Symmetry Properties of Clebsch—Gordan Coefficients

Abstract

Clebsch—Gordan coefficients in which the three angular momenta, j1, j2, and j = j3, are reordered may be simply related to each other. The most trivial case involves the exchange of the order of the quantum numbers, j1m1 and j2m2. The state vector ❘ j1, j2j2m2 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, j1j1j2m2❘j3j3 must be positive by our phase convention. Similarly, j2j2j1m1❘j3j3 must also be positive. On the contrary, the Clebsch—Gordan coefficient j1m1j2j2❘j3j3 has the sign with m1 = j3− j2 Hence, its sign is.