Addition of angular momenta

Let us consider two angular momenta J and J2 which are independent, that is, act on different parts of a system, and have eigenfunctions j, m 1) and j2, m2) respectively. If the two parts of the system interact through some physical mechanism, the two angular momenta become coupled and it is meaningful to define a resultant angular momentum, 

It is easy to show that the components of J obey the standard commutation relationships (5.13). The commuting operators J and J/ therefore have eigenfunctions j,m). In this section we describe how these eigenfunctions are related to those of j and j. We can describe the combined system by a simple product of the wavefunctions y i, m ) j2, m2) which we write as j mi j2m2) this is known as the uncoupled or decoupled representation. These products are still eigenfunctions of j, Jxz, J and J2z- They are also eigenfunctions of Jz with eigenvalue m = m +m2  

Remembering that the J operator acts only on the jx, mi) part of the decoupled representation and J2 acts only on the j2, m2) part, we see that J2 connects states with different values of mi and m2 but with constant (mi + m2). 

In order to determine the allowed values of the total angular momentum quantum number j, let us apply J in equation (5.70) to the uncoupled state jx, jx ji, ji), that is, with the maximum values for mi and m2  

This particular product wavefunction is therefore an eigenfunction of J2 with eigenvalue j = yi + 72  

For a many-electron atom, the operators for individual angular momenta of the electrons do not commute with the Hamiltonian operator, but their sum does. Hence we want to learn how to add angular momenta. 

We define the total angular momentum M of the system as the vector sum 

The vector equation (11.20) gives the three scalar equations 

K Ml and refer to different electrons, they will commute with each other, since each 

We now show that the components of the total angular momentum obey the usual angular-momentum conunutation relations. We have [Eq. (5.4)] 

If Ml and M2 refer to different electrons, they will commute with each other, since each will affect only functions of the coordinates of one electron and not the other. Even if Mi and M2 are the orbital and spin angular momenta of the same electron, they will commute, as one will affect only functions of the spatial coordinates while the other will affect functions of the spin coordinates. Thus (11.24) becomes 

Since all components of Mi commute with all components of M2, we have 

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Instead of solving these equations they can be analyzed according to the known rules for addition of angular momenta, applied to the orbital and spin degrees of freedom with the variables k and a. In general there are three different wave functions YjM corresponding to three orbital states, Yj(M. 

The process of addition of angular momenta can be continued. For four angular momenta... 

The Kotani spin functions for a six-electron singlet 0oo yt be constmcted by successive coupling of six one-electron spin functions (a or p) to an overall singlet according to the mles for addition of angular momenta. Each spin function is uniquely defined by the series of partial resultant spins of the consecutive groups of 1, 2,. .., 5 electrons, which can be used as an extended label for the spin function... 

Figure 6.17 Addition of angular momenta in a deformed odd A nucleus, fl is the projection of the total singular momentum of the odd nucleon. It is added vectorially to the rotational angular momentum of the core, R, to give the total angular momentum J whose projection on the symmetry axis is K.

For a proper addition of angular momenta certain vector coupling coefficients are required the Clebsch-Gordan coefficients, or alternatively the 3/-symbols, the 6/ -symbols and 9/-symbols, for the coupling of two, three and four angular momenta, respectively. The 3/-symbols can be calculated through the Racah formula the 6/ -symbols can be expressed in terms of the 3/-symbols and the 9/-symbols are evaluable with the help of either 3/-symbols or 6/ -symbols. For some special cases closed-form formulae also exist. 

Because of the vector nature of the angular momenta, addition of angular momenta in atoms is feasible. The total orbital angular momentum and its projection are expressed through the sums... 

In accordance with the addition of angular momenta the intermediate spin numbers are... 

We now find the possible values of the total-angular-moraentura quantum number J that arise from the addition of angular momenta with individual quantum numbers y l and y 2. 

Again, the physical links between the bodies and the rotation axis are supposed to be constant, which features the indeformability of the systan and ensures the same angular velocity for all parts. The composition of N poles into a single pole is based on this unique angular velocity (energy-per-entity) and on the addition of angular momenta (entity numbers) ... 

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