Clebsch-Gordan coefficients: Difference between revisions

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In [[quantum mechanics]], the '''Clebsch-Gordan coefficients''' (CG coefficients) are sets of numbers that arise in [[angular momentum coupling]].
{{subpages}}
 
 
:''This article presupposes knowledge of the [[angular momentum (quantum)|quantum theory of angular momentum]] and [[angular momentum coupling]].''
 
 
In [[quantum mechanics]], the '''Clebsch-Gordan coefficients''' (CG coefficients) are sets of numbers that arise in angular momentum coupling.
   
   
In [[mathematics]], the CG coefficients appear in  group [[representation theory]], particularly of [[compact Lie group]]s. They  arise in the explicit [[direct sum]] decomposition of the [[outer product]] of two [[irreducible representation]]s (irreps) of a group ''G''. In general the outer product representation (rep)—which is carried by a tensor product space—is reducible under ''G''. Decomposition of the outer product rep  into irreps of ''G'' requires a basis transformation of the tensor product space. The CG coefficients are the elements of the matrix of this basis transformation. In physics it is common to consider only orthonormal bases of the vector spaces involved, and  then CG coefficients constitute a unitary matrix.  
In [[mathematics]], the CG coefficients appear in  group [[representation theory]], particularly of [[compact Lie group]]s. They  arise in the explicit [[direct sum]] decomposition of the [[outer product]] of two [[irreducible representation]]s (irreps) of a group ''G''. In general the outer product representation (rep)—which is carried by a tensor product space—is reducible under ''G''. Decomposition of the outer product rep  into irreps of ''G'' requires a basis transformation of the tensor product space. The CG coefficients are the elements of the matrix of this basis transformation. In physics it is common to consider only orthonormal bases of the vector spaces involved, and  then CG coefficients constitute a unitary matrix.  


The name derives from the German mathematicians [[Alfred Clebsch]] (1833-1872) and [[Paul Gordan]] (1837-1912), who encountered an equivalent problem in [[invariant theory]].
The name derives from the German mathematicians [[Alfred Clebsch]] (1833–1872) and [[Paul Gordan]] (1837–1912), who encountered an equivalent problem in [[invariant theory]]. An alternative name for the CG coefficients is ''Wigner coefficients'', after [[Eugene Wigner]] (1902–1995).


The formulas below use [[Dirac's]] [[bra-ket notation]], i.e., the quantity <math>\langle \psi | \phi\rangle</math> stands for a positive definite inner product between the elements &psi; and &phi; of the same complex inner product space. We follow the physical convention  
The formulas below use [[Dirac's]] [[bra-ket notation]], i.e., the quantity <math>\scriptstyle \langle \psi | \phi\rangle</math> stands for a positive definite inner product between the elements &psi; and &phi; of the same complex inner product space. We follow the physical convention  
<math>\langle c\psi | \phi \rangle = c^* \langle \psi | \phi \rangle</math>, where <math>c^*\,</math> is the complex conjugate of the complex number ''c''.
<math>\scriptstyle \langle c\psi | \phi \rangle = c^* \langle \psi | \phi \rangle</math>, where <math>\scriptstyle c^*\,</math> is the complex conjugate of the complex number ''c''.


==Clebsch-Gordan coefficients==
==Angular momentum CG coefficients==
Although '''Clebsch-Gordan coefficients''' can be defined for arbitrary groups, we restrict our attention in this article to the groups associated with space and spin angular momentum, namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis.
Although Clebsch-Gordan coefficients  can be defined for arbitrary groups, we restrict our attention in this article to the groups associated with space and spin [[angular momentum (quantum)|angular momentum]], namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. See the article [[angular momentum (quantum)|angular momentum]] for the definition of angular momentum operators and their eigenstates. Since the definition of CG coefficients needs the definition of  tensor products of angular momentum eigenstates, this will be given first.
Below, this definition is made precise by defining angular momentum
operators, angular momentum eigenstates, and tensor products of these states.


