Photon operators calculations

Renaming the electron numbers shows that the first and fourth terms in the matrix element M i(M1,ms) and also the second and third terms are identical, and they can be combined. As a next step one calculates the action of the photon operator on the single-particle wavefunctions. Omitting for simplicity the wavefunction... 

Finally, it should be noted that treating one of the otherwise equivalent electrons in equ. (7.46) individually is frequently used for calculating matrix elements with a one-electron operator (e.g., the photon operator) acting on equivalent electrons. Similarly, if two-electron operators play a role, like in the Coulomb interaction between electrons, then it is convenient to separate two electrons from the equivalent electrons. This is done using the coefficients of fractional grandparentage (for more details see [Cow81]). 

A way to overcome the difficulties in the definition of the Hermitian phase operator has been proposed by Pegg and Barnett [40,45]. Their method is based on a contraction of the infinite-dimensional Hilbert-Fock space of photon states Within this method, the quantum phase variable is determined first in a finite 5-dimensional subspace of //, where the polar decomposition is allowed. The formal limit, v oc is taken only after the averages of the operators, describing the physical quantities, have been calculated. Let us stress that any restriction of dimension of the Hilbert-Fock space of photons is equivalent to an effective violation of the algebraic properties of the photon operators and therefore can lead to an inadequate picture of quantum fluctuations [46]. 

It is seen that the photon operators with m > 2 do not contribute to the polarization of radiation in the polar direction even if j > 2. It is straightforward to calculate the elements of the vacuum polarization matrix (143) at 0 = 0 ... 

Finally, the molecular absorption cross-section capture area of a molecule. Operationally, it can be calculated as the (Napierian) absorption coefficient divided by the number N of molecular entities contained in a unit volume of the absorbing medium along the light path ... 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

The correction of order a Za) induced by the polarization operator insertions in the external photon lines in Fig. 3.10 was obtained in [40, 41, 42] and may again be calculated in the skeleton integral approach. We will use the simplicity of the one-loop polarization operator, and perform this calculation in more detail in order to illustrate the general considerations above. For calculation of the respective contribution one has to insert the polarization operator in the skeleton integrand in (3.33)... 

Due to the simplicity of the photon polarization operator the calculation based on the scattering approximation [8] is so straightforward that we can... 

In the case of the polarization insertions the calculations may be simplified by simultaneous consideration of the insertions of both the electron and muon polarization loops [18, 19]. In such an approach one explicitly takes into account internal symmetry of the problem at hand with respect to both particles. So, let us preserve the factor 1/(1 - - m/M) in (9.9), even in calculation of the nonrecoil polarization operator contribution. Then we will obtain an extra factor m /m on the right hand side in (9.12). To facilitate further recoil calculations we could simply declare that the polarization operator contribution with this extra factor m /m is the result of the nonrecoil calculation but there exists a better choice. Insertion in the external photon lines of the polarization loop of a heavy particle with mass M generates correction to HFS suppressed by an extra recoil factor m/M in comparison with the electron loop contribution. Corrections induced by such heavy particles polarization loop insertions clearly should be discussed together with other radiative-recoil... 

The light-intensity (or photon flux) of the laser beam (/0) was calculated from the measured power output and from the known value of the energy of the photons at the desired wavelength 328 kJ.mol-1 for the emission line at 363.8 nm and 354 kJ.mol-1 for the emission line at 337.1 nm. The maximum value of /0 obtained at full power operation were respectively ... 

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