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The '''Wigner–Eckart theorem''' is a [[theorem]] of [[representation theory]] and [[quantum mechanics]]. It states that [[Matrix (mathematics)|matrix]] elements of [[spherical tensor]] [[Operator (physics)|operator]]s on the basis of [[angular momentum]] [[eigenstate]]s can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a [[Clebsch-Gordan coefficient]]. The name derives from physicists [[Eugene Wigner]] and [[Carl Eckart]] who developed the formalism as a link between the symmetry transformation groups of space (applied to the Schrödinger equations) and the laws of conservation of energy, momentum, and angular momentum.<ref name="Eckart Biography">[http://orsted.nap.edu/openbook.php?record_id=571&page=194 Eckart Biography]– The National Academies Press</ref>
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:<math>\langle jm|T^k_q|j'm'\rangle =\langle j||T^k||j'\rangle C^{jm}_{kqj'm'}</math>
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where ''T<sub>q</sub><sup>k</sup>'' is a rank ''k'' spherical tensor, <math>|jm\rangle</math> and <math>|j'm'\rangle</math> are eigenkets of total angular momentum ''J''<sup>2</sup> and its z-component ''J<sub>z</sub>'', <math>\langle j||T^k||j'\rangle</math> has a value which is independent of ''m'' and ''q'', and <math>C^{jm}_{kqj'm'}=\langle j'm';kq|jm \rangle</math> is the Clebsch-Gordan coefficient for adding ''j''&prime; and ''k'' to get ''j''.
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In effect, the Wigner–Eckart theorem says that operating with a spherical tensor operator of rank ''k'' on an angular momentum eigenstate is like adding a state with angular momentum ''k'' to the state. The matrix element one finds for the spherical tensor operator is proportional to a Clebsch-Gordan coefficient, which arises when considering adding two angular momenta.
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Starting with the definition of a [[spherical tensor]], we have that
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:<math>[J_{\pm}, T_q^{(k)}]=\hbar \sqrt{(k\mp q)(k\pm q+1)}T_{q\pm 1}^{(k)}</math>
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which we use to then calculate
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:<math>\langle \alpha',j'm'|[J_{\pm}, T_q^{(k)}]|\alpha,jm\rangle=\hbar \sqrt{(k\mp q)(k\pm q+1)}\langle \alpha',j'm'|T_{q\pm 1}^{(k)}|\alpha,jm\rangle </math>.
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If we expand the commutator on the LHS by calculating the action of the ''J''<sub>±</sub> on the bra and ket, then we get
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:<math> \begin{align}
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\langle \alpha',j'm'|[J_{\pm}, T_q^{(k)}]|\alpha,jm\rangle
 
& = \sqrt{(j'\pm m')(j'\mp m'+1)}\langle \alpha',j'm'\mp1 |T_{q}^{(k)}|\alpha,jm\rangle\\
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& \qquad -\sqrt{(j\mp m)(j\pm m+1)}\langle \alpha',j'm' |T_{q}^{(k)}|\alpha,jm\pm 1\rangle  
 
\end{align} </math>.
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We may combine these two results to get
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\sqrt{(j'\pm m')(j'\mp m'+1)}\langle \alpha',j'm'\mp1 |T_{q}^{(k)}|\alpha,jm\rangle
 
& = \sqrt{(j\mp m)(j\pm m+1)}\langle \alpha',j'm' |T_{q}^{(k)}|\alpha,jm\pm 1\rangle\\
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& \qquad +\sqrt{(k\mp q)(k\pm q+1)}\langle \alpha',j'm'|T_{q\pm 1}^{(k)}|\alpha,jm\rangle
 
