Bound states creation operator

Let us consider especially the bound state nP). Then we define the creation operator of a bound state pair by... 

There is a second, alternative approach. One could assume that the unpaired neutron and the unpaired proton form a quasibound state. The total number of components of the angular momenta of this quasi-bound state is given by n n v. Then we introduce a pair of new bosonic creation and annihilation operators associated with each level of this subsystem, cj, Cj, I,J =... 

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-... 

In this expression, a< and a] designate electron annihilation and creation operators, respectively. The projection operators in Eq. (71) are implemented by simply restricting the indices i, j, k, I to bound states and positive-energy continuum states, omitting contributions from negative-energy states entirely. In Eq. (74), the quantity Cj is the eigenvalue... 

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