Coupling of three angular momenta

The coupling of three angular momenta can be done by several routes (1) the third component is added to the result of the coupling of the first and the second component 

However, the final states A LM) and B LM) are dependent on the coupling path the different states of a system may correspond to the same values LM). In other words, for a full description of the states of a system two quantum numbers LM) are not sufficient intermediate quantum numbers should be included to form a complete set. This can be exemplified by 

Since the kets A LM) and B LM) span the same vector space as they are related by a unitary transformation such as 

The recoupling coefficients just introduced are independent of the projections of angular momenta M (they appear as a scalar product and we know that the scalar product does not depend upon the coordinate system). The recoupling coefficients can be arranged into 6y-symbols (having a number of symmetry properties) through 

These are interrelated to Racah W-coefficients via a different phase factor 

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Equation (B.22) can be used in the case of coupling of three angular momenta... 

We next consider the coupling of three angular momenta to give a scalar. We first couple j and j2 together to give a resultant of magnitude 73 and then couple this with j2 to produce a scalar, thus ... 

For the coupling of three angular momenta we have the basis set lxl2hmxm2mi) =... 

Fig. 1 shows the level diagram of antiprotonic helium which was experimentally established by observing several laser-induced transitions of the antiproton (see talk by T. Yamazaki [4]). Each level in Fig. 1 is split due to the presence of three angular momenta the orbital angular momentum L (mainly carried by the p), and the spins of the electron Se and the antiproton Sp. These momenta couple according to the following scheme ...

When the coupling of the angular momenta, exclusive of nuclear spin, is described by Hund s case (b), three ways of including the nuclear spin may be considered. In the first, known as case (b. v), the nuclear spin / is coupled to N, forming an intermediate F which is then coupled with. S to form F The corresponding basis kets take the form r], A N, A, I, F F, S, F), but they are unlikely to be used because the coupling of S to N is invariably much stronger than that between / and N. 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

These preliminary, but important clarifications, lead us to a discussion of the correlated motion within the Cl picture for both electrons in the ground state of helium. For this purpose it is illustrative to analyse the three-parameter Hylleraas function, the correlation properties of which have previously been described, in terms of Cl functions. Looking only for the individual components of orbital angular momenta r = i2 which couple to the desired Se state, one gets [GMM53]... 

In the more general case S 0 and the molecular angular momenta can be coupled in various ways. It is of primary importance to ascertain to what extent the interaction of the spin momentum S with the orbital momentum L is comparable to the rotation of the molecule, as well as to the interaction of each of the momenta L and S with the internuclear axis. An attempt to establish a hierarchy of interactions yields a number of possible, certainly idealized, coupling cases between angular momenta, first considered by Hund and known as Hund s coupling cases. Here we will discuss the three basic (out of five) cases of coupling of momenta in a linear molecule. 

Coupling of three or more angular momenta Racah algebra, Wigner 6-j and 9-j symbols... 

In spite of this, there does exist a general theoretical method for dealing with just this situation of the coupling of three (or more) angular momenta. It is the irreducible tensor method of Racah (29 and ITigwer (JO). 

For the coupling of three commuting angular momenta one can apply the following formula... 

This equation means that the operator corresponding to the dot product of the two source vectors is the same as a combination of the three operators that give the lengths of the three angular momenta. The coupled states, JMjjj2), are eigenfunctions of these three operators, and so they must be eigenfunctions of the combination in Equation 8.65. 

The ideas in Section IVA have been extended to three coupled angular momenta by Grondin et al. [3], in the context of a Hamiltonian of the form... 

Fig. 5. Vector model coupling of the spin angular momenta of two, three, and. four electrons...

However, all three particles in this equation are fermions with intrinsic spins S = jh. Therefore, we cannot balance the angular momentum in the reaction as written. The spins of the proton and the electron can be coupled to 0 or 1 ft and can also have relative angular momenta with any integral value from the emission process. This simple spin algebra will never yield the half-integral value on the left-hand side of the equation. Another fermion must be present among the products. 

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