Operators, angular momenta linear

The form of Lz in Cartesian coordinates is Eq. (7-la), and it is clear that orbital angular momentum is related to angular displacement in the same way as the linear operators are related. 

Without essential limitation of generality it may be assumed that the orientation of the molecule and its angular momentum are changed by collision independently, therefore F(JU Ji+, gt) = f (Jt, Ji+i)ip(gi). At the same time the functions /(/ , Ji+ ) and xp(gi) have common variables. There are two reasons for this. First, it may be due to the fact that the angle between / and u must be conserved for linear rotators for any transformation. Second, a transformation T includes rotation of the reference system by an angle sufficient to combine axis z with vector /. After substitution of (A7.16) and (A7.14) into (A7.13), one has to integrate over those variables from the set g , which are not common with the arguments of the function / (/ , /j+i). As a result, in the MF operator T becomes the same for all i and depends on the moments of tp as parameters. 

The effect of time reversal operator T is to reverse the linear momentum (L) and the angular momentum (J), leaving the position operator unchanged. Thus, by definition,... 

A little reflection shows that the commutation relationships, recognized as one of the fundamental differences between classical and quantum systems, are common to all forms of angular momentum, including orbital, polarization and spin. It is of interest to note that the eigenvalues for all forms of angular momentum can be obtained directly from the commutation rules, without using special differential operators. To emphasize the commonality, angular momentum M of all forms will be represented here by three linear operators Mx, My and Mz, that obey the commutation rules ... 

Instead of Cartesian coordinates it is convenient to use spherical coordinates. Properties of physical operators can be characterized according to the way they behave under rotation of the axes. These properties can be cast into a simple mathematical form by giving the commutation relations with the angular momentum. It is convenient to introduce the linear combinations... 

There is a natural representation of the Lie algebra so 3 using partial differential operators on We can define the three basic angular momentum operators as linear transformations on as follows ... 

Some readers may wonder why we make this restriction, especially if they have experience applying angnlar momentnm operators to discontinuous physical quantities. It is possible, with some effort, to make mathematical sense of the angular momentum of a discontinuous quantity hut, as the purposes of the text do not require the result, we choose not to make the effort. Compare spherical harmonics, which are effective because physicists know how to extrapolate from spherical harmonics to many cases of interest by taking linear combinations likewise, dense subspaces are useful because mathematicians know how to extrapolate from dense subspaces to the desired spaces. 

The molecular electronic wave functions ipe] are classified using the operators that commute with Hei. For diatomic (and linear polyatomic) molecules, the operator Lz for the component of the total electronic orbital angular momentum along the internuclear axis commutes with Hel (although L2 does not commute with tfel). The Lz eigenvalues are MLh,... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

New tensor operators can be obtain as linear combinations of strings of creation- and annihilation operators according to the usual angular momentum coupling rules. The coupling of two different sets of creation... 

Formally, orbital angular momentum operator L of a particle moving with linear momentum p = —itiV at a position r with respect to some... 

---