Energy level shift operator

In order to complete the calculation of the strong energy-level shift, one has to match at the accuracy 0(a) the particular combination of the non-relativistic coupling d d- Therefore we consider the scattering operator... 

A(E) and f(E) are Hilbert transform from one another [7]. Since the lifetime of the quasi-bound state cj) is much longer than the lifetime of the relevant part of the continuum QH (f)), the Hermitian operators A(E) and f (E) are smooth functions within the energy range of interest. It results that the energy dependence of the level-shift operator can be neglected and that the effective Hamiltonian becomes energy independent... 

This energy independence appears as well in the previous model Hamiltonian when the quasi-continuum tends to a continuum (16). The Hermitian part of the level-shift operator is often negligible and was neglected in Eq. (24). Using Eq. (7) the lineshape can be expressed in terms of the poles of the Green function (see Section 2.2.2)... 

The quantity R in Equation 2.10 is called the level shift operator or self-energy and is given by... 

The coupling to the continuum is implicitly contained in the second term in Eq. (376). The decay of the population in the bound state subspace is due to the imaginary part of P E — PHPY P. The contribution from the real part of P E — PHP) P, the so-called level shift due to the coupling to the continuum, can be neglected, if the energy dependence of the coupling term QHP is weak. In this case the second term in Eq. (376) can be regarded as a purely anti-Hermitian operator, and the effective Hamiltonian reduces to... 

A U(1,2)A with A the product of free positive -energy projection operators, as in (2.9). The operator U(l,2) = u + U must be chosen so that if the external potential is turned off, the one-photon and two-photon exchange scattering amplitudes are reproduced. (I note in passing that if one uses U = Uj-r rather that one must already include U to get the a Ry level shift correctly ) On comparing the eigenvalues of (4.3) with those obtained from,... 

A further term, which has no analogue in hydrogen, arises in the fine structure of positronium. This comes from the possibility of virtual annihilation and re-creation of the electron-positron pair. A virtual process is one in which energy is not conserved. Real annihilation limits the lifetimes of the bound states and broadens the energy levels (section 12.6). Virtual annihilation and re-creation shift the levels. It is essentially a quantum-electrodynamic interaction. The energy operator for the double process of annihilation and re-creation is different from zero only if the particles coincide, and have their spins parallel. There exists, therefore, in the triplet states, a term proportional to y 2(0). It is important only in 3S1 states, and is of the same order of magnitude as the Fermi spin-spin interaction. Humbach [65] has given an interpretation of this annihi-... 

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