~'Z'' = 2)~~, ~~the negative [[Hydrogen]] [[ion]] (H~~<~~sup>–</sup~~>, ~~''Z'' = 1), or the positive [[Lithium]] ion (Li<sup>+</sup>~~, ~~''Z'' = 3)~~ is~~:<ref name="Bransden">Physics of Atoms~~ and ~~Molecules, B~~.~~H. Bransden~~, C.J.~~Joachain, Longman, 1983, ISBN 0-582-44401-2~~<~~/ref~~> ~~For a more rigorous mathematical derivation of Schrödinger~~'s ~~equation~~, ~~see also~~.~~<ref name="Bransden"~~/>
	:<math> E\psi = -\hbar^2\left[\frac{1}{2\mu}\left(\nabla_1^2 +\nabla_2^2 \right) + \frac{1}{M}\nabla_1\cdot\nabla_2\right] \psi + \frac{e^2}{4\pi\epsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1}+\frac{1}{r_2} \right) \right] \psi </~~math>~~

	~~where '''r'''<sub>1<~~/~~sub>~~ is the ~~position of one electron (''r''~~<~~sub~~>1</~~sub> = \|'''r'''~~<~~sub~~>1<~~/sub~~>\| is ~~its magnitude)~~, ~~'''r'''<sub>2</sub>~~ is the ~~position of the other electron (''r''<sub>2<~~/~~sub> = \|'''r'''<sub>2<~~/~~sub>\| is the magnitude)~~, ~~''r''~~<~~sub~~>~~12</sub> = \|'''r'''<sub>12</sub>\|~~ is the ~~magnitude~~ of ~~the separation between them given by~~

	~~:<math> \|\bold{r}_{12}\| = \|\bold{r}_2~~ - ~~\bold{r}_1 \| \~~,~~\!</math>~~

	'~~'μ'' is again~~ the ~~two-body reduced mass~~ of ~~an electron with respect~~ to the ~~nucleus~~ of ~~mass ''M''~~, ~~so this time~~

	:<~~math~~> ~~\mu = \frac{m_e M}{m_e+M} \,\!~~<~~/math~~>

	and ~~''Z'' is the [[atomic number]] for the element (not a [[quantum number]])~~.

	~~The cross-term of two laplacians~~

	~~:<math>\frac{1}{M}\nabla_1\cdot\nabla_2\~~,~~\!</math>~~

	is ~~known~~ as the ~~''mass polarization term'', which arises due to the motion of [[Atomic nucleus\|atomic nuclei]]~~. ~~The wavefunction~~ is ~~a function of the two electron's positions:~~

	~~:<math> \psi = \psi(\bold{r}_1,\bold{r}_2) </math>~~

	~~There~~ is ~~no closed form solution for~~ this ~~equation~~.

	~~==Spectrum==~~
	~~The optical spectrum of the two electron atom~~ has ~~two systems of lines~~. ~~A para system of single lines~~, ~~and an ortho system of triplets (closely spaced group of three lines)~~. ~~The energy levels in the atom for the single lines are indicated by~~ <~~sup~~>1<~~/sup~~>~~S<sub>0</sub> <sup>1</sup>P<sub>1</sub> <sup>1</sup>D<sub>2</sub> <sup>1</sup>F<sub>3</sub> etc~~., and ~~for~~ the ~~triplets, some energy levels~~ are split: <sup>3</sup>S<sub>1</sub> <sup>3</sup>P<sub>2</sub> <sup>3</sup>P<sub>1</sub> <sup>3</sup>P<sub>0</sub> <sup>3</sup>D<sub>3</sub> <sup>3</sup>D<sub>2</sub> <sup>3</sup>D<sub>1</sub> <sup>3</sup>F<sub>4</sub> <sup>3</sup>F<sub>3</sub> <sup>3</sup>F<sub>2</sub>.<~~ref name="Herzberg"~~>{{cite book\|last=Herzberg\|first=Gerhard\|coauthors=J W T Spinks\|title=Atomic Spectra and Atomic Structure\|edition=2\|year=1944\|publisher=Dover Publications\|location=New York\|page=75}}</ref> Alkaline earths and Mercury also have ~~spectra with similar features~~, ~~due to the two outer valence electrons~~.~~<ref name="Herzberg"/>~~

