By Ewald Balcar, Stephen W. Lovesey
Application of quantum mechanics in physics and chemistry frequently involves manipulation and evaluate of sums and items of coupling coefficients for the idea of angular momentum. demanding situations encountered in such paintings will be tamed by way of graphical innovations that offer either the perception and analytical energy. The booklet is the 1st step by step exposition of a graphical procedure grounded in verified paintings. Copious routines get better general effects yet reveal the ability to head past.
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Additional info for Introduction to the Graphical Theory of Angular Momentum: Case Studies
Six 3jm-symbols in our deﬁnition of a 9j-symbol are arranged such that the intended line connections are anticipated All nodes are now connected to indicate the summation over all magnetic quantum numbers. + a c − a b c d e f k l m − d f k f k + + + m e − b ⇒ d l − ! c − b m a − e + l + ! Fig. 2. Some of the summations in these diagrams are purely formal. 2 Symmetry Properties of the 9j-Symbol 51 The graphical representation of the 9j-symbol, on the left-hand side of Fig. 2, has nodes arranged in shape of a hexagon with alternating positive and negative node signs.
6 would require a rotation that takes the positive node to the top. The angular momentum line b is then moved to the other side, leaving line c in the middle. 4 6j-Symbol with Three 0-Lines A very simple value is obtained for a 6j-symbol in which all angular momenta in the top row are zero. 0 0 0 d e f + + Φlr = (−1)2d+2e+2f 0 f =e f =e e=d e=d ⇒ 0 + 0 d + + d + Fig. 7. The angular momentum lines leaving the central node in Fig. 2 are all zero in this case. The line reversals arrange the 0-lines and the others as is appropriate for node removal.
11 if we keep the j = 0 line inside the closed loop: 20 3 Basic Tools for the Graphical Method a a c α −α γ (−1)a−α α √ = δc,0 δγ,0 2a + 1 a + a a ⇒ 0 (2a + 1) = δc,0 δγ,0 √ 2a + 1 a + ⇒ a + + a + Fig. 12. If only the 0-line is removed, the remaining diagram with just one twoline node equals entry (2) in Fig. 4 We note that the special case of the two-line node diagram (2) given in Fig. 4 must be obtained with the zero-line enclosed by the angular momentum line to guarantee the node removal without phase change.