Factorization and Solution of the Secular Equation

It will likewise be shown that when magnetic effects are neglected the square of the total orbital angular momentum must assume only the quantized values L(L + l)(A/2r) where L is an integer, while the square of the total spin angular momentum can take on only the values S(S + l)(h/2r) where S is integral or half-integral. (The letter L is usually used for the 

We shall now apply these ideas to the solution of the secular equation, taking the configuration np2 as an example. From Table 30-2 we see that Hn — IF is a linear factor of the equation, since f i alone has 2m = 2 and 2m, =0. A state with Ml = 2 must from the above considerations have L 2. Since 2 is the highest value of Ml in the table, it must correspond to L = 2. Furthermore the state must have S = 0, because otherwise there would appear entries in the table with Ml = 2 and Ms 9 0. This same root IF must appear five times in the secular equation, corresponding to the degenerate states L = 2, S = 0, Ms — 0, Ml = 2, 1, 0, —1, —2. From this it is seen that this root (which can be obtained from the linear factors) must occur in two of the linear factors (Ml = 2, —2 Ms = 0), in two of the quadratic factors (Ml = 1, —1 Ms = 0), and in the cubic factor (Ml = 0, Ms = 0). The linear factor — IF with Ml = 1, Ms = 1 must belong to the level L = 1, S = 1, because no terms with higher values of Ml and Ms appear in the table except those already accounted for. This level will correspond to the nine states with Ml = 1, 0, —1, and Ms = 1, 0, —1. Six of these are roots of linear factors (Ml = 1, Ms = +1 Ml = 0 Ms = 1), two of them are roots of the quadratic factors (Ml = +1, Ms = 0), and one is a root of the cubic factor (Ml = 0, Ms = 0). 

Without actually solving the quadratic equations or evaluating the integrals involved in them, we have determined their roots, since all the roots of the quadratics occur also in linear factors. 

1 This approximation, called (LS) or Russell-Saunders coupling, is valid for light atoms. Other approximations must be made for heavy atoms in which the magnetic effects are more important. 

The three energy levels for np2 which we have found are 

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