share/contrib/clebsh-gordan.tex

   1 \documentclass[11pt]{article}
   2
   3 \title{Clebsch-Gordan coefficients\footnote
   4 {Copyright 2007 by Edmond Orignac.
   5 This 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
  13 The 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
  15 in 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
  24 The Maxima function \texttt{wigner\_3j(j1,j2,m1,m2,j,m)} computes the $3j$
  25 coefficient of Wigner.
  26
  27 The Wigner $3j$ coefficient appears in the addition of a pair of angular
  28 momenta in Quantum Mechanics.
  29 It 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
  35 where $ \langle j_1,m_1; j_2, m_2 | j,m\rangle$ is the
  36 Clebsch-Gordan coefficient. The Clebsch-Gordan coefficient is used to
  37 construct the state of total angular momentum $j$ and total projection
  38 of angular momentum $m$ as a linear combination of states of angular momenta
  39 $j_1$ and $j_2$ and respective projections $m_1$ and $m_2$.
  40 One 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}
  44 The advantage of working with the $3j$ coefficients instead of the
  45 Clebsch-Gordan coefficients is that the former are more symmetric\cite{landau_mecaq}.
  46
  47
  48 The $3j$ coefficient is computed by application of
  49 Eq. (27.9.1) p. 1006 of \cite{abramowitz_math_functions}.
  50
  51
  52 \subsection{Wigner $6j$ coefficients}
  53
  54 The Maxima function \texttt{wigner\_6j(j1,j2,j3,j4,j5,j6)} computes the $6j$
  55 coefficient of Wigner.
  56
  57 The Wigner $6j$ coefficients appears in the addition of three angular momenta.
  58 When one is adding three angular momenta, one can form a first
  59 pair of angular momenta, add them together to form a new angular momentum
  60 using the $3j$ coefficients, and add the resulting angular momenta with the
  61 remaining angular momentum\cite{landau_mecaq,messiah_field_chapter}.
  62 There are 3 different ways of grouping the angular momenta, which leads to
  63 different representations of the total angular momentum.
  64 The Wigner $6j$ coefficients are used to pass from one representation to the
  65 other.
  66
  67 The 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
  72 The $6j$ 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
  77 The function\texttt{wigner\_9j(a,b,c,d,e,f,h,i,j)} computes
  78 the $9j$ coefficient of Wigner.
  79
  80 The $9j$ coefficients appears in the addition of four angular momenta.
  81 To add these angular momenta, one can first form two pairs of angular
  82 momenta and add them together to form the two resulting angular momenta
  83 and then add together the two resulting angular momenta.
  84 There are different ways to form the two pairs of angular momenta, and
  85 the $9j$ coefficient is used to transform from one representation to
  86 the other\cite{landau_mecaq,messiah_field_chapter}.
  87
  88 The 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
  93 The $9j$ coefficient is computed by applying Eq. (41) p. 917, of \cite{messiah_field_chapter}.
  94
  95
  96 \section{Limitations}
  97
  98 The $3nj$ with $n\ge 4$ (addition of $n+1$ angular momenta) are not
  99 implemented. The theory of these coefficients
 100 can be found in the book Edmonds Angular momentum
 101 in quantum Mechanics (Princeton University Press).
 102
 103 Various other coefficients can be defined that are related to the $3nj$
 104 coefficients such as the Racah $W$ or $X$ coefficients\cite{messiah_field_chapter}. These coefficients are not implemented.
 105
 106
 107 As the computation is done using exact formulas, it will break down if the
 108 angular momenta that are entered are too large. For these cases, one should
 109 implement 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}