	~~==References==~~
	~~<references/>~~

	~~==See also==~~
	*[[Hydrogen-like atom]]
	*[[Helium atom]]

	~~[[Category:Atoms]]~~
	~~[[Category:Quantum models]]~~

en>Tomruen: /* Subgroups */

2013-11-15T05:42:41Z

Subgroups

← Older revision		Revision as of 07:42, 15 November 2013
Line 1:		Line 1:
	I'~~m a 31 years old, married and study at the college (Japanese Studies).<br>In my spare time I~~'~~m trying to teach myself Chinese. I~~'~~ve been there and look forward to go there sometime in the future. I~~ like ~~to read, preferably on my beloved Kindle. I really love to watch 2 Broke Girls and Grey~~'~~s Anatomy as well as documentaries about nature. I like Computer programming.<br>xunjie 私たちはファッションの波に行くことができますだけでなく、~~		In [[atomic physics]], a '''two-electron atom''' or '''helium-like ion''' is a [[quantum mechanical]] system consisting of one [[Atomic nucleus\|nucleus]] with a [[Electrical charge\|charge]] of ''Z[[Elementary charge\|e]]'' and just two electrons. This is the first case of many-electron systems where the [[Pauli exclusion principle]] plays a central role.
	~~それが確立されたブランド、~~
	~~専門的な特殊化の概念やアイデアの共通の開発中のシャンチェンは、~~ [~~http://www.zgg.ch/data/video/shop/gaga.html ��ߥ�� rӋ ��ǥ��`�� ͨ؜~~] ~~今年の貿易は3つのステージで構成されていることが報告されている。~~
	~~2006年のCSTの段階的削減を承認した。~~
	~~TIP1：ハイウエストのパンツは、~~ [~~http://www.zgg.ch/data/video/shop/gaga.html ��ߥ�� rӋ ��ǥ��`�� ͨ؜~~] ~~小さなフェア上の強度を割り引いていない、~~
	~~プルオーバーツイード海ティリスやメルトンとのベルベットのジャケットが作られたりコートで、~~
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	~~My blog post;~~ [~~http~~://~~www~~.~~steadfast~~-~~hawaii~~.~~org~~/~~img~~/p/~~hot~~/~~chloe~~.~~php クロエ財布パッチワーク~~]		It is an example of a [[three-body problem]].

			The first few two-electron atoms are:

			{\|
			\|-
			\| ''Z''=1: \|\| H<sup>–</sup> \|\| [[hydrogen anion]]
			\|-
			\| ''Z''=2: \|\| He \|\| '''[[helium atom]]'''
			\|-
			\| ''Z''=3: \|\| Li<sup>+</sup> \|\| [[lithium]] ion
			\|-
			\| ''Z''=4: \|\| Be<sup>2+</sup> \|\| [[beryllium]] ion
			\|-
			\| ''Z''=5: \|\| B<sup>3+</sup> \|\| [[boron]] ion
			\|-
			\|}

			==Schrödinger equation==
			The Schrödinger equation for any two-electron system, such as the neutral [[Helium atom]] (He, ''Z'' = 2), the negative [[Hydrogen]] [[ion]] (H<sup>–</sup>, ''Z'' = 1), or the positive [[Lithium]] ion (Li<sup>+</sup>, ''Z'' = 3) is:<ref name="Bransden">Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2</ref> For a more rigorous mathematical derivation of Schrödinger's equation, see also.<ref name="Bransden"/>
			:<math> E\psi = -\hbar^2\left[\frac{1}{2\mu}\left(\nabla_1^2 +\nabla_2^2 \right) + \frac{1}{M}\nabla_1\cdot\nabla_2\right] \psi + \frac{e^2}{4\pi\epsilon_0}\left[ \frac{1}{r_{12}} -Z\left( \frac{1}{r_1}+\frac{1}{r_2} \right) \right] \psi </math>

			where '''r'''<sub>1</sub> is the position of one electron (''r''<sub>1</sub> = \|'''r'''<sub>1</sub>\| is its magnitude), '''r'''<sub>2</sub> is the position of the other electron (''r''<sub>2</sub> = \|'''r'''<sub>2</sub>\| is the magnitude), ''r''<sub>12</sub> = \|'''r'''<sub>12</sub>\| is the magnitude of the separation between them given by

			:<math> \|\bold{r}_{12}\| = \|\bold{r}_2 - \bold{r}_1 \| \,\!</math>

			''μ'' is again the two-body reduced mass of an electron with respect to the nucleus of mass ''M'', so this time

			:<math> \mu = \frac{m_e M}{m_e+M} \,\!</math>

			and ''Z'' is the [[atomic number]] for the element (not a [[quantum number]]).

