Coefficients of Clebsch-Gordan

In Physical, the coefficients of Clebsch-Gordan are numbers which appear at the time of the study of the couplings of Angular momentum subjected to the laws of the quantum Mécanique. They bear the name of the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912), which encountered a similar problem in Théorie of the invariants.

In theory of the representation, in particular of the compact groups of Dregs , these coefficients are used to carry out the direct Somme produced tensorial of two irreducible representations.

One can define the coefficients of Clebsch-Gordan associated with the group '' SO ('' 3 '') '' in a more direct way, like product of harmonic spherical. The addition of Spin S in quantum Mécanique is included/understood by this approach. In this article, one will use the Notation bra-ket of Dirac.

Preliminary notations

Operators of angular momentum

The Opérateur S of Angular momentum are the square operators $j\_1,\; j\_2$ and $j\_3$ which checks the following relations:

$\backslash \; left\; j\_k,\; j\_l\; \backslash \; right\; =\; I\; \backslash \; sum\_\; \{m=1\}\; ^3\; \backslash \; varepsilon\_\; \{klm\}\; j\_m\; \backslash ,$

with $\backslash \; varepsilon\_\; \{klm\}$ the Symbol of Levi-Civita. These three terms can be regarded as the components of a vectorial operator $\backslash \; mathbf\; \{J\}$. The square of the standard of $\backslash \; mathbf\; \{J\}$ is defined by:

$\backslash \; mathbf\; \{J\}\; ^2\; =\; j\_1^2+j\_2^2+j\_3^2$

One also defines the operators $(j\_+)$ and $(j\_-)$ by:

$j\_\; \backslash \; pm\; =\; j\_1\; \backslash \; pm\; I\; j\_2.\; \backslash ,$

States of angular momentum

One can show that $\backslash \; mathbf\; \{J\}\; ^2$ commutates with $j\_1,\; j\_2$ and $j\_3$:

$\backslash \; left\; \backslash \; mathbf\; \{J\}\; ^2,\; j\_k\; \backslash \; right\; =\; 0$ with K = 1,2,3.

When two square operators commutate, they have a common whole of clean functions. By convention, one chooses $\backslash \; mathbf\; \{J\}\; ^2$ and $j\_3$. According to the relations of commutation, one determines the eigenvalues:

j_3|J m \ rangle = m |J m \ rangle & \; \; \; m = - J, - j+1, \ ldots, J. \end{alignat}

The operators $(j\_+)$ and $(j\_-)$ change the value of $m$:

$j\_\backslash pm\; |jm\; \backslash \; rangle\; =\; C\_\; \backslash \; pm\; \backslash \; left\; (J,\; m\; \backslash \; right)\; |J\; m\; \backslash \; pm\; 1\; \backslash \; rangle$

with

$C\_\; \backslash \; pm\; \backslash \; left\; (J,\; m\; \backslash \; right)\; =\; \backslash \; sqrt\; \{J\; \backslash \; left\; (j+1\; \backslash \; right)\; -\; m\; \backslash \; left\; (m\; \backslash \; pm\; 1\; \backslash \; right)\}\; =\; \backslash \; sqrt\; \{\backslash \; left\; (J\; \backslash \; mp\; m\; \backslash \; right)\; \backslash \; left\; (J\; \backslash \; pm\; m\; +\; 1\; \backslash \; right)\}.$

A factor of dephasing (complex) can be added to the definition of $C\_\; \backslash \; pm\; \backslash \; left\; (J,\; m\; \backslash \; right)$. The states of Angular momentum must be orthogonal — because their eigenvalues are distinct — and are supposed to be standardized:

$\backslash langle\; j\_1\; m\_1\; |\; j\_2\; m\_2\; \backslash \; rangle\; =\; \backslash \; delta\_\; \{j\_1,\; j\_2\}\; \backslash \; delta\_\; \{m\_1,\; m\_2\}.$

