3j symbol

3j symbols are a notation for coupling two angular momenta in quantum mechanics and were introduced by Eugene Wigner . They can be used to transform states between the coupled and uncoupled basis. The 3j symbols are an alternative to the Clebsch-Gordan coefficients .

There are also 6j symbols according to Wigner corresponding to the coupling of three angular momenta and 9j symbols when coupling four angular momentum.

use

In order to write the state of a system consisting of two components with angular momentum and an overall system, two orthonormal bases are used in quantum mechanics , each of which is the eigenbase of a complete set of commuting observables . On the one hand, the eigen basis of the operators of the two subsystems: the square of the magnitude of the two angular momentum vectors and the respective components ( ); the respective eigenvalues ​​are denoted by and the corresponding base states are written as. On the other hand, the angular momentum of the entire system, i.e. i.e., and (the corresponding quantum numbers are denoted by and ) in addition to the angular momenta of the subsystems (but not the ); here one writes the eigen-states as . ${\displaystyle j_{1}}$ ${\displaystyle j_{2}}$ ${\displaystyle {\vec {J}}_{i}^{2}}$ ${\displaystyle z}$ ${\displaystyle J_{i}^{z}}$ ${\displaystyle i=1,2}$ ${\displaystyle j_{i},m_{i}}$ ${\displaystyle |j_{1}m_{1};j_{2}m_{2}\rangle }$ ${\displaystyle {\vec {J}}^{2}=({\vec {J}}_{1}+{\vec {J}}_{2})^{2}}$ ${\displaystyle J^{z}=J_{1}^{z}+J_{2}^{z}}$ ${\displaystyle j}$ ${\displaystyle m}$ ${\displaystyle {\vec {J}}_{i}^{2}}$ ${\displaystyle J_{i}^{z}}$ ${\displaystyle |j_{1},j_{2};jm\rangle }$

Then the component of the state can be written with the 3j symbol as follows: ${\displaystyle j_{1}m_{1};j_{2}m_{2}}$ ${\displaystyle |j_{1},j_{2};jm\rangle }$

${\displaystyle \langle j_{1}m_{1};j_{2}m_{2}|j_{1}j_{2};jm\rangle =(-1)^{j_{1}-j_{2}+m}{\sqrt {2j+1}}{\begin{pmatrix}j_{1}&j_{2}&j\\m_{1}&m_{2}&-m\end{pmatrix}}.}$

The left side of the equation is also known as the Clebsch-Gordan coefficient . Compared with these, the coupling with 3j symbols is formulated more symmetrically and the symmetry properties of the 3j symbols can therefore be formulated more easily.

Relationship to Clebsch-Gordan coefficients

As a function of the Clebsch-Gordan coefficients, the following expression results for the 3j symbols:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\equiv {\frac {(-1)^{j_{1}-j_{2}-m_{3}}}{\sqrt {2j_{3}+1}}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,(-m_{3})\rangle .}$

Here j and m stand for the angular momentum quantum numbers.

The addition of two angular momenta with Clebsch-Gordan coefficients

${\displaystyle |j_{3}\,m_{3}\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{3}\,m_{3}\rangle |j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle .}$

For the 3j symbols, the formulation corresponds to the addition of three angular momenta to zero:

${\displaystyle \sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}\sum _{m_{3}=-j_{3}}^{j_{3}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle |j_{3}m_{3}\rangle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=|0\,0\rangle .}$

The state corresponds to vanishing angular momentum quantum numbers ( ). Since the 3j symbols treat all angular momenta on the same level, the formulation is more symmetrical than with Clebsch-Gordan coefficients and is manifestly rotationally invariant. ${\displaystyle |0\,0\rangle }$ ${\displaystyle j=m=0}$

Selection rules

The 3j symbols disappear except for:

${\displaystyle {\begin{aligned}&m_{i}\in \{-j_{i},-j_{i}+1,-j_{i}+2,\ldots ,j_{i}\},\quad (i=1,2,3).\\&m_{1}+m_{2}+m_{3}=0\\&|j_{1}-j_{2}|\leq j_{3}\leq j_{1}+j_{2}\\&(j_{1}+j_{2}+j_{3}){\text{ ist eine ganze Zahl (und sogar gerade falls }}m_{1}=m_{2}=m_{3}=0{\text{)}}\\\end{aligned}}}$

Symmetry properties

The 3j symbol is invariant with an even permutation of the columns:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{pmatrix}}={\begin{pmatrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{pmatrix}}.}$