From the formal definition recursion relations for the Clebsch-Gordan coefficients
From the formal definition recursion relations for the Clebsch-Gordan coefficients
can be found. In order to settle the numerical values for the coefficients, a phase convention
can be found. In order to settle the numerical values for the coefficients, a phase convention
must be adopted. Below the Condon and Shortley phase convention is chosen.
must be adopted. Below the Condon and Shortley phase convention is chosen.
==Angular momentum operators==
Angular momentum operators are Hermitian operators ''j''<sub>1</sub>, ''j''<sub>2</sub>, and ''j''<sub>3</sub>,that satisfy the commutation relations
:<math>
  [j_k,j_l] = i \sum_{m=1}^3 \varepsilon_{klm}j_m,
</math>
where <math>\varepsilon_{klm}</math> is the [[Levi-Civita permutation symbol|Levi-Civita symbol]]. Together the
three components define a vector operator <math>\mathbf{j}</math>. The
square of the length of <math>\mathbf{j}</math> is defined as
:<math>
\mathbf{j}^2 = j_1^2+j_2^2+j_3^2.
</math>
We also define raising <math>(j_+)</math> and lowering <math>(j_-)</math> operators
:<math>
j_\pm = j_1 \pm i j_2. \,
</math>
==Angular momentum states==
It can be shown from the above definitions that <math>\mathbf{j}^2</math> commutes with <math>j_1, j_2</math>
and <math>j_3</math>
:<math>
  [\mathbf{j}^2, j_k] = 0\ \mathrm{for}\ k = 1,2,3
</math>
When two Hermitian operators commute a common set of eigenfunctions exists.
Conventionally <math>\mathbf{j}^2</math> and <math>j_3</math> are chosen.
From the commutation relations the possible eigenvalues can be found.
The result is
:<math>
  \mathbf{j}^2 |j m\rangle = j(j+1) |j m\rangle,  \qquad j=0, 1/2, 1, 3/2, 2, \ldots
</math>
:<math>
  j_3|j m\rangle = m |j m\rangle,  \qquad\quad m = -j, -j+1, \ldots , j.
</math>
The raising and lowering operators change the value of <math>m</math>
:<math>
  j_\pm |jm\rangle = C_\pm(j,m) |j m\pm 1\rangle
</math>
with
:<math>
  C_\pm(j,m) = \sqrt{j(j+1)-m(m\pm 1)} = \sqrt{(j\mp m)(j\pm m + 1)}.
</math>
A (complex) phase factor could be included in the definition of <math>C_\pm(j,m)</math>
The choice made here is in agreement with the Condon and Shortley phase conventions.
The angular momentum states must be orthogonal (because their eigenvalues with
respect to a Hermitian operator are distinct) and they are assumed to be normalized
:<math>
  \langle j_1 m_1 | j_2 m_2 \rangle = \delta_{j_1,j_2}\delta_{m_1,m_2}.
</math>


