Clebsch-Gordan coefficients

In physics, the Clebsch-Gordan coefficients are sets of numbers that arise in angular momentum coupling under the laws of quantum mechanics.

In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912), who encountered an equivalent problem in invariant theory.

In terms of classical mathematics, the CG coefficients, or at least those associated to the group SO(3), may be defined much more directly, by means of formulae for the multiplication of spherical harmonics. The addition of spins in quantum-mechanical terms can be read directly from this approach. The formulas below use Dirac's bra-ket notation.

Clebsch-Gordan coefficients

Clebsch-Gordan coefficients are the expansion coefficients oftotal angular momentum eigenstates in anuncoupled tensor product basis.

Below, this definition is made precise by defining angular momentumoperators, angular momentum eigenstates, and tensor products of these states.

From the formal definition recursion relations for the Clebsch-Gordan coefficientscan be found. To find numerical values for the coefficients a phase conventionmust be adopted. Below the Condon-Shortley phase convention is chosen.

Angular momentum operators

Angular momentum operators are Hermitian operators $j\_1,\; j\_2$,and $j\_3$ that satisfy the commutation relations:$[j\_k,j\_l]\; =\; i\; sum\_\{m=1\}^3\; varepsilon\_\{klm\}j\_m,$where $varepsilon\_\{klm\}$ is the Levi-Civita symbol. Together thethree components define a vector operator $\{mathbf\; j\}$. Thesquare of the length of $\{mathbf\; j\}$ is defined as:$mathbf\{j\}^2\; =\; j\_1^2+j\_2^2+j\_3^2.$We also define raising $(j\_+)$ and lowering $(j\_-)$ operators:$j\_pm\; =\; j\_1\; pm\; i\; j\_2.\; ,$

Angular momentum states

It can be shown from the above definitions that $mathbf\{j\}^2$ commutes with $j\_1,\; j\_2$and $j\_3$:$[mathbf\{j\}^2,\; j\_k]\; =\; 0\; mathrm\{for\}\; k\; =\; 1,2,3$When two Hermitian operators commute a common set of eigenfunctions exists.Conventionally $mathbf\{j\}^2$ and $j\_3$ are chosen.From the commutation relations the possible eigenvalues can be found.The result is:The raising and lowering operators change the value of $m$:$j\_pm\; |j,m\; angle\; =\; C\_pm(j,m)\; |j,mpm\; 1\; angle$with:$C\_pm(j,m)\; =\; sqrt\{j(j+1)-m(mpm\; 1)\}\; =\; sqrt\{(jmp\; m)(jpm\; m\; +\; 1)\}.$A (complex) phase factor could be included in the definition of $C\_pm(j,m)$The choice made here is in agreement with the Condon and Shortley phase conventions.The angular momentum states must be orthogonal (because their eigenvalues withrespect to a Hermitian operator are distinct) and they are assumed to be normalized:$langle\; j\_1,m\_1\; |\; j\_2,m\_2\; angle\; =\; delta\_\{j\_1,j\_2\}delta\_\{m\_1,m\_2\}.$

