Vrpfjr) = ±i>fm(-r) . The total angular momentum quantum number j is half an odd integer, and the commutation relations between the components of J permit us to introduce raising and lowering operators in the usual way:

Hence the y states can be explicitly constructed using Clebsch-Gordon (CG) coefficients yjm(f) = (t m-i-, i i\jm)aY^m_i(f)

where the CG coefficients come from Table 6.1.

As Eq. (6.17) and Table 6.1 show, there are precisely two y's for each j (and m, which we ignore in the following discussion). These have values of the orbital angular momentum t equal to j + \ or j - The parity of the }>'s depends on whether this value of i is even or odd. Once j and the parity are specified, £ is uniquely determined. However, instead of designating these states by parity, which is ±, we introduce a new quantum number k defined in the following way:

*A general discussion of angular momentum eigenfunctions and the addition of angular momentum can be found, for example, in Rose (1957).

6.1 SPHERICALLY SYMMETRIC POTENTIALS

Hence

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