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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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The Wigner–Eckart theorem reads:
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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''.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


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. When stated another way, one can say that the Wigner-Eckart theorem is a theorem that tells you how vector operators behave in a subspace. Within a given subspace, a component of a vector operator will behave in a way proportional to the same component of the angular momentum operator. This definition is given in the book "Quantum Mechanics" by Cohen-Tannoudji, Diu and Laloe.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==Proof==
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


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>
==Demos==


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>.


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}
==Test pages ==
\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\\
& \qquad -\sqrt{(j\mp m)(j\pm m+1)}\langle \alpha',j'm' |T_{q}^{(k)}|\alpha,jm\pm 1\rangle 
\end{align} </math>.


We may combine these two results to get
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<math> \begin{align}
*[[Inputtypes|Inputtypes (private Wikis only)]]
\sqrt{(j'\pm m')(j'\mp m'+1)}\langle \alpha',j'm'\mp1 |T_{q}^{(k)}|\alpha,jm\rangle
*[[Url2Image|Url2Image (private Wikis only)]]
& = \sqrt{(j\mp m)(j\pm m+1)}\langle \alpha',j'm' |T_{q}^{(k)}|\alpha,jm\pm 1\rangle\\
==Bug reporting==
& \qquad +\sqrt{(k\mp q)(k\pm q+1)}\langle \alpha',j'm'|T_{q\pm 1}^{(k)}|\alpha,jm\rangle
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
\end{align} </math>.
 
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
 
:<math>\sum_j a_{ij}x_j=0,\qquad \sum_j a_{ij}y_j=0</math>
 
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
 
:<math>\frac{x_j}{x_k}=\frac{y_j}{y_k}</math>
 
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
 
:<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>.
 
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.
 
==Example==
 
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.)
 
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/>
 
==See Also==
 
*[[Tensor operator]]
 
 
==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 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

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