Total angular momentum wavefunction

Assume that an experiment has been carried out on an atom to measure its total angular momentum L. According to quantum mechanics, only values equal to L(L+1) h will be observed. Further assume, for the particular experimental sample subjected to observation, that values of equal to 2 and 04f were detected in relative amounts of 64 % and 36%, respectively. This means that the atom s original wavefunction / could be represented as ... 

The angular momenta of atoms are described by the quantum numbers L, S or J. When spin-orbit coupling is important, it is the total angular momentum J which is a constant of the system. A group of atomic wavefunctions with a common J value - akin to a term, as described in Section 3.6 - comprise (27 -i- 1) members with Mj... 

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]... 

Fig. 20. A resonance state for the H + HD system at Ec = 1.2 eV at a total angular momentum of J = 25. In (a) the wavefunction is shown in the H + HD Jacobi coordinates for the collinear subspace. In (b) the wavefunction has been sliced perpendicular to the minimum energy path and is plotted in symmetric stretch and bend normal mode coordinates.

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... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Bound states are readily included in the line shape formalism either as initial or final state, or both. In Eq. 6.61 the plane wave expression(s) are then replaced by the dimer bound state wavefunction(s) and the integration(s) over ky and/or kjj2 are replaced by a summation over the n bound state levels with total angular momentum J n or J . The kinetic energy is then also replaced by the appropriate eigen energy. In this way the bound-free spectral component is expressed as [358]... 

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... 

We assume that each nucleon has a pseudo-spin i and pseudo-orbital angular momentum k. These couple to form the single particle angular momenta J,J (in [j]) of the two interacting nucleons. The wavefunction of a pair of nucleons coupled to a total angular momentum L (and z component p) is then given by ... 

Finally, often a representation is required in which the orbital angular momentum i and the electron spin s are coupled to form the total angular momentum j. For the wavefunction given in equ. (7.15) this is provided using the Clebsch-Gordan coupling formula (equ. (7.41b)), giving... 

A rigorous modelling of thermal broadening is — in practice — quite cumbersome and tedious. Let us consider a general asymmetric top molecule such as H2O, for example. Each total angular momentum state, specified by the quantum number J, splits into (2 J + 1) nondegenerate substates with energies E 0f (K = 1,..., 2J + 1). Every one of these (2J + 1) rotational states corresponds to a different type of rotational motion and is described by a distinct rotational wavefunction (see Section 11.3). 

The theory outlined above can be used to calculate the exact bound-state energies and wavefunctions for any triatomic molecule and for any value J of the total angular momentum quantum number. We can solve the set of coupled equations (11.7) subject to the boundary conditions Xjfi (R Jp) —> 0 in the limits R —> 0 and R — oo (Shapiro and Balint-Kurti 1979). Alternatively we may expand the radial wavefunctions in a suitable set of one-dimensional oscillator wavefunctions ipm(R),... 

Multiplicity Spin Multiplicity) The number of possible orientations, calculated as 2S -L 1, of the spin angular momentum corresponding to a given total spin quantum number (S), for the same spatial electronic wavefunction. A state of singlet multiplicity has S = 0 and 2S -i- 1 = 1. A doublet state has S = 1/2, 2S -i- 1 = 2, etc. Note that when S > L (the total orbital angular momentum quantum number) there are only 21 -t 1 orientations of total angular momentum possible. 

The functions (r. R.O.t = 0) involve a product of the initial wavefunction and the internal coordinate dejjendent vector components of the transition dipole moment (see Ref. [43] and Appendix B of Ref. [33] ). As the total angular momentum is a conserved quantity during the time propagation of the wavepacket, we may divide up the initied wavepacket (Eq. (23)) into three components [43], one for each of the allowed valuers of J. Thus E(p (23) may be rewritten as ... 

Figure 9 compares transitions probabilities for zero total angular momentum J resulting from (2S-DIABATIC) and (IS-GP) calculations with initial and final states chosen to be representative of two cases (A) initial and/or final states have low rovibrational excitation, (B) initial and final states have both high rovibrational excitation. In case (A), both (2S-DIABATIC) and (IS-GP) results almost coincide, even for energies close to the three body dissociation limit. Case (A) corresponds to the kind of transitions already considered in Sect. 4.1 where it was shown that it is valid to use the single rovibrational wavefunction associated to the ground electronic state... 

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