Levi-Civita symbol for vectors
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11: 34.10 Zeros
§34.10 Zeros
►In a symbol, if the three angular momenta do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the symbol is zero. Similarly the symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four symbols in the summation. …However, the and symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. Such zeros are called nontrivial zeros. …12: 34.12 Physical Applications
§34.12 Physical Applications
►The angular momentum coupling coefficients (, , and symbols) are essential in the fields of nuclear, atomic, and molecular physics. …, and symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).13: 34.13 Methods of Computation
§34.13 Methods of Computation
►Methods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …14: 1.1 Special Notation
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real variables. | |
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inner, or scalar, product for real or complex vectors or functions. | |
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, | column vectors. |
the space of all -dimensional vectors. | |
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15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►A complex linear vector space is called an inner product space if an inner product
is defined for all with the properties: (i) is complex linear in ; (ii) ; (iii) ; (iv) if then .
… becomes a normed linear vector space.
If then is normalized.
Two elements and in are orthogonal if .
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►The adjoint of does satisfy where .
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16: 34.9 Graphical Method
§34.9 Graphical Method
►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For specific examples of the graphical method of representing sums involving the , and symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).17: 21.1 Special Notation
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►Lowercase boldface letters or numbers are -dimensional real or complex vectors, either row or column depending on the context.
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positive integers. | |
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-dimensional vectors, with all elements in , unless stated otherwise. | |
th element of vector . | |
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scalar product of the vectors and . | |
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set of -dimensional vectors with elements in . | |
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18: 21.6 Products
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►that is, is the number of elements in the set containing all -dimensional vectors obtained by multiplying on the right by a vector with integer elements.
Two such vectors are considered equivalent if their difference is a vector with integer elements.
…where and are arbitrary -dimensional vectors.
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►Then
…Thus is a -dimensional vector whose entries are either or .
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19: 34.1 Special Notation
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►The main functions treated in this chapter are the Wigner
symbols, respectively,
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►An often used alternative to the
symbol is the Clebsch–Gordan coefficient
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nonnegative integers. | |
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34.1.1
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►For other notations for , ,
symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).