Infinitesimal rotation

The T corresponding to various infinitesimal transformations (e.g., an infinitesimal rotation about the 2-axis, or an infinitesimal Lorentz transformation about the x-axis) can be explicitly computed from this representation. The finite transformations can then be obtained by exponentiation. For example, for a pure rotation about the 1-direction (x-axis) through the angle 6,8 is given by... 

This operator can now be shown to be identical with the operator for an infinitesimal rotation of the vector field multiplied by i, i.e. J = — M. The components of the angular momentum operator satisfy the commutation relations... 

This algebra can be expressed in terms of the infinitesimal rotation generators of the 0(3) group [42] in three dimensional space ... 

There is also a relation between polar unit vectors, boost generators, and electric fields. An electric field is a polar vector, and unlike the magnetic field, cannot be put into matrix form as in Eq. (724). The cross-product of two polar unit vectors is however an axial vector k, which, in the circular basis, is e<3>. In spacetime, the axial vector k becomes a 4 x 4 matrix related directly to the infinitesimal rotation generator /3) of the Poincare group. A rotation generator is therefore the result of a classical commutation of two matrices that play the role of polar vectors. These matrices are boost generators. In spacetime, it is therefore... 

Figure 4.1(b)). One can see in Figure 4.1(c) that reflection in a plane normal to the axis of rotation does not change the direction of rotation, but that it is reversed (Figure 4.1 (d)) on reflection in a plane that contains the axis of rotation. Specification of a rotation requires a statement about both the axis of rotation and the amount of rotation. We define infinitesimal rotations about the axes OX, OY, and OZ by (note the cyclic order)... 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

I hhh inversion operator operators that generate infinitesimal rotations about x y z, respectively (Chapter 11)... 

Compare this with Eq. (4.329), where Jn = (n x 9/0u) is the operator of infinitesimal rotations over n. We mention that, in our attempts to obtain low-frequency susceptibilities, we have omitted in Eq. (4.332) the term responsible for gyromagnetic effects. 

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... 

Analogously, for an infinitesimal rotation through 5cp about a general axis it, one obtains... 

In the earlier section relating to an infinitesimal rotation, we introduced a relation between the operator for an infinitesimal rotation through 8cp about a general axis n and the angular momentum operator We can carry this connection further and find a corresponding expression for a finite rotation through a finite angle y... 

The subscripts s and m denote that space or molecule-fixed electron coordinates are held constant for example, (3/39)s means (3/39)r x X ,y",z" The desirability of rewriting the total Hamiltonian in terms of (3/3 9)m, (3/3)m and (3/3x)m is thus evident (3/39)s corresponds to the effect on the wavefunction of an infinitesimal rotation of the nuclei alone, with the electrons held fixed in space, whereas (3/30)m corresponds to the effect of an infinitesimal rotation of the molecule-fixed axis system and all the particles with it. 

Since successive rotations about a given axis are additive, we can form thefinite rotation operator R(ah) from a series of m infinitesimal rotations, each through an angle aim... 

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... 

Expanding equation (5.10) as a power series in the infinitesimal rotations, we obtain... 

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 ... 

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