By Professor Ingvar Lindgren, Dr. John Morrison (auth.)
This ebook has constructed via a sequence of lectures on atomic idea given those final 8 years at Chalmers collage of know-how and several other oth er learn facilities. those classes have been meant to make the elemental parts of atomic conception to be had to experimentalists operating with the hyperfine constitution and the optical houses of atoms and to supply a few perception into fresh advancements within the thought. the unique goal of this e-book has steadily prolonged to incorporate a variety of themes. we've attempted to supply an entire description of atomic conception, bridging the distance among introductory books on quantum mechanics - resembling the e-book by way of Merzbacher, for example - and current day learn within the box. Our presentation is proscribed to static atomic prop erties, comparable to the potent electron-electron interplay, however the formalism could be prolonged with out significant problems to incorporate dynamic houses, comparable to transition chances and dynamic polarizabilities.
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Extra resources for Atomic Many-Body Theory
So we can write 1(11)22) = 111, 11) . 32 2. 19b), we obtain 1(11)21> = 1(11)20> = J 2 (1 11 ,10> J + 110,11» 6 (Ill, 1 -1> 1(11)2 - 1) = 1(11)2 - 2> = J + 2110, 10> + 11 -1, 11» 2 (110,1 -1> II = L(1) + L(2), and using + 11 -1,10» -I, 1 -1>. There are two product functions for which ML = 1, namely Ill, 10> and 110, 11 > . As we have seen, one linear combination of these two states corresponds to 1(11)21> . The other linear combination must correspond to the state 1(11) 11> , and we write it 1(11)11> = al(ll, 10> + bllO, 11>.
44 2. 115), the operator L2 = (J - S)2 = J2 + S2 - 2J. 119) when operating within an SLJ manifold. The identity of the diagonal elements leads to L(L + 1) = J(J + 1) + S(S + 1) - 2as J(J + 1) . 122) If we take g. 123) This result, which is usually derived in elementary courses by the nonrigorous vector model, is a direct consequence of the Wigner-Eckart theorem. 3. Reduced Matrix Elements of the C Tensor The Winger-Eckart theorem is a fundamental result, which we shall use frequently in the following to evaluate the matrix elements of interaction operators.
We would like now to express the function t! I yjm) in terms of the coupled basis functions Iy' J M). 4 The Wigner-Eckart Theorem t;1 yjm) = L. ¢>(PJM) (JMlkq,jm) 1M 41 . 105) this last equation becomes t~lyjm) = L. 'II/JM c(y"J)ly"JM) (JMlkq,jm). We now take the scalar product with (YJ'm' 1 and use the orthogonality of the functions to obtain (y'j'm' 1t; 1 yjm) = c(y'j') (j'm' 1kq,jm) . 106) This shows that the matrix element can be factorized into a vector-coupling coefficient and a factor independent of the magnetic quantum numbers.