Kostant polynomial: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>The Disambiguator
en>Mark viking
→‎Background: Added wlm one red link for ring of exponentials
 
Line 1: Line 1:
In mathematics and mathematical physics, '''Slater integrals''' are certain integrals of products of three [[spherical harmonic]]s. They occur naturally when applying an [[orthonormal basis]] of functions on the [[unit sphere]] that transform in a particular way under rotations in three dimensions. Such integrals are particularly useful when computing properties of atoms which have natural spherical symmetry. These integrals are defined below along with some of their mathematical properties.


==Formulation==


In connection with the [[Quantum mechanics|quantum theory]] of [[atomic structure]], [[John C. Slater]] defined the integral of three spherical harmonics as a coefficient <math>c</math>.<ref>John C. Slater, Quantum Theory of Atomic Structure, McGraw-Hill (New York, 1960), Volume I</ref> These coefficients are essentially the product of two [[Wigner 3-j symbol|Wigner 3jm symbols]].
They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. The Word - Press Dashboard : an administrative management tool that supports FTP content upload  2. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. It allows Word - Press users to easily use HTML5  the element enable native video playback within the browser. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated.  If you are you looking for more about [http://urlon.com.br/wordpress_backup_plugin_189197 wordpress backup] take a look at our internet site. Many times the camera is following Mia, taking in her point of view in almost every frame. <br><br>Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. To turn the Word - Press Plugin on, click Activate on the far right side of the list. Socrates: (link to  ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.
 
:<math>c^k(\ell,m,\ell',m')=\int d^2\Omega \ Y_\ell^m(\Omega)^* Y_{\ell'}^{m'}(\Omega) Y_k^{m-m'}(\Omega)</math>
 
These integrals are useful and necessary when doing atomic calculations of the [[Hartree–Fock]] variety where matrix elements of the [[Coulomb operator]] and [[Exchange operator]] are needed. For an explicit formula, one can use Gaunt's formula for [[associated Legendre polynomials]].
 
Note that the product of two spherical harmonics can be written in terms of these coefficients. By expanding such a product over a [[spherical harmonic]] basis with the same order
:<math>
Y_\ell^m Y_{\ell'}^{m'} = \sum_{\ell''} \hat{A}^{\ell''}(\ell,m,\ell',m',) Y_{\ell''}^{m+m'},
</math>
one may then multiply by <math>Y^*</math> and integrate, using the conjugate property and being careful with phases and normalisations:
:<math>
\int Y_\ell^m Y_{\ell'}^{m'} Y_{L}^{-M} d^2\Omega = (-1)^{m+m'}\hat{A}^{L}(\ell,m,\ell',m') = (-1)^{m}c^L(\ell,-m,\ell',m').
</math>
Hence
:<math>
Y_\ell^m Y_{\ell'}^{m'} = \sum_{\ell''}(-1)^{m'} c^{\ell''}(\ell,-m,\ell',m',) Y_{\ell''}^{m+m'},
</math>
These coefficient obey a number of identities.  They include
 
::<math>
\begin{align}
c^k(\ell,m,\ell',m') &= c^k(\ell,-m,\ell',-m')\\
&=(-1)^{m-m'}c^k(\ell',m',\ell,m)\\
&=(-1)^{m-m'}\sqrt{\frac{2\ell+1}{2k+1}}c^\ell(\ell',m',k,m'-m)\\
& = (-1)^{m'}\sqrt{\frac{2\ell'+1}{2k+1}}c^{\ell'}(k,m-m',\ell,m).\\
\sum_{m=-\ell}^{\ell} c^k(\ell,m,\ell,m)  &=  (2\ell+1)\delta_{k,0}.\\
\sum_{m=-\ell}^\ell \sum_{m'=-\ell'}^{\ell'} c^k(\ell,m,\ell',m')^2  &=  \sqrt{(2\ell+1)(2\ell'+1)}\cdot c^k(\ell,0,\ell',0).\\
\sum_{m=-\ell}^\ell c^k(\ell,m,\ell',m')^2 & =  \sqrt{\frac{2\ell+1}{2\ell'+1}}\cdot c^k(\ell,0,\ell',0).\\
\sum_{m=-\ell}^\ell c^k(\ell,m,\ell',m')c^k(\ell,m,\tilde\ell,m')  &=  \delta_{\ell',\tilde\ell}\cdot\sqrt{\frac{2\ell+1}{2\ell'+1}}\cdot c^k(\ell,0,\ell',0).\\
\sum_m c^k(\ell,m+r,\ell',m) c^k(\ell,m+r,\tilde\ell,m)  &=  \delta_{\ell,\tilde\ell} \cdot \frac{\sqrt{(2\ell+1)(2\ell'+1)}}{2k+1}\cdot c^k(\ell,0,\ell',0).\\
\sum_m c^k(\ell,m+r,\ell',m)c^q(\ell,m+r,\ell',m)  &= \delta_{k,q}\cdot\frac{\sqrt{(2\ell+1)(2\ell'+1)}}{2k+1}\cdot c^k(\ell,0,\ell',0).
\end{align}
</math>
 
==References==
<references/>
 
[[Category:Atomic physics]]
[[Category:Quantum chemistry]]
[[Category:Rotational symmetry]]

Latest revision as of 22:15, 24 September 2014


They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. All this is very simple, and the best thing is that it is totally free, and you don't need a domain name or web hosting.

Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. If you are you looking for more about wordpress backup take a look at our internet site. Many times the camera is following Mia, taking in her point of view in almost every frame.

Just ensure that you hire experienced Word - Press CMS developer who is experienced enough to perform the task of Word - Press customization to get optimum benefits of Word - Press CMS. It was also the very first year that the category of Martial Arts was included in the Parents - Connect nationwide online poll, allowing parents to vote for their favorite San Antonio Martial Arts Academy. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. To turn the Word - Press Plugin on, click Activate on the far right side of the list. Socrates: (link to ) Originally developed for affiliate marketers, I've used this theme again and again to develop full-fledged web sites that include static pages, squeeze pages, and a blog.

Additionally Word - Press add a default theme named Twenty Fourteen. The SEOPressor Word - Press SEO Plugin works by analysing each page and post against your chosen keyword (or keyword phrase) and giving a score, with instructions on how to improve it. The templates are designed to be stand alone pages that have a different look and feel from the rest of your website. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white.

There is no denying that Magento is an ideal platform for building ecommerce websites, as it comes with an astounding number of options that can help your online business do extremely well. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. Thus, Word - Press is a good alternative if you are looking for free blogging software. Definitely when you wake up from the slumber, you can be sure that you will be lagging behind and getting on track would be a tall order.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Kostant_polynomial&oldid=264215"