==Tensor product space==
==Tensor product space==
Let <math>V_1</math> be the <math>2j_1+1</math> dimensional
Let <math>V_1</math> be the <math>2j_1+1</math> dimensional
vector space spanned by the states
vector space spanned by the eigenstates of <math>\scriptstyle \mathbf{j}^2\otimes 1</math> and <math>\scriptstyle j_z\otimes 1</math>
:<math>
:<math>
   |j_1 m_1\rangle,\quad m_1=-j_1,-j_1+1,\ldots j_1
   |j_1 m_1\rangle,\quad m_1=-j_1,-j_1+1,\ldots j_1
</math>
</math>
and <math>V_2</math> the <math>2j_2+1</math> dimensional
and <math>V_2</math> the <math>2j_2+1</math> dimensional
vector space spanned by
vector space spanned by the eigenstates of <math>\scriptstyle 1\otimes\mathbf{j}^2</math> and <math>\scriptstyle 1\otimes j_z</math>
:<math>
:<math>
   |j_2 m_2\rangle,\quad m_2=-j_2,-j_2+1,\ldots j_2.
   |j_2 m_2\rangle,\quad m_2=-j_2,-j_2+1,\ldots j_2.
Line 84: Line 39:
Angular momentum operators acting on <math>V_{12}</math> can be defined by
Angular momentum operators acting on <math>V_{12}</math> can be defined by
:<math>
:<math>
   (j_i \otimes 1)|j_1 m_1\rangle|j_2 m_2\rangle \equiv (j_i|j_1m_1\rangle) \otimes |j_2m_2\rangle
   (j_i \otimes 1)|j_1 m_1\rangle|j_2 m_2\rangle \equiv (j_i|j_1m_1\rangle) \otimes |j_2m_2\rangle,
\quad i = x,y,z,
</math>
</math>
and
and
:<math>
:<math>
   (1 \otimes j_i) |j_1 m_1\rangle|j_2 m_2\rangle) \equiv |j_1m_1\rangle \otimes j_i|j_2m_2\rangle.
   (1 \otimes j_i) |j_1 m_1\rangle|j_2 m_2\rangle) \equiv |j_1m_1\rangle \otimes j_i|j_2m_2\rangle,\quad i = x,y,z.
</math>
</math>
Total angular momentum operators are defined by
Total angular momentum operators are defined by
:<math>
:<math>
   J_i = j_i \otimes 1 + 1 \otimes j_i\quad\mathrm{for}\quad i = 1,2,3
   J_i = j_i \otimes 1 + 1 \otimes j_i\quad\mathrm{for}\quad i = x,y,z.
</math>
</math>
The total angular momentum operators satisfy the required commutation relations
The total angular momentum operators satisfy the commutation relations
:<math>
:<math>
   [J_k,J_l] = i \sum_{m=1}^3 \epsilon_{klm}J_m
   [J_k,J_l] = i \sum_{m=x,y,z} \epsilon_{klm}J_m \,
</math>
</math>
and hence total angular momentum eigenstates exist
showing that  '''J''' is indeed an angular momentum operator. The quantity <math>\scriptstyle \epsilon_{klm} </math> is the [[Levi-Civita symbol]].
Hence total angular momentum eigenstates exist
:<math>
:<math>
   \mathbf{J}^2 |(j_1j_2)JM\rangle = J(J+1) |(j_1j_2)JM\rangle  
   \mathbf{J}^2 |(j_1j_2)JM\rangle = J(J+1) |(j_1j_2)JM\rangle  
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   J_z |(j_1j_2)JM\rangle = M |(j_1j_2)JM\rangle,\quad \mathrm{for}\quad M=-J,\ldots,J
   J_z |(j_1j_2)JM\rangle = M |(j_1j_2)JM\rangle,\quad \mathrm{for}\quad M=-J,\ldots,J
</math>
</math>
It can be derived [see, e.g., Messiah (1981) pp. 556-558] that <math>J</math> must satisfy the triangular condition
It can be derived&mdash;see, e.g., [[Angular_momentum_coupling#Triangular_conditions|this article]]&mdash;that <math>J</math> must satisfy the triangular condition
:<math>
:<math>
   |j_1-j_2| \leq J \leq j_1+j_2
   |j_1-j_2| \leq J \leq j_1+j_2
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   \sum_{J=|j_1-j_2|}^{j_1+j_2} (2J+1) = (2j_1+1)(2j_2+1)
   \sum_{J=|j_1-j_2|}^{j_1+j_2} (2J+1) = (2j_1+1)(2j_2+1)
</math>
</math>
The total angular momentum states form an orthonormal basis of <math>V_{12}</math>
The total angular momentum states form an orthonormal basis of <math>V_{12}</math><ref>The kets |(''j''<sub>1</sub>''j''<sub>2</sub>)''JM''&rang; and |''JM''&rang; denote the same state; the latter is used if it is not important that this state arose by coupling two states with angular momenta ''j''<sub>1</sub> and ''j''<sub>2</sub>, or if the state arose in another way.</ref>
:<math>
:<math>
   \langle J_1 M_1 | J_2 M_2 \rangle = \delta_{J_1J_2}\delta_{M_1M_2}
   \langle J M | J' M' \rangle = \delta_{J\, J'}\delta_{M\,M'}
</math>
</math>


Line 131: Line 88:
Applying the operator
Applying the operator
:<math>
:<math>
   J_3 = j_3 \otimes 1 + 1 \otimes j_3
   J_z = j_z \otimes 1 + 1 \otimes j_z
</math>
</math>
to both sides of the defining equation shows that the Clebsch-Gordan coefficients
to both sides of the defining equation shows that the Clebsch-Gordan coefficients
Line 140: Line 97:


==Recursion relations==
==Recursion relations==
Applying the total angular momentum raising and lowering operators
Applying the total angular momentum raising and lowering operators, see e.g., [[Angular momentum (quantum)#Angular momentum states|this article]],
:<math>
:<math>
   J_\pm = j_\pm \otimes 1 + 1 \otimes j_\pm
   J_\pm = j_\pm \otimes 1 + 1 \otimes j_\pm
Line 191: Line 148:


