1\documentclass[11pt]{article} 2 3\title{Clebsch-Gordan coefficients\footnote 4{Copyright 2007 by Edmond Orignac. 5This file is released under the terms of the GNU General Public License, version 2.}} 6 7\begin{document} 8 9\maketitle 10 11\section{Wigner recoupling coefficients} 12 13The Maxima script \texttt{clebsch\_gordan.mac} defines the $3j,6j$ and $9j$ 14 coefficients that are used in the theory of addition of angular momenta 15in quantum mechanics\cite{landau_mecaq,messiah_field_chapter}. 16 17% The Book of Messiah is available from Dover in an english translation. 18% Angular momenta and Wigner coefficients can be found in Appendix C. 19% An online reference is: 20% Weisstein, Eric W. "Wigner 9j-Symbol." From MathWorld--A Wolfram Web Resource.% http://mathworld.wolfram.com/Wigner9j-Symbol.html 21 22\subsection{Wigner $3j$ coefficients} 23 24The Maxima function \texttt{wigner\_3j(j1,j2,m1,m2,j,m)} computes the $3j$ 25coefficient of Wigner. 26 27The Wigner $3j$ coefficient appears in the addition of a pair of angular 28momenta in Quantum Mechanics. 29It is defined as\cite{landau_mecaq,messiah_field_chapter}: 30\begin{equation} 31 \label{eq:3j-def} 32 \left(\begin{array}{ccc} j_1 & j_2 & j \\ m_1 & m_2 & m\end{array} \right) = (-1)^{j_1-j_2-m} \frac 1 {\sqrt{2j+1}} \langle j_1,m_1; j_2, m_2 | j,-m\rangle, 33\end{equation} 34 35where $ \langle j_1,m_1; j_2, m_2 | j,m\rangle$ is the 36Clebsch-Gordan coefficient. The Clebsch-Gordan coefficient is used to 37construct the state of total angular momentum $j$ and total projection 38of angular momentum $m$ as a linear combination of states of angular momenta 39$j_1$ and $j_2$ and respective projections $m_1$ and $m_2$. 40One has: 41\begin{equation} 42 |j,m\rangle = \sum_{m_1,m_2} \langle j_1,m_1;j_2,m_2|j,m\rangle |j_1,m_1\rangle |j_2,m_2\rangle 43\end{equation} 44The advantage of working with the $3j$ coefficients instead of the 45Clebsch-Gordan coefficients is that the former are more symmetric\cite{landau_mecaq}. 46 47 48The $3j$ coefficient is computed by application of 49Eq. (27.9.1) p. 1006 of \cite{abramowitz_math_functions}. 50 51 52\subsection{Wigner $6j$ coefficients} 53 54The Maxima function \texttt{wigner\_6j(j1,j2,j3,j4,j5,j6)} computes the $6j$ 55coefficient of Wigner. 56 57The Wigner $6j$ coefficients appears in the addition of three angular momenta. 58When one is adding three angular momenta, one can form a first 59pair of angular momenta, add them together to form a new angular momentum 60using the $3j$ coefficients, and add the resulting angular momenta with the 61remaining angular momentum\cite{landau_mecaq,messiah_field_chapter}. 62There are 3 different ways of grouping the angular momenta, which leads to 63different representations of the total angular momentum. 64The Wigner $6j$ coefficients are used to pass from one representation to the 65other. 66 67The notation for the $6j$ symbols is: 68\begin{equation} 69 \left\{\begin{array}{ccc}j_1 & j_2 & j_3 \\ j_4 & j_5 & j_6\end{array} \right\} 70\end{equation} 71 72The $6j$ coefficient is computed by application of the formula p. 513 Eq. (108,10) of \cite{landau_mecaq} or the equivalent formula p. 915, Eq. (36) of \cite{messiah_field_chapter}. 73 74 75\subsection{Wigner $9j$ coefficients} 76 77The function\texttt{wigner\_9j(a,b,c,d,e,f,h,i,j)} computes 78the $9j$ coefficient of Wigner. 79 80The $9j$ coefficients appears in the addition of four angular momenta. 81To add these angular momenta, one can first form two pairs of angular 82momenta and add them together to form the two resulting angular momenta 83and then add together the two resulting angular momenta. 84There are different ways to form the two pairs of angular momenta, and 85the $9j$ coefficient is used to transform from one representation to 86the other\cite{landau_mecaq,messiah_field_chapter}. 87 88The notation for the $9j$ coefficient is: 89\begin{equation} 90 \left\{\begin{array}{cccc} a & b & c \\ d & e & f \\ h & i & j \end{array} \right\} 91\end{equation} 92 93The $9j$ coefficient is computed by applying Eq. (41) p. 917, of \cite{messiah_field_chapter}. 94 95 96\section{Limitations} 97 98The $3nj$ with $n\ge 4$ (addition of $n+1$ angular momenta) are not 99implemented. The theory of these coefficients 100can be found in the book Edmonds Angular momentum 101in quantum Mechanics (Princeton University Press). 102 103Various other coefficients can be defined that are related to the $3nj$ 104coefficients such as the Racah $W$ or $X$ coefficients\cite{messiah_field_chapter}. These coefficients are not implemented. 105 106 107As the computation is done using exact formulas, it will break down if the 108angular momenta that are entered are too large. For these cases, one should 109implement recurrence formulas or use asymptotic expansions. 110 111 112 113\begin{thebibliography}{1} 114 115\bibitem{abramowitz_math_functions} 116{\sc Abramowitz, M., and Stegun, I.} 117\newblock {\em Handbook of mathematical functions}. 118\newblock Dover, New York, 1972. 119 120\bibitem{landau_mecaq} 121{\sc Landau, L.~D., and Lifshitz, E.~M.} 122\newblock {\em Quantum Mechanics : non-relativistic theory}. 123\newblock perg, New York, 1962. 124 125\bibitem{messiah_field_chapter} 126{\sc Messiah, A.} 127\newblock {\em M\'ecanique Quantique}, vol.~2. 128\newblock Dunod, Paris, 1995. 129 130\end{thebibliography} 131 132 133\end{document} 134