Introduction to Quantum Mechanics · Problem 7.4

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · two_level_system

Problem

Apply perturbation theory to the most general two-level system. The unperturbed Hamiltonian is H0=(Ea000Eb0)\mathsf{H}^0 = \begin{pmatrix} E_a^0 & 0 \\ 0 & E_b^0 \end{pmatrix} and the perturbation is H′=λ(VaaVabVbaVbb)\mathsf{H}' = \lambda \begin{pmatrix} V_{aa} & V_{ab} \\ V_{ba} & V_{bb} \end{pmatrix} with Vba=Vab∗V_{ba} = V_{ab}^*, VaaV_{aa} and VbbV_{bb} real, so that H\mathsf{H} is hermitian. As in Section 7.1.1, λ\lambda 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\lambda (and then set λ\lambda 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 Eb>EaE_b > E_a. (c) Setting Vaa=Vbb=0V_{aa} = V_{bb} = 0, show that the series in (b) only converges if ∣VabEb0−Ea0∣<12.\left|\frac{V_{ab}}{E_b^0 - E_a^0}\right| < \frac{1}{2}. *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.

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