==Explicit expression==
==Explicit expression==
For an explicit expression of the Clebsch-Gordan coefficients
The first derivation of an algebraic formula for CG coefficients was given by Wigner in his famous 1931 book. The following expression for the CG coefficients is due to Van der
and tables with numerical values see
Waerden (1932) and is the most symmetric one of the various existing forms, see, e.g., Biedenharn and Louck (1981) for a derivation,
[[table of Clebsch-Gordan coefficients]].
:<math>
\langle jm | j_1 m_1 ;j_2 m_2 \rangle  = \delta_{m,m_1+m_2}
                  \Delta (j_1 ,j_2 ,j) 
</math>
 
:<math>
\times \sum_t (-1)^t
{\textstyle \frac{
\left[(2j+1) (j_1 +m_1 )! (j_1 -m_1)! (j_2 +m_2 )! (j_2 -m_2 )! (j+m)!
      (j-m)! \right]^{\frac{1}{2}}}{t! (j_1 +j_2 -j-t)! (j_1 -m_1 -t)! (j_2 +m_2 -t)!
      (j-j_2 +m_1 +t)!  (j-j_1 -m_2 +t)!} }
</math>
where
:<math>
\Delta (j_1 ,j_2 ,j) \equiv \left[ \frac{
(j_1 +j_2 -j)! (j_1 -j_2 +j)! (-j_1 +j_2 +j)!}
{ (j_1 +j_2 +j+1)!}\right]^{{\frac{1}{2}}},
</math>
and the sum runs over all values of ''t'' which do not lead to
negative factorials.


==Orthogonality relations==
==Orthogonality relations==
Line 215: Line 191:


==Special cases==
==Special cases==
For <math>J=0</math> the Clebsch-Gordan coefficients are given by
For <math>J=0\,</math> the Clebsch-Gordan coefficients are given by
 
:<math>
:<math>
   \langle j_1 m_1 j_2 m_2 | 0 0 \rangle = \delta_{j_1,j_2}\delta_{m_1,-m_2}
   \langle j_1 m_1 j_2 m_2 | 0 0 \rangle = \delta_{j_1,j_2}\delta_{m_1,-m_2}
\frac{(-1)^{j_1-m_1}}{\sqrt{2j_2+1}}.
\frac{(-1)^{j_1-m_1}}{\sqrt{2j_2+1}}.
</math>
</math>
For <math>J=j_1+j_2</math> and <math>M=J</math> we have
 
:<math>
  \langle j_1 m_1 0 0 | J M \rangle = \delta_{j_1 J}\delta_{m_1,M}, \qquad j_1 \ge 0.
</math>
 
For <math>J=j_1+j_2\,</math> and <math>M=J\,</math> we have
:<math>
:<math>
   \langle j_1 j_1 j_2 j_2 | (j_1+j_2) (j_1+j_2) \rangle = 1.
   \langle j_1 j_1 j_2 j_2 | (j_1+j_2) (j_1+j_2) \rangle = 1.
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</math>
</math>


==See also==
==Notes==
 
<small><references/></small>[[Category:Suggestion Bot Tag]]
* [[3-jm symbol]]
* [[Racah W-coefficient]]
* [[6-j symbol]]
* [[9-j symbol]]
* [[Spherical harmonics]]
* [[Associated Legendre polynomials]]
* [[Angular momentum]]
* [[Angular momentum coupling]]
* [[Total electronic angular momentum quantum number]]
* [[Azimuthal quantum number]]
* [[Table of Clebsch-Gordan coefficients]]
 
==External links==
* [http://www.gleet.org.uk/cleb/cgjava.html Java<sup>TM</sup> Clebsch-Gordan Coefficient Calculator]
* [http://www.volya.net/vc/vc.php Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator]
 
==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= [[Elsevier|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 }}

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This article presupposes knowledge of the quantum theory of angular momentum and angular momentum coupling.


In quantum mechanics, the Clebsch-Gordan coefficients (CG coefficients) are sets of numbers that arise in angular momentum coupling.

In mathematics, the CG coefficients appear in group representation theory, particularly of compact Lie groups. They arise in the explicit direct sum decomposition of the outer product of two irreducible representations (irreps) of a group G. In general the outer product representation (rep)—which is carried by a tensor product space—is reducible under G. Decomposition of the outer product rep into irreps of G requires a basis transformation of the tensor product space. The CG coefficients are the elements of the matrix of this basis transformation. In physics it is common to consider only orthonormal bases of the vector spaces involved, and then CG coefficients constitute a unitary matrix.

