Apply perturbation theory to the most general two-level system. The unperturbed Hamiltonian is

and the perturbation is

with V_ba = V_ab, V_aa and V_bb real, so that H is hermitian. As in section 7.1.1, λ is a constant that will later be set to 1.
(a) Find the exact energies for this two-level system.
(b) Expand your result from (a) to second order in λ (and then set λ to 1).
Verify that the terms in the series agree with the results from perturbation theory in Sections 7.1.2 and 7.1.3. Assume that E_b > E_a.
(c) Setting V_aa = V_bb = 0, show that the series in (b) only converges if

Comment: In general, perturbation theory is only valid if the matrix elements of the perturbation are small compared to the energy level spacings. Otherwise, the first few terms (which are all we ever calculate) will give a poor approximation to the quantity of interest and, as shown here, the series may fail to converge at all, in which case the first few terms tell us nothing.

Transcribed Image Text:

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