Operators Infinitesimal rotation

For later convenience, it is useful to rewrite the infinitesimal rotation operator in terms of the angular momentum operator. If the rotation axis happens to coincide with one of the Cartesian axes, say the z axis, we can write... 

Thus the infinitesimal rotation operator Jz is the same as the familiar angular momentum operator h Jz (note that we use dimensionless angular momentum operators in this book). 

A generalisation of this result allows us to express the infinitesimal rotation operator about any axis it in the form... 

We next seek the irreducible representations of the full rotation group, formed by the infinite number of finite rotations R(ait). Because all such rotations can be expressed in terms of the infinitesimal rotation operators Jx, Jy and Jz (or equivalently J+, J and Jz), we start from these. 

Here 0 and are the spherical polar angles (only two angles are required to define the orientation of the vector r in space). Since these operators are the same as the infinitesimal rotation operators, all the results of the previous sections apply. The eigenfunctions of L2 and Lz are known as the spherical harmonics,... 

We can also define infinitesimal rotation operators Jx, Jy and. /- for rotations about the body-fixed axes in accordance with equation (5.7). These commute with the usual space-fixed infinitesimal rotation operators Jx, Jy and Jy. In addition, because of equation (5.40), they obey anomalous commutation relationships with each other ... 

Any rotation in x, y, z space can be described in terms of the infinitesimal rotation operators /, ly, and 7 whose matrix representations were defined in Eq. 3.31. These operators satisfy certain commutation relationships Eq. 3,32, and they add like ordinary unit vectors. Any small rotation may be expressed as a linear combination of 7i, ly, I,. As before, the basic problem is to find the reps of the full rotation group. 

The diatomic molecule Hamiltonian has in it a potential energy term that transforms like z. The presence of this term means that the Hamiltonian remains invariant under those transformations for which Rz = z. Only one of the infinitesimal rotation operators Z, I , Iz can generate irreducible representations of the group (C , or As before,... 

In order to describe the effects of the reorientation on a function f (JJ) of the orientation of a particular molecule necessary to introduce the infinitesimal rotation operators I, ... 

In this model the reorientation of any rigid molecule is described by a rotational diffusion tensor. Using the same infinitesimal rotation operators as in the discussion of cumulants, and choosing the principal axes of the diffusion tensor, the orientational propagator, Ctt), for this model is... 

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