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K. T. Hecht - One of the best experts on this subject based on the ideXlab platform.

  • Symmetry Properties of Clebsch—Gordan Coefficients
    Quantum Mechanics, 2020
    Co-Authors: K. T. Hecht
    Abstract:

    Clebsch—Gordan Coefficients in which the three angular momenta, j 1, j 2, and j = j 3, are reordered may be simply related to each other. The most trivial case involves the exchange of the order of the quantum numbers, j 1 m 1 and j 2 m 2. The state vector ❘ j 1, j 2 j 2 m 2〉 is a direct product of two vectors involving separate subspaces of the full Hilbert space, or in terms of the coordinate representation, the wave function is a product of functions involving different variables. For example, might be a function of orbital variables and might be a function of spin variables. Thus, the product of these two functions should not depend on the order in which we write the two functions. Therefore, when we expand this product function in terras of the total angular momentum eigenfunctions , the result must be independent of the order in which we write the original product function, , or , with the possible exception of an over-all phase factor. This phase factor comes in because our phase convention fixing the overall sign of the Clebsch—Gordan Coefficients gives preference to the angular momenta sitting in the number 1 and number 3 positions of the Clebsch—Gordan Coefficient. Thus, 〈j 1 j 1 j 2 m 2❘j 3 j 3〉 must be positive by our phase convention. Similarly, 〈j 2 j 2 j 1 m 1❘j 3 j 3〉 must also be positive. On the contrary, the Clebsch—Gordan Coefficient 〈j 1 m 1 j 2 j 2❘j 3 j 3〉 has the sign with m 1 = j 3− j 2 Hence, its sign is.

N. Aizawa - One of the best experts on this subject based on the ideXlab platform.

  • Tensor operators and Clebsch–Gordan Coefficients of the quantum algebra suq(1,1)
    Journal of Mathematical Physics, 1993
    Co-Authors: N. Aizawa
    Abstract:

    Two kinds of suq(1,1) tensor operators are considered. One of them carries a nonunitary finite dimensional representation of suq(1,1), and the other carries a unitary infinite dimensional representation. Explicit formulas of the former tensor operators are constructed and a useful formula to calculate the matrix elements of rank 1 tensors is derived. By making use of the q analog of the Wigner–Eckart’s theorem, the Clebsch–Gordan Coefficient can be extracted from the matrix element of a tensor operator. Explicit formulas of three kinds of the suq(1,1) Clebsch–Gordan Coefficients are given, that is, the Clebsch–Gordan Coefficient which couples two (non) unitary representations to get the third (non) unitary representation, and the one which couples a nonunitary and a unitary representations to get a new unitary representation. It is shown that these Clebsch–Gordan Coefficients and the one of suq(2) can transmute one another by the appropriate replacement of their variables. It is further shown that, by usi...

Zhao-xian Yu - One of the best experts on this subject based on the ideXlab platform.