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Indeed, to the 1st order, the sum is 1. Note that $|c_b(t)|^2$ is on the 2nd order of the perturbation.

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This answer is basically what I learned from the Lubos Motl's answer(hopefully I understand it right), with a little bit generalization. Question raised We want to solve a Hamiltonian $H=H_0+\lambda V$ perturbatively . What we know is the solution of $H_0$, whose eigenstates are $|n^0\rangle$ with eigenenergies $E_n^0$ has no degeneracy; it also has a ...

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Even if it's not really appropriate to derivate quantitative results about electrons properties in solids, Jellium model still have some interesting qualitative features. As you correctly pointed, Jellium still a mean-field theory and so fails about dealing with strongly correlated systems. Lets remind a few about jellium model. The starting point is the ...

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The unrenormalized vertex diagram is the dimensionally regularized Feynman integral corresponding to the following single Feynman diagram: It is called ‘unrenormalized’ because it is not accompanied by a diagram in which the momentum loop is replaced by a counterterm vertex generated by the QED Lagrangian. Adding a second Feynman diagram with the ...

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