ABSTRACT
So far, most of the discussion was concerned with states and matrix elements of nucleons in the same orbit. As emphasized in Section 15, it is in such configurations that the antisymmetrization may cause complications which were discussed in detail. Coupling of states in which nucleons occupy different orbits does not add any essential complication. Any antisymmetric state of the j 1 n 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3149.tif"/> configuration with given Ji (and additional quantum number α 1, if necessary) may be coupled to any allowed state with J 2 (and α 2) of the j 2 n 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3150.tif"/> configuration with Clebsch-Gordan coefficients to yield states with all possible values of the total J. Any such state may be subsequently antisymmetrized with respect to the n 1 nucleons in the j 1-orbit and the n 2 nucleons in the j 2-orbit. This antisymmetrization yields always an allowed state with the given value of J. The expectation value of the Hamiltonian in such a state is made of three parts. The first is the single nucleon energies and two-body interactions of the j 1-nucleons in the state with α 1 J 1. The second includes these terms in the state with α 2 J 2 of the j 2 n 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3151.tif"/> configuration. These two parts have the same values in all states with the various J values, |J 1 − J 2| ≤ J ≤ J 1 + J 2. Their evaluation has 646been extensively discussed in preceding sections. The third part is the interaction between the n 1 nucleons in the j 1-orbit and the n 2 ones in the j 2-orbit. We shall presently consider the calculation of this part in the expression () 〈 j 1 n 1 ( α 1 J 1 ) j 2 n 2 ( α 2 J 2 ) J M | ∑ i < h n 1 + n 2 V i h | j 1 n 1 ( α 1 J 1 ) j 2 n 2 ( α 2 J 2 ) J M 〉 = 〈 j 1 n 1 α 1 J 1 M 1 | ∑ i < h n 1 V i h | j 1 n 1 α 1 J 1 M 1 〉 + 〈 j 2 n 2 α 2 J 2 M 2 | ∑ i > h = n 1 + 1 n 1 + n 2 V i h | j 2 n 2 α 2 J 2 M 2 〉 + 〈 j 1 n 1 ( α 1 J 1 ) j 2 n 2 ( α 2 j 2 ) J M | ∑ i = 1 n 1 ∑ h = n 1 + 1 n 1 + n 2 V i h | j 1 n 1 ( α 1 J 1 ) j 2 n 2 ( α 2 J 2 ) J M 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq3152.tif"/>