With odd permutation there is a phase factor:

${\displaystyle {\begin{aligned}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}&=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{2}&j_{1}&j_{3}\\m_{2}&m_{1}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{1}&j_{3}&j_{2}\\m_{1}&m_{3}&m_{2}\end{pmatrix}}\\&=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{3}&j_{2}&j_{1}\\m_{3}&m_{2}&m_{1}\end{pmatrix}}.\end{aligned}}}$

Changing the sign of the quantum numbers (corresponding to a time reversal ) also gives a phase factor: ${\displaystyle m}$

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-m_{1}&-m_{2}&-m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}.}$

There are also so-called regge symmetries:

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}={\begin{pmatrix}j_{1}&{\frac {j_{2}+j_{3}-m_{1}}{2}}&{\frac {j_{2}+j_{3}+m_{1}}{2}}\\j_{3}-j_{2}&{\frac {j_{2}-j_{3}-m_{1}}{2}}-m_{3}&{\frac {j_{2}-j_{3}+m_{1}}{2}}+m_{3}\end{pmatrix}}.}$

${\displaystyle {\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}=(-1)^{j_{1}+j_{2}+j_{3}}{\begin{pmatrix}{\frac {j_{2}+j_{3}+m_{1}}{2}}&{\frac {j_{1}+j_{3}+m_{2}}{2}}&{\frac {j_{1}+j_{2}+m_{3}}{2}}\\j_{1}-{\frac {j_{2}+j_{3}-m_{1}}{2}}&j_{2}-{\frac {j_{1}+j_{3}-m_{2}}{2}}&j_{3}-{\frac {j_{1}+j_{2}-m_{3}}{2}}\end{pmatrix}}.}$

There are a total of 72 symmetries that can be represented by a regge symbol :

${\displaystyle R={\begin{array}{|ccc|}\hline -j_{1}+j_{2}+j_{3}&j_{1}-j_{2}+j_{3}&j_{1}+j_{2}-j_{3}\\j_{1}-m_{1}&j_{2}-m_{2}&j_{3}-m_{3}\\j_{1}+m_{1}&j_{2}+m_{2}&j_{3}+m_{3}\\\hline \end{array}}}$

The 72 symmetries correspond to the interchanging of rows and columns and the transposition of the matrix.

Orthogonality relations

The orthogonality relations follow from the fact that the 3j symbols are a unitary transformation of the various angular momentum bases (the bases for the angular momentum j1, j2 and that of the coupled system with angular momentum j3).

${\displaystyle (2j_{3}+1)\sum _{m_{1}m_{2}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j'_{3}\\m_{1}&m_{2}&m'_{3}\end{pmatrix}}=\delta _{j_{3},j'_{3}}\delta _{m_{3},m'_{3}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}}.}$

${\displaystyle \sum _{j_{3}m_{3}}(2j_{3}+1){\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}'&m_{2}'&m_{3}\end{pmatrix}}=\delta _{m_{1},m_{1}'}\delta _{m_{2},m_{2}'}.}$

The triangular delta is equal to 1 if the triangle condition is fulfilled and 0 otherwise. The triangle condition is that it assumes one of the values . ${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}}}$ ${\displaystyle j_{3}}$ ${\displaystyle j_{1}+j_{2},j_{1}+j_{2}-1,\cdots ,|j_{1}-j_{2}|}$

Connection to spherical functions

The 3j symbols are the integral of the product of three spherical surface functions :

${\displaystyle {\begin{aligned}&{}\quad \int Y_{l_{1}m_{1}}(\theta ,\varphi )Y_{l_{2}m_{2}}(\theta ,\varphi )Y_{l_{3}m_{3}}(\theta ,\varphi )\,\sin \theta \,\mathrm {d} \theta \,\mathrm {d} \varphi \\&={\sqrt {\frac {(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi }}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\0&0&0\end{pmatrix}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\end{aligned}}}$

where , and are whole numbers. ${\displaystyle l_{1}}$ ${\displaystyle l_{2}}$ ${\displaystyle l_{3}}$

Similarly, with spin-weighted spherical surface functions and with half-integer angular momentum for : ${\displaystyle s_{1}+s_{2}+s_{3}=0}$