\end{align} </math>.
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This recursion relation for the matrix elements closely resembles that of the [[Clebsch-Gordan coefficient]]. In fact, both are of the form <math>\sum_j a_{ij}x_j=0</math>. We therefore have two sets of linear homogeneous equations
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:<math>\sum_j a_{ij}x_j=0,\qquad \sum_j a_{ij}y_j=0</math>  
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one for the Clebsch-Gordan coefficients (''x<sub>j</sub>'') and one for the matrix elements (''y<sub>j</sub>''). It is not possible to exactly solve for the ''x<sub>j</sub>''. We can only say that the ratios are equal, that is
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:<math>\frac{x_j}{x_k}=\frac{y_j}{y_k}</math>
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or that ''x<sub>j</sub>'' = ''cy<sub>j</sub>'', where ''c'' is a coefficient of proportionality independent of the indices. Hence, by comparing recursion relations, we can identify the Clebsch-Gordan coefficient <math>\langle j_1 j_2; m_1,m_2\pm 1|j_1 j_2; jm \rangle</math> with the matrix element <math>\langle \alpha', j'm'|T_{q\pm 1}^{(k)}|\alpha, jm\rangle</math>, then we may write
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:<math>\langle \alpha', j'm'|T_{q\pm 1}^{(k)}|\alpha, jm\rangle=\text{(proportionality constant)}\langle jk; mq\pm 1|jk;j'm'\rangle</math>.
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By convention the proportionality constant is written as <math>\langle \alpha'j'||T^{(k)}||\alpha j\rangle \frac{1}{\sqrt{2j+1}}</math>, where the denominator is a normalizing factor.
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==Example==
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Consider the position expectation value <math>\langle njm|x|njm\rangle</math>. This matrix element is the expectation value of a Cartesian operator in a spherically-symmetric hydrogen-atom-eigenstate [[Basis (linear algebra)|basis]], which is a nontrivial problem. However, using the Wigner–Eckart theorem simplifies the problem. (In fact, we could obtain the solution quickly using [[Parity (physics)|parity]], although a slightly longer route will be taken.)
</ul>
 
We know that ''x'' is one component of {{vec|''r''}}, which is a vector. Vectors are rank-1 tensors, so ''x'' is some linear combination of ''T''<sup>1</sup><sub>q</sub> for ''q'' = -1, 0, 1. In fact, it can be shown that  
 
:<math>x=\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}\,,</math>
 
where we defined the  
[[spherical tensor]]s<ref name="J. Sakurai 1994">J. J. Sakurai: "Modern quantum mechanics" (Massachusetts, 1994, Addison-Wesley)</ref>
''T''<sup>1</sup><sub>0</sub> = ''z''
and
:<math>T^1_{\pm1}=\mp (x \pm i y)/{\sqrt{2}}</math>
(the pre-factors have to be chosen according to the definition<ref name="J. Sakurai 1994"/> of a [[spherical tensor]] of rank ''k''. Hence, the ''T''<sup>1</sup><sub>''q''</sub> are only proportional to the [[ladder operators]]).
Therefore
:<math>\langle njm|x|n'j'm'\rangle = \langle njm|\frac{T_{-1}^{1}-T^1_1}{\sqrt{2}}|n'j'm'\rangle = \frac{1}{\sqrt{2}}\langle nj||T^1||n'j'\rangle (C^{jm}_{1(-1)j'm'}-C^{jm}_{11j'm'})</math>
The above expression gives us the matrix element for ''x'' in the <math>|njm\rangle</math> basis.  To find the expectation value, we set ''n''&prime; = ''n'', ''j''&prime; = ''j'', and ''m''&prime; = ''m''.  The selection rule for ''m''&prime; and ''m'' is <math>m\pm1=m'</math> for the <math>T_{\mp1}^{(1)}</math> spherical tensors. As we have ''m''&prime; = ''m'', this makes the Clebsch-Gordan Coefficients zero, leading to the expectation value to be equal to zero.
 
==References==
 
<references/>
 
==External links==
 
*J. J. Sakurai, (1994). "Modern Quantum Mechanics", Addison Wesley, ISBN 0-201-53929-2.
*{{mathworld|urlname=Wigner-EckartTheorem|title= Wigner–Eckart theorem}}
*[http://electron6.phys.utk.edu/qm2/modules/m4/wigner.htm Wigner–Eckart theorem]
*[http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/TensorOperators.htm Tensor Operators]
 
{{DEFAULTSORT:Wigner-Eckart theorem}}
[[Category:Quantum mechanics]]
[[Category:Representation theory of Lie groups]]
[[Category:Theorems in quantum physics]]
[[Category:Theorems in representation theory]]

Latest revision as of 10:24, 1 May 2014

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"That transaction isn't approved," Moore told journalists when asked about a suggested Telus deal with Mobilicity.

Moore confirmed he had been Nike Shox referring to a second attempt by Telus to acquire spectrum won by Mobilicity in a 2008 auction which was set aside for challengers to dominant operators such as Telus.

Ottawa blocked Telus' initial C$380 million bid for Mobilicity in June, but allow it Nike Blazer Mid Uk to needlessly buy another small rider, Public Mobile, last week. Community bought airwaves in the similar auction, but its spectrum had never been considered off limits to the massive operators.

