The Hamiltonian matrix for a two-state system can be written as

Clearly, the energy eigenfunctions for the unperturbed problems (λ = 0) are given by

(a) Solve this problem exactly to find the energy eigenfunctions Ψ₁ and Ψ₂ and the energy eigenvalues E₁ and E₂.

(b) Assuming thatsolve the same problem using timeindependent perturbation theory up to first order in the energy eigenfunctions and up to second order in the energy eigenvalues. Compare with the exact results obtained in (a).

(c) Suppose the two unperturbed energies are "almost degenerate"; that is,

Show that the exact results obtained in (a) closely resemble what you would expect by applying degenerate perturbation theory to this problem with E⁰₁ set exactly equal to E⁰₂.

Transcribed Image Text:

H = [E ΑΔΙ Δ Ε