6j symbol

The 6y symbol of Eugene Wigner is a notation for the coupling of angular momentum in the quantum mechanics . It plays a role in the coupling of three quantum mechanical angular momenta .

definition

It is defined as the sum over products of four 3j symbols as follows:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}}$

${\displaystyle =\sum _{m_{1},\dots ,m_{6}}(-1)^{\sum _{k=1}^{6}(j_{k}-m_{k})}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-m_{1}&-m_{2}&-m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{5}&j_{6}\\m_{1}&-m_{5}&m_{6}\end{pmatrix}}{\begin{pmatrix}j_{4}&j_{2}&j_{6}\\m_{4}&m_{2}&-m_{6}\end{pmatrix}}{\begin{pmatrix}j_{4}&j_{5}&j_{3}\\-m_{4}&m_{5}&m_{3}\end{pmatrix}}.}$

It should be noted that not all of them make non-disappearing contributions (selection rules for the 3j symbols, see there). ${\displaystyle m_{i}}$

Symmetries

The 6j symbol is invariant when its columns are swapped:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{3}&j_{2}\\j_{4}&j_{6}&j_{5}\end{Bmatrix}}={\begin{Bmatrix}j_{3}&j_{2}&j_{1}\\j_{6}&j_{5}&j_{4}\end{Bmatrix}}=\cdots }$

It is also invariant with the simultaneous exchange of symbols in two columns:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{4}&j_{5}&j_{3}\\j_{1}&j_{2}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{5}&j_{6}\\j_{4}&j_{2}&j_{3}\end{Bmatrix}}={\begin{Bmatrix}j_{4}&j_{2}&j_{6}\\j_{1}&j_{5}&j_{3}\end{Bmatrix}}.}$

There are a total of 24 symmetries.

The 6j symbol

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}}$

vanishes unless the triangle condition is met: ${\displaystyle j_{1},j_{2},j_{3}}$

${\displaystyle j_{1}=|j_{2}-j_{3}|,\ldots ,j_{2}+j_{3}}$

Because of the above-mentioned symmetries must also , , satisfy the triangle condition. In addition, the sum of all elements of these three-tuples must be an integer. ${\displaystyle j_{1},j_{5},j_{6}}$ ${\displaystyle j_{4},j_{2},j_{6}}$ ${\displaystyle j_{4},j_{5},j_{3}}$

Special case

The following formula applies to the 6j symbol: ${\displaystyle j_{6}=0}$

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&0\end{Bmatrix}}={\frac {\delta _{j_{2},j_{4}}\delta _{j_{1},j_{5}}}{\sqrt {(2j_{1}+1)(2j_{2}+1)}}}(-1)^{j_{1}+j_{2}+j_{3}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}}}$

The triangular delta is equal to 1 if the triangle condition is met and 0 otherwise. ${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}}}$ ${\displaystyle j_{1},j_{2},j_{3}}$

Orthogonality relation

The 6j symbols satisfy the orthogonality relation:

${\displaystyle \sum _{j_{3}}(2j_{3}+1){\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}'\end{Bmatrix}}={\frac {\delta _{j_{6}^{}j_{6}'}}{2j_{6}+1}}{\begin{Bmatrix}j_{1}&j_{5}&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{4}&j_{2}&j_{6}\end{Bmatrix}}.}$

Asymptotic development

If all are big in the 6j symbol: ${\displaystyle j_{i}}$

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}\sim {\frac {1}{\sqrt {12\pi |V|}}}\cos {\left(\sum _{i=1}^{6}J_{i}\theta _{i}+{\frac {\pi }{4}}\right)}.}$

The formula was conjectured by Tullio Regge and G. Ponzano and was proven by Justin Roberts. and uses the asymptotically resulting tetrahedron geometry. Where V is the volume of the tetrahedron, the length of the side and the angle of the sides that meet the i-th edge. ${\displaystyle J_{i}=j_{i}+{\frac {1}{2}}}$ ${\displaystyle i}$ ${\displaystyle \theta _{i}}$

Relation to Racah-W coefficients

They are related to the Racah W coefficients, which are also used to couple three angular momenta:

${\displaystyle {\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}=(-1)^{j_{1}+j_{2}+j_{4}+j_{5}}W(j_{1}j_{2}j_{5}j_{4};j_{3}j_{6}).}$

The Racah W coefficients are coefficients:

${\displaystyle W(j_{1}j_{2}Jj_{3};J_{12}J_{23})\equiv {\frac {\langle (j_{1},(j_{2}j_{3})J_{23})J|((j_{1}j_{2})J_{12},j_{3})J\rangle }{\sqrt {(2J_{12}+1)(2J_{23}+1)}}}.}$

at the transition from a basis in which and to are coupled and this then with the total angular momentum and a basis in which first and to are coupled and this then with to : ${\displaystyle j_{1}}$ ${\displaystyle j_{2}}$ ${\displaystyle J_{12}}$ ${\displaystyle j_{3}}$ ${\displaystyle J}$ ${\displaystyle j_{2}}$ ${\displaystyle j_{3}}$ ${\displaystyle J_{23}}$ ${\displaystyle j_{3}}$ ${\displaystyle J}$

${\displaystyle |((j_{1}j_{2})J_{12}j_{3})JM\rangle =\sum _{J_{23}}\langle (j_{1},(j_{2}j_{3})J_{23})J|((j_{1}j_{2})J_{12}j_{3})J\rangle \,|(j_{1},(j_{2}j_{3})J_{23})JM\rangle }$

${\displaystyle ={\sqrt {(2J_{12}+1)}}\sum _{J_{23}}{\sqrt {(2J_{23}+1)}}\,W(j_{1}j_{2}Jj_{3};J_{12}J_{23})|(j_{1},(j_{2}j_{3})J_{23})JM\rangle }$

literature

• Alan Robert Edmonds : Angular momentum in quantum mechanics, BI university pocket books 1964 (English original Princeton UP 1957)
• A. Messiah : Quantum Mechanics , Volume 2, De Gruyter 1985, Appendix C.

Web links

• 6j symbol, Mathworld

Individual evidence

1. ^ Ponzano, Regge: Semiclassical Limit of Racah Coefficients, in: Spectroscopy and Group Theoretical Methods in Physics, Amsterdam, 1968, pp. 1–58
2. ↑ J. Roberts: Classical 6j-symbols and the tetrahedron, Geometry and Topology, Volume 3, 1998, pp 21-66, Arxiv