Angular momentum body-fixed

From these relations it follows that is related to the angular momentum modulus, and that the pairs of angle a, P and y, 8 are the azimuthal, and the polar angle of the (J ) and the (L ) vector, respectively. The angle is associated with the relative orientation of the body-fixed and space-fixed coordinate frames. The probability to find the particular rotational state IMK) in the coherent state is... 

The original semiclassical version of the centrifugal sudden approximation (SCS) developed by Strekalov [198, 199] consistently takes into account adiabatic corrections to IOS. Since the orbital angular momentum transfer is supposed to be small, scattering occurs in the collision plane. The body-fixed correspondence principle method (BFCP) [200] was used to write the S-matrix for f — jf Massey parameter a>xc. At low quantum numbers, when 0)zc —> 0, it reduces to the usual non-adiabatic expression, which is valid for any Though more complicated, this method is the necessary extension of the previous one adapted to account for adiabatic corrections at higher excitation... 

The theory behind body-fixed representations and the associated angular momentum function expansions of the wavefunction (or wave packet) in terms of bases parameterized by the relevant constants of the motion and approximate constants of the motion is highly technical. Some pertinent results will simply be stated. The two good constants of the motion are total angular momentum, J, and parity, p = +1 or 1. An approximate constant of the motion is K, the body-fixed projection of total angular momentum on the body-fixed axis. For simplicity, we will restrict attention to the helicity-decoupled or centrifugal sudden (CS) approximation in which K can be assumed to be a constant of the motion. In terms of aU its components, and the iteration number k, the real wave packet is taken to be [21]... 

The TD wavefunction satisfying the Schrodinger equation ih d/dt) F(f) = // (/,) can be expanded in a basis set whose elements are the product of the translational basis of R, vibrational wavefunctions for r, r2, and the body-fixed (BF) total angular momentum eigenfunctions as41... 

In order to transform to the body-fixed representation, we will need to relate the angular functions Wj (R,r) to angular functions defined relative to the body-fixed axes [L., J,K,M,p)QjK ), where J,K,M,p) are the parity-adapted total angular momentum eigenfunctions of Eq. (4.5) and x(0) normalized associated Legendre polynomials of the body-fixed Jacobi angle]. 

The identity operator within the space of functions with a hxed value of J and the parity (denoted by p), and that are associated assymptotically with a quantum number K of the body-fixed z component of the total angular momentum, is... 

The projection of the electronic orbital angular momentum is neglected in this adiabatic representation, and the parity of the electronic function under reflection through the x — z body-fixed plane, (Txz, is given by... 

No one wants to work with 18 -y symbols. There are two work-arounds for the numerical implementation of the coupled channel theory in the total angular basis. First - as suggested by Tscherbul and Dalgarno [21] - one can use a basis of angular momentum states defined in the body-fixed coordinate frame. For example, for two molecules in a S electronic state with nonzero electron spin, the eigenstates of the full Hamiltonian can be written as... 

Following the quenching process further, the system is described best in the body-fixed frame, at least for the smaller internuclear distances. Then no orbital angular momentum can be transferred to the relative internuclear motion lz = 0, and the question arises as to how the electronic orbital angular momentum Lz = 1 of the 3p II) state of Na should be disposed of since the final state has to be 3s 2). The obvious solution is to change the orientation of the molecular angular momentum j, that is, to induce a Aiz= 1 transition in the molecule and thus maintain a constant... 

The eigenfunctions of the atom must be eigenfunctions of the total angular momentum and its projection on a space fixed z axis. If we ignore the electron spins for simplicity, the total angular momentum L and its projection M on the space fixed z axis are conserved. The spatial wavefunctions iM(r,R) are related to the body fixed wavefunctions by the Euler transformation... 

The first point reflects the fact that the dynamics is independent of the orientation in space. Points 2 and 3 manifest that both the total angular momentum and the parity are conserved. The Coriolis coupling arises from the continuous rotation of the body-fixed system with the scattering vector. Finally we stress that for J = Q, = 0 Equation (11.7) goes over into (3.20). 

The unpaired pn electron has an orbital angular momentum L with projections A = 1 on the body-fixed 2-axis which — in a classical sense — describe rotation of the electron in opposite directions about the internuclear axis. Furthermore, it possesses a spin S with components = 1/2. Coupling of A and leads to two (2 1 = 2) manifolds with A = 1/2 and 3/2 which are represented by 2nx/2 and 2n3/2, respectively. The splitting is in the range of several hundredths cm-1. 

Here, the operator J is the total angular momentum operator in the space-fixed frame, and Tx, X=A and B, is defined by Eq. (1-262). Note, that the present coordinate system corresponds to the so-called two-thirds body-fixed system of Refs. (7-334). Therefore, the internal angular momentum operators jA and jB, and the pseudo angular momentum operator J do not commute, so the second term in Eq. (1-265) cannot be factorized. 

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