Commutation relations shift operators

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

The commutation relations among their components differ slightly from those of the shift operators. From Eqs. [36] and [39], it follows that... 

The decoupling of the Dirac equation to the two-component form is a rather complicated process. It manipulates the resolvant operator which is transferred from the denominator to the numerator and then exposed to consecutive commutator relations in order to be shifted to the far right. Finally, a number of one-electron terms of the order 1/c2 is obtained some of them have no classical analogy and cannot be derived from non-relativistic theories. 

The commutation relation replaces BB by B B - - 1. The B B part made that in the sequence of operators B in jJ/y the operator B has been shifted right by one position (see the first term) and on its right side, it has only u — 1 of B operators. On the other hand, the presence of the unit operator made that we got the second term in the form I ijfy = Now we will use once more the commutation relation. Also this time it will cause a one-position... 

We now proceed to consider the total spin of Slater determinants. Using the relation (1.8.14), we obtain the following commutators between the spin-orbital ON operators and the shift operators ... 

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