Examples of spherical tensor operators

The most obvious example of a spherical tensor operator is the angular momentum itself (a spherical tensor of rank one)  

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Just as angular momentum wave functions can be coupled together using the Clebsch Gordan coefficients, so too can spherical tensors. Two spherical tensors Rkl and S 2 can be combined to form a tensor of rank K which takes all possible values from (k + ki) to (k — k2), assuming k kp. 

This result may be compared with equation (5.77). It often happens that k = /c2, in which case a zeroth-rank tensor K = 0, as well as others, can be produced by their 

This differs from equation (5.110) in both phase and normalisation factors. We have seen that, for k = 1, the spherical tensor corresponds to a cartesian vector the spherical scalar product in this case is the same as the cartesian scalar product of two vectors  

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