A spokesman pertaining to Mobilicity said the company remains inside talks with Industry Canada officials. The company declined to supply further comment.

Mobilicity, which has a lot less than 200,000 customers, received creditor protection last month simply because it sought approval for a deal that would allow it to keep operating.

"We believe that either the Government is going to be forced to change its brain and Mobilicity will be acquired by way of Telus, or it will cease to Cheap Nike Roshe Run Uk work," Canaccord Genuity analyst Dvai Ghose wrote inside a note to clients.

Telus can be separately seeking a judicial review of the government's Ralph Lauren Sale Uk restrictions on the sale of variety licenses, which Ottawa says should limit the market dominance involving Telus, Rogers Communications Inc and BCE Corporation, which together control many 90% of the wireless market.

Comet Ison Cheap Nike Trainers Uk

Comet Ison (named after the Russian International Scientific Optical Network, which will discovered it) hurtled towards the sun's rays at 845,000 miles per hour until, on Nov. 28, the item came less than one solar diameter from our star's surface (with regards to 724,000 miles); the heat (pretty much 3,000C), and tidal forces shown too much for the so called "comet with the century."Ison (probably) disintegrated. Nevertheless the excitement and attention it pulled in proved that, while do not seen as omens of doom or messengers from the gods, comets retain the power to jolt and awe us.'AS Mystical AS THEY COME'"Comets are about as unexplained as they come," said Dr. Stephen Lowry, a cometary scientist on the centre for astrophysics and planetary technology at the University of Kent in England.Comets, like planet's, orbit our sun, but these "dirty snowballs" originate from one of two locations within our solar system: the actual Kuiper Belt, beyond the orbit of Neptune; along with the extremely distant shell regarding debris called the Oort Cloud, which usually extends far into interstellar room."Many of the comets that formed one of several outer planets were tossed out of the planetary system to very distant orbits to form the Oort Reasoning; Ison was one of these," explained Prof. Alan Fitzsimmons from the astrophysics research heart at Queen's University in Belfast.Comet Ison, he said, was a first time visitor, but on an orbit Cheap Nike Trainers Uk that would make it a sungrazer, skimming just above the surface of the sunshine.A FIRST IN MODERN HISTORY"We haven't ever seen such a comet before throughout modern history, and examining its behaviour should uncover more clues to the framework of comets and how they were originally put together."Cometary orbits can vary dramatically. Individuals from the Oort Cloud can take an incredible number of years to complete one Cheap Hollister Hoodies elliptical exerciser circuit of the sun.Some others, like Halley's Comet, can return in just a lifetime provided you live beyond 75.Halley's Comet was first noted by way of the Chinese in 240 BC, and appears in the Bayeux Tapestry; it is branded after the English astronomer Edmund Halley, that first calculated its orbit.It truly is their distinctive tail which makes comets so instantly recognizable. That forms when a comet is near the sun, the warmth heating up the particular frozen gases, ice and dirt inside the comet's nucleus to form a coma, or surroundings, and a tail.TWO TYPES OF TAIL"There are two main types of tail: the particular dust tail and the petrol tail," said Prof. Level Bailey of the Armagh Observatory in Northern Cork."The gas is blown by the solar wind and the airborne debris is blown out by radiation pressure from the sunrays."The gas tail is like this smoke from a ship within a gale. It doesn't really matter that way the comet is going as the speed of solar blowing wind from the sun is much quicker than the comet."So the gas end is more or less radially directed away from the solar."If the Earth passes through the trail of a comet Nike Shoes Nz such as happened by using comet Swift Tuttle in August we go through a meteor shower.But it's just what lies inside a comet that is best.TIME CAPSULE"It's the fact that you've got these types of time capsule of primordial goods left over from the birth on the solar system," claimed Fitzsimmons, "and by studying it you have an idea of the conditions and surroundings that was around when the planets and the sun formed."Next twelve months, scientists will be doing this when, after a 10 year voyage, the European Space Agency spacecraft Rosetta finds comet 67P/Churyumov Gerasimenko."Rosetta is by far the largest cometary mission introduced to date," said Lowry, a member of the team behind the Osiris optical camera on board Rosetta."Studying comet 67P/Churyumov Gerasimenko will allow people to understand specifically what Kuiper Weight loss belt Cheap Longchamp Uk bodies are made of and how these people formed."

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