Angular momentum projection quantum number

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... 

The problem is not simplified by Eq. (15), since there exists a closed-form expression for the multi-scattering matrix for n spheres in terms of spherical Bessel and Hankel functions, spherical harmonics and 3j-symbols, where l, l and to, m are total angular momentum and z-projection quantum numbers, respectively (Henseler, Wirzba and Guhr, 1997) ... 

Here y is the component of the transition-dipole operator in the direction of the light s electric field vector E, Jj, M, and p are the energy, total angular momentum, its space-fixed projection, and the parity of the initial bound state k, v, j, and irij are the relative momentum, vibrational quantum number, rotational angular momentum, and its space-fixed projection for the scattering state. 

It is not difficult to surmise that the final expression for the fully resolved differential photodissociation cross section is extremely complicated (Balint-Kurti and Shapiro 1981). It contains vast quantities of sums and angular momentum coupling elements. Note that the cross section depends explicitly on the magnetic quantum numbers Mi and mj. Somewhat simpler cross section expressions can be derived by averaging over the initial projection quantum number Mi and summing over the final projection quantum number of the rotor, mj. As shown by Balint-Kurti and Shapiro, the angle-resolved cross section then takes on the general form... 

Each initial rotational state yields a distinct final rotational state distribution. This holds true even if the total angular momentum J is the same and merely the projection quantum numbers K- and K+, are different. 

To proceed further, detailed analysis of the RF PADs is required. The outgoing free electron partial waves are decomposed into symmetry-adapted spherical harmonics [51, 55], as given by Eq. (3.4). For Civ, these harmonics are described by their Civ symmetry and by / >, where l, A,, are the orbital angular momentum and projection quantum numbers, respectively. Values of / = 0,1,2... are labeled s, p, d. .. whereas values of >. = 0,1,2... are labeled... 

The state of a spinless hydrogen atom is completely specified by the principal quantum number n, the orbital angular momentum quantum number and the magnetic (projection) quantum number m. The Schrodinger equation is... 

Scattering theory concerns a collision of two bodies, that may change the state of one or both of the bodies. In our application one body (the projectile) is an electron, whose internal state is specified by its spin-projection quantum number v. The other body (the target) is an atom or an atomic ion, whose internal bound state is specified by the principal quantum number n and quantum numbers j, m and / for the total angular momentum, its projection and the parity respectively. We... 

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... 

Unlike the electron spin, the photon spin cannot be interpreted as the total angular momentum in the rest reference frame there is no such a frame for the photon. Thus the photon spin cannot be separated from the orbital angular momentum. However the photon spin projection quantum number /i may have... 

Nikitin and Zare (1994) also define two different sets of molecule frame projection quantum numbers according to whether the molecule frame projection of each component angular momentum is onto the internuclear axis, z Lz, Sz, jz, Rz, Jz, —> A, E, Cl, 0, Cl,, or onto the total angular momentum axis,... 

For high rotational levels, or for a molecule like OH, for which the spin-orbit splitting is small, even for low J, the pattern of rotational/fme-structure levels approaches the Hund s case (b) limit. In this situation, it is not meaningful to speak of the projection quantum number Q. Rather, we first consider the rotational angular momentum N exclusive of the electron spin. This is then coupled with the spin to yield levels with total angular momentum J = A-I- land A -1. As before, there are two nearly degenerate pairs of levels associated... 

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... 

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