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