Operators square

The contribution to the Lamb shift is given by the insertion of the one-loop polarization operator squared Ii k) in the skeleton integral in (3.33), and taking into account the multiplicity factor 3 one easily obtains [46, 47, 48]... 

The simplest correction is induced by the diagrams in Fig. 9.8(a) with two insertions of the one-loop vacuum polarization in the external photon lines. The respective contribution to HFS is obtained from the skeleton integral in (9.9) by the substitution of the polarization operator squared... 

The angular momentum operator squared L, expressed in spherical polar coordinates, is... 

Simultaneous operations are placed in parentheses to distinguish them from sequential operations. Square brackets distinguish separate polymerizations, where their products are later mixed and/or reacted. In a formal sense, the parentheses and brackets constitute operations in their own right. 

Furthermore large basis sets are needed for an accurate description of the region close to the nucleus where relativistic effects become important. Methods based on the replacement of the Dirac operator by approximate bound operators (square of the Dirac operator, its absolute value etc...) have not been very successful as can been understood from the fact that they break the Lorentz invariance for fermions. 

In order to determine the two-center bond order, it is reasonable to attempt to obtain the expansion similar to (9.20), but for the squares of the density-operator components. Unfortunately, in the general case, the spur of the density operator squared depends on both the AO basis set and the computational tech-... 

The Flamiltonian commutes widi the angular momentum operator as well as that for the square of the angular momentum I . The wavefiinctions above are also eigenfiinctions of these operators, with eigenvalues tndi li-zland It should be emphasized that the total angular momentum is L = //(/ + )/j,... 

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

The following operations involving square matriees eaeh of the same dimension are useful to express in terms of the individual matrix elements ... 

This new operator is referred to as the square of the total angular momentum operator. 

Again, the square of the total rotational angular momentum operator appears in Hj-ot... 

In both Eqs. (1.3) and (1.4), the summations are carried out over all classes of data. From a computational point of view, standard deviation may be written in a more convenient form by carrying out the following operations. First both sides of Eq. (1.4) are squared then the difference Mi - M is squared to give... 

In multiplying by we use, again, examples of the vibrations of NH3. The result depends on whether we require when (a) one quantum of each of two different e vibrations is excited (i.e. a combination level) or (b) two quanta of the same e vibration are excited (i.e. an overtone level). In case (a), such as for the combination V3 - - V4, the product is written E x E and the result is obtained by first squaring the characters under each operation, giving... 

Applying the operator I to both sides and squaring, we obtain... 

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