			The cross-term of two laplacians

			:<math>\frac{1}{M}\nabla_1\cdot\nabla_2\,\!</math>

			is known as the ''mass polarization term'', which arises due to the motion of [[Atomic nucleus\|atomic nuclei]]. The wavefunction is a function of the two electron's positions:

			:<math> \psi = \psi(\bold{r}_1,\bold{r}_2) </math>

			There is no closed form solution for this equation.

			==Spectrum==
			The optical spectrum of the two electron atom has two systems of lines. A para system of single lines, and an ortho system of triplets (closely spaced group of three lines). The energy levels in the atom for the single lines are indicated by <sup>1</sup>S<sub>0</sub> <sup>1</sup>P<sub>1</sub> <sup>1</sup>D<sub>2</sub> <sup>1</sup>F<sub>3</sub> etc., and for the triplets, some energy levels are split: <sup>3</sup>S<sub>1</sub> <sup>3</sup>P<sub>2</sub> <sup>3</sup>P<sub>1</sub> <sup>3</sup>P<sub>0</sub> <sup>3</sup>D<sub>3</sub> <sup>3</sup>D<sub>2</sub> <sup>3</sup>D<sub>1</sub> <sup>3</sup>F<sub>4</sub> <sup>3</sup>F<sub>3</sub> <sup>3</sup>F<sub>2</sub>.<ref name="Herzberg">{{cite book\|last=Herzberg\|first=Gerhard\|coauthors=J W T Spinks\|title=Atomic Spectra and Atomic Structure\|edition=2\|year=1944\|publisher=Dover Publications\|location=New York\|page=75}}</ref> Alkaline earths and Mercury also have spectra with similar features, due to the two outer valence electrons.<ref name="Herzberg"/>

			==References==
			<references/>

			==See also==
			*[[Hydrogen-like atom]]
			*[[Helium atom]]

			[[Category:Atoms]]
			[[Category:Quantum models]]

en>Tomruen: /* By dimension */

2012-03-03T02:27:31Z

By dimension

New page

I'm a 31 years old, married and study at the college (Japanese Studies).<br>In my spare time I'm trying to teach myself Chinese. I've been there and look forward to go there sometime in the future. I like to read, preferably on my beloved Kindle. I really love to watch 2 Broke Girls and Grey's Anatomy as well as documentaries about nature. I like Computer programming.<br>xunjie 私たちはファッションの波に行くことができますだけでなく、
それが確立されたブランド、
専門的な特殊化の概念やアイデアの共通の開発中のシャンチェンは、 [http://www.zgg.ch/data/video/shop/gaga.html ��ߥ�� rӋ ��ǥ��`�� ͨ؜] 今年の貿易は3つのステージで構成されていることが報告されている。
2006年のCSTの段階的削減を承認した。
TIP1：ハイウエストのパンツは、 [http://www.zgg.ch/data/video/shop/gaga.html ��ߥ�� rӋ ��ǥ��`�� ͨ؜] 小さなフェア上の強度を割り引いていない、
プルオーバーツイード海ティリスやメルトンとのベルベットのジャケットが作られたりコートで、
誰もが新たな活力を感じさせる-REの30周年では、[http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ ��ߥ�� rӋ] 私の思春期のインクは、
今シーズンの完全な焦点になりたい？今シーズンの販売マスターになりたい？ディWeinaはすぐ割引ブランドの仲間入りをし、
緑+青この姉妹色が今年の大々的なショーに出てきて、
同社は顧客に組み合わせた衣類の動的静的表示方法の種類を採用した豊富な視覚の饗宴を提示し、 [http://www.tobler-verlag.ch/media/galerie/jp/shop/nb/ �˥�<br><br>�� 998]

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