Angular momentum projection quantum

Launay and Le Dourneuf use essentially, although not exactly, the same PA hyperspherical coordinates as Pack and Parker. In particular, they choose the quantisation axis for the internal motion as the axis of least inertia, so that the rotational couplings about this axis are minimised in a variational way [49, 52]. This leads to rapid convergence of reaction probabilities with respect to the total angular momentum projection quantum number i , and so allows one to use far smaller basis sets for large J values than are required when all possible projections are retained. 

C. Distributions of Angular-Momentum-Projection Quantum Number... 

In the above equations , I and m are the main quantum number and the quantum numbers associated with angular momentum and angular momentum projection, respectively. The eigenfunction is defined... 

Vmax(i ) is the maximum value of the rotational quantum number in vibrational level v included in the vibrational-rotational-orbital basis. In these calculations we did not eliminate higher values of the body-frame angular momentum projection fL... 

In each term of this expression the quantum numbers must satisfy the conservation requirement for fixed total angular momentum projection, Aftot = mi + m2 + mi. In addition, applying exchange symmetry to each term requires that Fi miR, k - 0) = s — ) Fi mi R,9) for bosons, and5(-l) + F i,m/(. 0) for fermions. 

Protons and neutrons have an intrinsic spin and an intrinsic magnetic moment. Experiments have revealed that there are just two possible orientations of their spin vectors with respect to a z-axis defined by an external field. This means that their intrinsic spin quantum number is 1/2 they are fermions. Whereas the letter S is commonly used for the electron spin quantum number, the letter I is commonly used for nuclear spin quantum numbers. Thus, I = 1/2 for a proton, and the allowed values for the quantum number giving the nuclear spin angular momentum projection on the z-axis, Mj, are + 1/2 and -1/2. 

Now, we have besides the vibrational, the electronic angular momentum the latter is characterized by the quantum number A corresponding to the magnitude of its projection along the molecular axis, L. Here we shall consider A as a unsigned quantity, that is, for each A 7 0 state there will be two possible projections of the electronic angular momentum, one corresponding to A and the other to —A. The operator Lj can be written in the form... 

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... 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

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... 

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