The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory. An alternative name for the CG coefficients is Wigner coefficients, after Eugene Wigner (1902–1995).

The formulas below use Dirac's bra-ket notation, i.e., the quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \langle \psi | \phi\rangle} stands for a positive definite inner product between the elements ψ and φ of the same complex inner product space. We follow the physical convention Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \langle c\psi | \phi \rangle = c^* \langle \psi | \phi \rangle} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle c^*\,} is the complex conjugate of the complex number c.

Angular momentum CG coefficients

Although Clebsch-Gordan coefficients can be defined for arbitrary groups, we restrict our attention in this article to the groups associated with space and spin angular momentum, namely the groups SO(3) and SU(2). In that case CG coefficients can be defined as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. See the article angular momentum for the definition of angular momentum operators and their eigenstates. Since the definition of CG coefficients needs the definition of tensor products of angular momentum eigenstates, this will be given first.

From the formal definition recursion relations for the Clebsch-Gordan coefficients can be found. In order to settle the numerical values for the coefficients, a phase convention must be adopted. Below the Condon and Shortley phase convention is chosen.

Tensor product space

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_1} be the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2j_1+1} dimensional vector space spanned by the eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{j}^2\otimes 1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle j_z\otimes 1}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_1 m_1\rangle,\quad m_1=-j_1,-j_1+1,\ldots j_1 }

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_2} the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2j_2+1} dimensional vector space spanned by the eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 1\otimes\mathbf{j}^2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 1\otimes j_z}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_2 m_2\rangle,\quad m_2=-j_2,-j_2+1,\ldots j_2. }

The tensor product of these spaces, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}\equiv V_1\otimes V_2} , has a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2j_1+1)(2j_2+1)} dimensional uncoupled basis

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_1 m_1\rangle|j_2 m_2\rangle \equiv |j_1 m_1\rangle \otimes |j_2 m_2\rangle, \quad m_1=-j_1,\ldots j_1, \quad m_2=-j_2,\ldots j_2. }

Angular momentum operators acting on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}} can be defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (j_i \otimes 1)|j_1 m_1\rangle|j_2 m_2\rangle \equiv (j_i|j_1m_1\rangle) \otimes |j_2m_2\rangle, \quad i = x,y,z, }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1 \otimes j_i) |j_1 m_1\rangle|j_2 m_2\rangle) \equiv |j_1m_1\rangle \otimes j_i|j_2m_2\rangle,\quad i = x,y,z. }

Total angular momentum operators are defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_i = j_i \otimes 1 + 1 \otimes j_i\quad\mathrm{for}\quad i = x,y,z. }

The total angular momentum operators satisfy the commutation relations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [J_k,J_l] = i \sum_{m=x,y,z} \epsilon_{klm}J_m \, }

showing that J is indeed an angular momentum operator. The quantity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \epsilon_{klm} } is the Levi-Civita symbol. Hence total angular momentum eigenstates exist

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{J}^2 |(j_1j_2)JM\rangle = J(J+1) |(j_1j_2)JM\rangle }

It can be derived—see, e.g., this article—that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} must satisfy the triangular condition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |j_1-j_2| \leq J \leq j_1+j_2 }

The total number of total angular momentum eigenstates is equal to the dimension of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{12}} [1]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle J M | J' M' \rangle = \delta_{J\, J'}\delta_{M\,M'} }

Formal definition of Clebsch-Gordan coefficients

The total angular momentum states can be expanded in the uncoupled basis

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |(j_1j_2)JM\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} |j_1m_1\rangle|j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle }

The expansion coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle j_1m_1j_2m_2|JM\rangle} are called Clebsch-Gordan coefficients.