Tensor product space

Let $V\_1$ be the $2j\_1+1$ dimensionalvector space spanned by the states:$$
j_1 m_1 angle,quad m_1=-j_1,-j_1+1,ldots j_1and $V\_2$ the $2j\_2+1$ dimensionalvector space spanned by:$$
j_2 m_2 angle,quad m_2=-j_2,-j_2+1,ldots j_2.The tensor product of these spaces, $V\_\{12\}equiv\; V\_1otimes\; V\_2$,has a $(2j\_1+1)(2j\_2+1)$ dimensional uncoupled basis:$$
j_1 m_1 angle|j_2 m_2 angle equiv |j_1 m_1 angle otimes |j_2 m_2 angle, quad m_1=-j_1,ldots j_1, quad m_2=-j_2,ldots j_2.Angular momentum operators acting on $V\_\{12\}$ can be defined by:$(j\_i\; otimes\; 1)|j\_1\; m\_1\; angle|j\_2\; m\_2\; angle\; equiv\; (j\_i|j\_1m\_1\; angle)\; otimes\; |j\_2m\_2\; angle$and:$(1\; otimes\; j\_i)\; |j\_1\; m\_1\; angle|j\_2\; m\_2\; angle\; equiv\; |j\_1m\_1\; angle\; otimes\; j\_i|j\_2m\_2\; angle.$Total angular momentum operators are defined by:$J\_i\; =\; j\_i\; otimes\; 1\; +\; 1\; otimes\; j\_iquadmathrm\{for\}quad\; i\; =\; 1,2,3.$The total angular momentum operators satisfy the required commutation relations:$[J\_k,J\_l]\; =\; i\; sum\_\{m=1\}^3\; epsilon\_\{klm\}J\_m$and hence total angular momentum eigenstates exist:It can be derived that $J$ must satisfy the triangular condition:$$
j_1-j_2| leq J leq j_1+j_2.The total number of total angular momentum eigenstates is equal to the dimensionof $V\_\{12\}$:$sum\_\{J=|j\_1-j\_2^\{j\_1+j\_2\}\; (2J+1)\; =\; (2j\_1+1)(2j\_2+1).$The total angular momentum states form an orthonormal basis of $V\_\{12\}$:$langle\; J\_1\; M\_1\; |\; J\_2\; M\_2\; angle\; =\; delta\_\{J\_1J\_2\}delta\_\{M\_1M\_2\}.$

Formal definition of Clebsch-Gordan coefficients

The total angular momentum states can be expanded in the uncoupled basis:$$
(j_1j_2)JM angle = sum_{m_1=-j_1}^{j_1} sum_{m_2=-j_2}^{j_2}
j_1m_1 angle|j_2m_2 angle langle j_1m_1j_2m_2|JM angleThe expansion coefficients $langle\; j\_1m\_1j\_2m\_2|JM\; angle$are called Clebsch-Gordan coefficients.

Applying the operator:$J\_3\; =\; j\_3\; otimes\; 1\; +\; 1\; otimes\; j\_3$to both sides of the defining equation shows that the Clebsch-Gordan coefficientscan only be nonzero when:$M\; =\; m\_1\; +\; m\_2.,$

Recursion relations

Applying the total angular momentum raising and lowering operators:$J\_pm\; =\; j\_pm\; otimes\; 1\; +\; 1\; otimes\; j\_pm$to the left hand side of the defining equation gives:$J\_pm|(j\_1j\_2)JM\; angle\; =\; C\_pm(J,M)\; |(j\_1j\_2)JMpm\; 1\; angle\; =\; C\_pm(J,M)sum\_\{m\_1m\_2\}|j\_1m\_1\; angle|j\_2m\_2\; angle\; langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; Mpm\; 1\; angle.$Applying the same operators to the right hand side gives:Combining these results gives recursion relations for the Clebsch-Gordancoefficients:$C\_pm(J,M)\; langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; Mpm\; 1\; angle\; =\; C\_pm(j\_1,m\_1mp\; 1)\; langle\; j\_1\; \{m\_1mp\; 1\}\; j\_2\; m\_2|J\; M\; angle\; +\; C\_pm(j\_2,m\_2mp\; 1)\; langle\; j\_1\; m\_1\; j\_2\; \{m\_2mp\; 1\}|J\; M\; angle.$Taking the upper sign with $M=J$ gives:$0\; =\; C\_+(j\_1,m\_1-1)\; langle\; j\_1\; \{m\_1-1\}\; j\_2\; m\_2|J\; J\; angle\; +\; C\_+(j\_2,m\_2-1)\; langle\; j\_1\; m\_1\; j\_2\; m\_2-1|J\; J\; angle.$In the Condon and Shortley phase convention the coefficient$langle\; j\_1\; j\_1\; j\_2\; J-j\_1|J\; J\; angle$ is takenreal and positive. With the last equation all otherClebsch-Gordan coefficients $langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; J\; angle$can be found. The normalization is fixed by the requirement thatthe sum of the squares, which corresponds to the norm of thestate $|(j\_1j\_2)JJ\; angle$ must be one.

