By H.A.; De Hoffmann, F. Bethe

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2 Write down the scalar product of two vectors in terms of their cartesian and spherical components. 3 If r is the position vector, express it in terms of its spherical components and hence show that where is a spherical harmonic of order 1 and r is the modulus of the vector r. 4 Given any two vectors A and B, construct a vector product and a tensor product of rank 1. How are their spherical components related? 5 If C = A x B , show that the spherical component of the vector C is given by where is a component of the spherical tensor of rank 1 formed by taking the tensor product of the two vectors A and B.

14) in Eq. 15) Once again, we can subject the operators in the coordinate system X1 Y1 Z1 to a unitary transformation and obtain the corresponding operators in the coordinate system XYZ. 17) for R( α,β,γ ) all the rotations are carried out in the original coordinate system and its usefulness will be seen in the next section. 2. The Matrix The rotation matrix has been defined in Eq. 1) of the previous chapter and now we can express its elements as the matrix elements of the rotation operator R ( α,β,γ ).

First let us make a rotation through an angle a about the Z axis as illustrated in Fig. 1. 4) To know how the spherical components transform, we need to express the spherical components in terms of the Cartesian components. The transfor- 36 CHAPTER 4 mation of the Cartesian components is already given in Eq. 4). 7) The transformation of the spherical components can now be conveniently written in a matrix form. 9) where MZ ( α ) is the transformation matrix for rotation about the Z axis through an angle α.