${\displaystyle {\begin{aligned}&{}\quad \int \mathrm {d} {{\vec {e}}_{n}}\,{}_{s_{1}}Y_{j_{1}m_{1}}({{\vec {e}}_{n}})\,{}_{s_{2}}Y_{j_{2}m_{2}}({{\vec {e}}_{n}})\,{}_{s_{3}}Y_{j_{3}m_{3}}({{\vec {e}}_{n}})\\[8pt]&={\sqrt {\frac {(2j_{1}+1)(2j_{2}+1)(2j_{3}+1)}{4\pi }}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-s_{1}&-s_{2}&-s_{3}\end{pmatrix}}\end{aligned}}}$

Recursion relations

${\displaystyle {\begin{aligned}&{}\quad -{\sqrt {(l_{3}\mp s_{3})(l_{3}\pm s_{3}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}&s_{3}\pm 1\end{pmatrix}}\\&={\sqrt {(l_{1}\mp s_{1})(l_{1}\pm s_{1}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}\pm 1&s_{2}&s_{3}\end{pmatrix}}+{\sqrt {(l_{2}\mp s_{2})(l_{2}\pm s_{2}+1)}}{\begin{pmatrix}l_{1}&l_{2}&l_{3}\\s_{1}&s_{2}\pm 1&s_{3}\end{pmatrix}}\end{aligned}}}$

Asymptotic development

For applies to a non-vanishing 3j symbol (AR Edmonds): ${\displaystyle l_{1}\ll l_{2},l_{3}}$

${\displaystyle {\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {2l_{3}+1}}}}$

with and Wigner's small D-matrix . A better approximation that satisfies Regge symmetry is: ${\displaystyle \cos(\theta )=-2m_{3}/(2l_{3}+1)}$ ${\displaystyle d_{mn}^{l}}$

${\displaystyle {\begin{pmatrix}l_{1}&l_{2}&l_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}\approx (-1)^{l_{3}+m_{3}}{\frac {d_{m_{1},l_{3}-l_{2}}^{l_{1}}(\theta )}{\sqrt {l_{2}+l_{3}+1}}}}$

with . ${\displaystyle \cos(\theta )=(m_{2}-m_{3})/(l_{2}+l_{3}+1)}$

Metric tensor

The following quantity plays the role of a metric tensor in theory and is also called the Wigner 1-jm symbol :

${\displaystyle {\begin{pmatrix}j\\m\quad m'\end{pmatrix}}:={\sqrt {2j+1}}{\begin{pmatrix}j&0&j\\m&0&m'\end{pmatrix}}=(-1)^{j-m'}\delta _{m,-m'}}$

It is used to express time reversal in angular momentum.

Other properties

${\displaystyle \sum _{m}(-1)^{j-m}{\begin{pmatrix}j&j&J\\m&-m&0\end{pmatrix}}={\sqrt {2j+1}}~\delta _{J,0}}$

${\displaystyle {\frac {1}{2}}\int _{-1}^{1}P_{l_{1}}(x)P_{l_{2}}(x)P_{l}(x)\,\mathrm {d} x={\begin{pmatrix}l&l_{1}&l_{2}\\0&0&0\end{pmatrix}}^{2}}$

with the Legendre function . ${\displaystyle P_{l}}$

Relationship to Racah V coefficients

The relationship to the Racah -V coefficients is a simple phase factor:

${\displaystyle V(j_{1}j_{2}j_{3};m_{1}m_{2}m_{3})=(-1)^{j_{1}-j_{2}-j_{3}}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{pmatrix}}}$

literature

• Alain Robert Edmonds : angular momenta in quantum mechanics , BI university paperback 1964 (English: Angular Momentum in Quantum Mechanics, Princeton UP, 1960)
• A. Messiah : Quantum Mechanics , Volume 2, De Gruyter 1985, Appendix C.

Web links

• Mathworld, 3j symbol
• Anthony Stone, Wigner coefficient calculator
• 369j symbol calculator at the Plasma Laboratory of Weizmann Institute of Science (numerical calculation)
• Frederik J Simons: Matlab Software, Code THREEJ.M

Individual evidence

1. ↑ Wigner: On the Matrices Which Reduce the Kronecker Products of Representations of SR Groups , in: LC Biedenharn, H. van Dam (Ed.): Quantum theory of angular momentum, Academic Press 1965, pp. 87-133. Reprinted in Wigner, Collected Works, Springer, Volume 1, 1993, pp. 608-654
2. ^ Wigner: Group Theory and its application to atomic spectra, Academic Press 1959
3. ^ Tullio Regge , Symmetry Properties of Clebsch-Gordan Coefficients, Nuovo Cimento, Volume 10, 1958, p. 544
4. ^ Racah V coefficient, Mathworld