Applying the operator

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_z = j_z \otimes 1 + 1 \otimes j_z }

to both sides of the defining equation shows that the Clebsch-Gordan coefficients can only be nonzero when

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = m_1 + m_2.\, }

Recursion relations

Applying the total angular momentum raising and lowering operators, see e.g., this article,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\pm = j_\pm \otimes 1 + 1 \otimes j_\pm }

to the left hand side of the defining equation gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\pm|(j_1j_2)JM\rangle = C_\pm(J,M) |(j_1j_2)JM\pm 1\rangle = C_\pm(J,M)\sum_{m_1m_2}|j_1m_1\rangle|j_2m_2\rangle \langle j_1 m_1 j_2 m_2|J M\pm 1\rangle. }

Applying the same operators to the right hand side gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_\pm \sum_{m_1m_2} |j_1m_1\rangle|j_2m_2\rangle \langle j_1m_1j_2m_2|JM\rangle }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\sum_{m_1m_2}\left[ C_\pm(j_1,m_1)|j_1 m_1\pm 1\rangle |j_2m_2\rangle +C_\pm(j_2,m_2)|j_1 m_1\rangle |j_2 m_2\pm 1\rangle \right] \langle j_1 m_1 j_2 m_2|J M\rangle }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \sum_{m_1m_2} |j_1m_1\rangle|j_2m_2\rangle \left[ C_\pm(j_1,m_1\mp 1) \langle j_1 {m_1\mp 1} j_2 m_2|J M\rangle +C_\pm(j_2,m_2\mp 1) \langle j_1 m_1 j_2 {m_2\mp 1}|J M\rangle \right]. }

Combining these results gives recursion relations for the Clebsch-Gordan coefficients

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\pm(J,M) \langle j_1 m_1 j_2 m_2|J M\pm 1\rangle = C_\pm(j_1,m_1\mp 1) \langle j_1 {m_1\mp 1} j_2 m_2|J M\rangle + C_\pm(j_2,m_2\mp 1) \langle j_1 m_1 j_2 {m_2\mp 1}|J M\rangle. }

Taking the upper sign with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=J} gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = C_+(j_1,m_1-1) \langle j_1 {m_1-1} j_2 m_2|J J\rangle + C_+(j_2,m_2-1) \langle j_1 m_1 j_2 m_2-1|J J\rangle. }

In the Condon and Shortley phase convention the coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle j_1 j_1 j_2 J-j_1|J J\rangle} is taken real and positive. With the last equation all other Clebsch-Gordan coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle j_1 m_1 j_2 m_2|J J\rangle} can be found. The normalization is fixed by the requirement that the sum of the squares, which corresponds to the norm of the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |(j_1j_2)JJ\rangle} must be one.

The lower sign in the recursion relation can be used to find all the Clebsch-Gordan coefficients with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=J-1} . Repeated use of that equation gives all coefficients.

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

Explicit expression

The first derivation of an algebraic formula for CG coefficients was given by Wigner in his famous 1931 book. The following expression for the CG coefficients is due to Van der Waerden (1932) and is the most symmetric one of the various existing forms, see, e.g., Biedenharn and Louck (1981) for a derivation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle jm | j_1 m_1 ;j_2 m_2 \rangle = \delta_{m,m_1+m_2} \Delta (j_1 ,j_2 ,j) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times \sum_t (-1)^t {\textstyle \frac{ \left[(2j+1) (j_1 +m_1 )! (j_1 -m_1)! (j_2 +m_2 )! (j_2 -m_2 )! (j+m)! (j-m)! \right]^{\frac{1}{2}}}{t! (j_1 +j_2 -j-t)! (j_1 -m_1 -t)! (j_2 +m_2 -t)! (j-j_2 +m_1 +t)! (j-j_1 -m_2 +t)!} } }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta (j_1 ,j_2 ,j) \equiv \left[ \frac{ (j_1 +j_2 -j)! (j_1 -j_2 +j)! (-j_1 +j_2 +j)!} { (j_1 +j_2 +j+1)!}\right]^{{\frac{1}{2}}}, }

and the sum runs over all values of t which do not lead to negative factorials.

Orthogonality relations

These are most clearly written down by introducing the alternative notation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle J M|j_1 m_1 j_2 m_2\rangle \equiv \langle j_1 m_1 j_2 m_2|J M \rangle }

The first orthogonality relation is

and the second

Special cases

For the Clebsch-Gordan coefficients are given by

For and we have

Symmetry properties

Relation to 3-jm symbols

Clebsch-Gordan coefficients are related to 3-jm symbols which have more convenient symmetry relations.

Notes

  1. ↑ The kets |(j1j2)JM⟩ and |JM⟩ denote the same state; the latter is used if it is not important that this state arose by coupling two states with angular momenta j1 and j2, or if the state arose in another way.
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