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The assumption is that the perturbation will break the degeneracy and there will be some linear combinations of states that diagonalise the perturbation. For a two-level system, one (diagonalized) matrix element will be larger than the other. The linear combination that has the larger matrix element is the upper state and the other is the lower state. Now ...


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After some hours of pondering, I finally realize two things: If I were to diagonalize $H'$, I must do it in the subspace of fixed $n$ and $J$, instead of fixed $n$ and $L$. This is because the energy eigenvalues of the unperturbed Hamiltonian $H_0$ are specified by $(n,J)$ (states with same $n$ and $J$ but different $L$'s can have the same energy). The ...


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In case the potential is a polynomial, the perturbative expansion for the energy eigenvalues can be generated using very simple recursion relations as shown first by Bender and Wu in Phys. Rev. 184, 1231 (1969) and Phys. Rev. D7, 1620 (1973), see page 29 and further here for the details. This has allowed people to obtain thousands of terms to test ...


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OP asks about the algebraic structure (rather than the actual value) of the $m$'th order $E^{(m)}_n$ of the $n$'th energy level in non-degenerate perturbation theory, see Wikipedia to fix notation. It is natural to introduce a type of "Feynman diagrams" to indicate the algebraic structure. The actual value is encoded in an integer coefficient/weight in ...


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Rather tautologically, a non-perturbative effect is one that is invisible to perturbation theory. An effect is invisible to perturbation theory exactly if it is in a non-analytic part of the partition function with respect to the coupling constant $g$. Observe that perturbation theory is essentially the Taylor expansion of the partition function $Z$ (or ...


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1) As you know, $$ \tag 1 \theta\epsilon^{\mu \nu \alpha \beta}F_{\mu \nu}F_{\alpha \beta} = \theta\partial_{\mu}K^{\mu}, $$ where $K_{\mu}$ is the so-called Chern-Simons class. The Feynman diagrams method tells us that the term $(1)$ defines the diagram which corresponds to the two-photon (or two-three-four non-abelian bosons) vertex $V_{A}$ (where the ...



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