The lower sign in the recursion relation can be used to findall the Clebsch-Gordan coefficients with $M=J-1$.Repeated use of that equation gives all coefficients.

This procedure to find the Clebsch-Gordan coefficients shows thatthey are all real (in the Condon and Shortley phase convention).

Explicit expression

For an explicit expression of the Clebsch-Gordan coefficientsand tables with numerical values see
table of Clebsch-Gordan coefficients.

Orthogonality relations

These are most clearly written down by introducing thealternative notation:$langle\; J\; M|j\_1\; m\_1\; j\_2\; m\_2\; angle\; equiv\; langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; M\; angle$The first orthogonality relation is:$sum\_\{J=|j\_1-j\_2^\{j\_1+j\_2\}\; sum\_\{M=-J\}^\{J\}\; langle\; j\_1\; m\_1\; j\_2\; m\_2|J\; M\; angle\; langle\; J\; M|j\_1\; m\_1\text{'}\; j\_2\; m\_2\text{'}\; angle\; =\; delta\_\{m\_1,m\_1\text{'}\}delta\_\{m\_2,m\_2\text{'}\}$and the second:$sum\_\{m\_1m\_2\}\; langle\; J\; M|j\_1\; m\_1\; j\_2\; m\_2\; angle\; langle\; j\_1\; m\_1\; j\_2\; m\_2|J\text{'}\; M\text{'}\; angle\; =\; delta\_\{J,J\text{'}\}delta\_\{M,M\text{'}\}.$

pecial cases

For $J=0$ the Clebsch-Gordan coefficients are given by:$langle\; j\_1\; m\_1\; j\_2\; m\_2\; |\; 0\; 0\; angle\; =\; delta\_\{j\_1,j\_2\}delta\_\{m\_1,-m\_2\}frac\{(-1)^\{j\_1-m\_1\{sqrt\{2j\_2+1.$For $J=j\_1+j\_2$ and $M=J$ we have:$langle\; j\_1\; j\_1\; j\_2\; j\_2\; |\; (j\_1+j\_2)\; (j\_1+j\_2)\; angle\; =\; 1.$

ymmetry properties

:

Relation to 3-jm symbols

Clebsch-Gordan coefficients are related to 3-jm symbols which havemore convenient symmetry relations.:

Relation to Wigner D-matrices

:

ee also

* 3-jm symbol
* Racah W-coefficient
* 6-j symbol
* 9-j symbol
* Spherical harmonics
* Associated Legendre polynomials
* Angular momentum
* Angular momentum coupling
* Total angular momentum quantum number
* Azimuthal quantum number
* Table of Clebsch-Gordan coefficients
* Wigner D-matrix

References

* cite book |last= Edmonds |first= A. R. |title= Angular Momentum in Quantum Mechanics |year= 1957
publisher= Princeton University Press |location= Princeton, New Jersey |isbn= 0-691-07912-9

* cite book |last= Condon |first= Edward U. |coauthors= Shortley, G. H. |title= The Theory of Atomic Spectra |year= 1970
publisher= Cambridge University Press |location= Cambridge |isbn= 0-521-09209-4 |chapter= Chapter 3

* cite book |last= Messiah |first= Albert |title= Quantum Mechanics (Volume II) |year= 1981 | edition= 12th edition
publisher= North Holland Publishing |location= New York |isbn= 0-7204-0045-7

* cite book |last= Brink |first= D. M. |coauthors= Satchler, G. R. |title= Angular Momentum
year= 1993 |edition= 3rd edition |publisher= Clarendon Press |location= Oxford |isbn= 0-19-851759-9 |chapter= Chapter 2

* cite book |last= Zare |first= Richard N. |title= Angular Momentum |year=1988
publisher= John Wiley & Sons |location= New York |isbn= 0-471-85892-7 |chapter= Chapter 2

* cite book |last= Biedenharn |first= L. C. |coauthors= Louck, J. D. |title= Angular Momentum in Quantum Physics
year= 1981 |publisher= Addison-Wesley |location= Reading, Massachusetts |isbn= 0201135078

External links

* [http://www.volya.net/vc/vc.php Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator]