ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
The known general formula for the Clebsch–Gordan coefficients of the three-dimensional rotation group involves one summation that results in explicit summation-free expressions for the coefficients where either one of the angular momenta is the sum of the other two or the magnetic quantum number corresponding to one of the angular momenta takes its maximum value in magnitude. By using very different techniques, explicit expressions for the coefficients 〈 j1, 0; j2, 0|| j, 0〉, 〈 j1, (1)/(2) ; j2, − (1)/(2) || j, 0〉 are also obtained where the integral or half-integral nature of the j's is indicated by the magnetic quantum number involved. Here the expressions depend upon whether j1+ j2+ j is an even or an odd integer. For these coefficients, the magnetic quantum numbers involved take their minimum value in magnitude. By using the recursion relation for the coefficients of the form 〈 j1, m, j2, −m, || j, 0〉, these coefficients can be calculated in terms of the above known ones provided the explicit value of the coefficient 〈 j1, 1; j2, −1|| j, 0〉 is known, where j1+ j2+ j is an odd integer. (The recursion relation for these coefficients in terms of 〈 j1, 0; j2, −1|| j, 0〉 becomes a triviality since 〈 j1, 0; j2, 0|| j, 0〉 vanishes when j1+ j2+ j is an odd integer.)The main purpose of this paper is to give an explicit expression for the coefficients 〈 j1, 1; j2, −1|| j, 0〉, where j1+ j2+ j is an odd integer. This expression is obtained by using a complicated transformation between hypergeometric functions, which seems to have been neglected so far. For the coefficients where the magnetic quantum numbers have their minimum value in magnitude, this transformed expression becomes summation-free and the explicit values of the three already known coefficients and the fourth so far unknown are obtained. Further study of this transformation may be useful on its own because it provides a link between very different types of expressions. For completeness, explicit expressions for the coefficient 〈 j1, 1; j2, −1|| j, 0〉, where j1+ j2+ j is an even integer, and of 〈 j1, 1; j2, 1|| j, 2〉 and 〈 j1, (1)/(2) ; j2, − (1)/(2) || j, 0〉, where j1+ j2+ j is an even or an odd integer, are given.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.527204
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