Hyperoctahedral group: Difference between revisions

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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.
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It is an example of a [[three-body problem]].
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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]]

Latest revision as of 18:03, 18 December 2014

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The light is known as a summer shade, as white reflects it. However, white is very hard to keep clean than other colours and shows filth. The most emotionally intense color, red symbolizes fire and love. Reddish clothing is found and makes the wearer appear big time.

Wearing clothing that was red ought to be avoided during discussions or confrontations as it's an intense colour. As an accent, reddish colour is generally used in home decorating. Next time you happen to be considering bringing people's attention be sure cheap ugg boots that red top is going to do the trick.
Green coloring represents development and life. It is the http://tinyurl.com/nqbkz6z color that is most soothing and tranquilizing. Hospital staff wears green uniforms since it relaxes individuals. People that wear clothing that was green are thought to be generous and very kind.

Green is extremely popular amongst branding colors in financial organizations, legal practices and businesses that are environmental as it correlated with harmony, energy, wealth and efficiency. Yellow is the shade that is most joyful. http://tinyurl.com/nqbkz6z It symbolizes optimism, happiness and warmth.
There is a purpose why the smiley-face icon is yellowish. Those who wear yellow clothing are thought to be sensible very communicative and cheerful. Nevertheless, anxiety can be caused by a lot of exposure to yellow - so go easy with that color. Based on Dr. Alexander Schauss, who's manager of the American Institute of Biosocial and Medical Research in Tacoma, WA, if a man attempts to be upset or intense in the existence of pink, he cannot.

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Military officers and police use uniforms in blue. Royal purple exudes sensuality class and extravagance. Clothing that is purple indicates the person is romantic. This color is associated with sophistication. People who wear clothing that was purple are regarded as strange or creative.
Understanding it or maybe not, colours say more than you have ever believed. Next time you get your clothing, bear that in mind.

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