Definition and properties

Definition

The states of Angular momentum can be developed by supposing them not-coupled:

$|\backslash \; left\; (j\_1j\_2\; \backslash \; right)\; JM\; \backslash \; rangle\; =\; \backslash \; sum\_\; \{m\_1=-j\_1\}\; ^\; \{j\_1\}\; \backslash \; sum\_\; \{m\_2=-j\_2\}\; ^\; \{j\_2\}\; |j\_1m\_1\; \backslash \; rangle|j\_2m\_2\backslash rangle\; \backslash langle\; j\_1m\_1j\_2m\_2|JM\backslash rangle$

The coefficients which appear in the development, noted $\backslash \; langle\; j\_1m\_1j\_2m\_2|JM\; \backslash \; rangle$, are the coefficients of Clebsch-Gordan.

By applying the operator:

$J\_3\; =\; j\_3\; \backslash \; otimes\; 1\; +\; 1\; \backslash \; otimes\; j\_3$

on the two sides of the equality, one shows that the coefficients of Clebsch-Gordan can not be null only when:

$M\; =\; m\_1\; +\; m\_2.\; \backslash ,$

Relations of orthogonality

One can introduce the alternative, but equivalent, following notation:

$\backslash \; langle\; J\; M|j\_1\; m\_1\; j\_2\; m\_2\backslash rangle\; \backslash equiv\; \backslash langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; M\; \backslash \; rangle$

It is then possible to establish two relations of orthogonality:

$\backslash sum\_\{J=|j\_1-j\_2|\}\; ^\; \{j\_1+j\_2\}\; \backslash \; sum\_\; \{M=-J\}\; ^\; \{J\}\; \backslash \; langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; M\; \backslash \; rangle\; \backslash \; langle\; J\; M|j\_1\; m\_1\text{'}\; j\_2\; m\_2\text{'}\; \backslash \; rangle\; =\; \backslash \; delta\_\; \{m\_1,\; m\_1\text{'}\}\; \backslash \; delta\_\; \{m\_2,\; m\_2\text{'}\}$

$\backslash \; sum\_\; \{m\_1m\_2\}\; \backslash \; langle\; J\; M|j\_1\; m\_1\; j\_2\; m\_2\backslash rangle\; \backslash langle\; j\_1\; m\_1\; j\_2\; m\_2|I\; \backslash \; rangle\; =\; \backslash \; delta\_\; \{J,\; I\}\; \backslash \; delta\_\; \{M,\; Me\}$

Properties of symmetry

The relation of following Symétrie is always valid:

$\backslash langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; M\; \backslash \; rangle\; =\; \backslash \; left\; (-\; 1\; \backslash \; right)\; ^\; \{j\_1+j\_2-j\_3\}\; \backslash \; langle\; j\_1\; \{-\; m\_1\}\; j\_2\; \{-\; m\_2\}|J\; \{-\; M\}\; \backslash \; rangle=\; \backslash \; left\; (-\; 1\; \backslash \; right)\; ^\; \{j\_1+j\_2-j\_3\}\; \backslash \; langle\; j\_2\; m\_2\; j\_1\; m\_1|J\; M\; \backslash \; rangle.$

Bond with the symbols 3-jm

The coefficients of Clebsch-Gordan are connected to the Symboles 3-jm, which are more pleasant to handle because of Symétrie S simpler. This relation is expressed by the following equation:

$\backslash langle\; j\_1\; m\_1\; j\_2\; m\_2\; |\; j\_3\; m\_3\; \backslash \; rangle\; =\; \backslash \; left\; (-\; 1\; \backslash \; right)\; ^\; \{j\_1-j\_2+m\_3\}\; \backslash \; sqrt\; \{2j\_3+1\}$

\begin{pmatrix} j_1 & j_2 & j_3 \ \ m_1 & m_2 & - m_3 \end{pmatrix}.

See too

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