Total angular momentum quantum

Nuclear spin 1 = Total angular momentum quantum number 7 = 0,1,2,., ... 

This same analysis ean be used to deseribe how a set of funetions /j,m (0, X) (labeled by a total angular momentum quantum number that determines the number of funetions in the set and an M quantum number that labels the Z-axis projeetion of this... 

Figure 3. The ground and first two e ncjted states or the Na valence electron. The F values are the total angular momentum quantum number (.Peys et ill.. 1991).

The primary reason it is difficult to treat angular momentum rigorously is due to the angular momentum catastrophe [58]. As noted in Section III, cross sections and other experimental observables are sums over all relevant total angular momentum quantum numbers, J. Each J represents a quantum dynamics problem to be solved, and the size of the problem increases dramatically with J. For each J, there are Nk projections of K, where Nts = fmax — min + 1- For a fliree-atom system, the minimum value of K, is a function of both J and p, such that = 0 when J and p are... 

The operator Tang contains the cross-terms that give rise to the Coriolis coupling that mixes states with different fl (the projection of the total angular momentum quantum number J onto the intermolecular axis). This term contains first derivative operators in y. On application of Eq. (22), these operators change the matrix elements over ring according to... 

Finally, the magnetic moments resulting from the spin and the orbital motion interact. This spin-orbit coupling is taken into account by the total angular momentum quantum number J (Russel-Saunders coupling) ... 

Fig. 4. Computed partial cross-sections in A2 for the F + HD (v = 0, j = 0) —> HF + D reaction as a function of the total angular momentum quantum number, J, up to collision energies of 3 kcal/mol.

Finally, we can demonstrate that the degeneracy is even when the total angular momentum quantum number F is half-integer, again via a reductio ad absurdum method. Suppose that the degeneracy of the eigenstates is k, then we... 

SO on. The ground state is the state with the lowest energy. It contains the maximum number of unpaired electrons (highest spin multiplicity) and if more than one state has the same value for net spin (S), the ground state is that with the highest net orbital angular momentum (L). In addition, the total angular momentum quantum number is J = L -i- S if the shell is more than half-filled and J = L - S if not. 

In an orbital, the orbital angular momentum L as well as the electron spin must be considered. In a free ion, L and S are tightly coupled to give a total angular momentum quantum number J which interacts with an applied field. In... 

Fig. 2-12.—The interaction of the orbital angular momentum vectors for two p electrons (h = 1, l — 1) to form the resultant total angular momentum vector, with values corresponding to 0, 1, and 2 for the total angular momentum quantum number L.

One can show that the total angular-momentum quantum number J takes on the possible values... 

For 2 molecules, we used J as the rotational angular-momentum quantum number since 2 molecules have no electronic spin or orbital angular momentum, J is also the total angular-momentum quantum number, exclusive of nuclear spin, for such molecules. Recall that for atoms J is also used as the total angular-momentum quantum number apart from nuclear spin.) The rotational energy in case (a) is given approximately by... 

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